------------------------------------------------- Regions of the Interval Spectrum Some Concepts and Names ------------------------------------------------- In naming categories of intervals, or regions of the spectrum in which they are found, there may be many valid and desirable schemes reflecting the diversity of viewpoints and styles to be found in world musics. What I describe here is merely one possible solution, and one influenced by my own musicmaking experience and philosophy which seeks an equitable and inclusive balance between intervals at or near simple integer ratios, and those having a more complex or active nature. An invaluable resource in considering my own scheme is David Keenan's "A Note on the Naming of Musical Intervals" (1999, 2001) available at: . I am much indebted to this thoughtful paper, and likewise to the system of interval names used in Manuel Op de Coul's free software program Scala with its many features for scale creation, analysis, and MIDI file generation or retuning . -------------------------------------- 1. Some ratios and points of reference -------------------------------------- In the approach followed here, the spectrum of harmonic intervals is a continuum with each location or subtle shading having its own charm. At the same time, certain ratios both simple and more complex can serve as helpful reference points in viewing this spectrum in terms of certain regions, e.g. "a large major third" or "a small minor seventh." A main caution is that the borders are inevitably "fuzzy," so that one region shades into another and suggested values in cents are more illustrative than definitive. Also, interval perceptions can be influenced by musical style and context. Often the specific interval "maps" and concepts which have developed for a given world musical tradition can be more informative than any generalized scheme such as this. For our generalized scheme, six types of ratios may assist in mapping regions of interval space. A _Pythagorean_ ratio is generated from a chain, often comparatively short, of pure 3:2 fifths or 4:3 fourths, or prime factors of (2,3). The Pythagorean major third at 81:64 (408 cents) and minor third at 32:27 (294 cents) are familiar examples. Broadly speaking, of course, a "Pythagorean third" might be _any_ size of third generated from a chain of pure fifths, no matter how long. Thus in a circulating Pythagorean system with 53 notes per octave (a circle of 52 pure fifths plus one narrow by about 3.62 cents) we would find many sizes of thirds. Normally, however, a "Pythagorean" ratio implies the shortest available chain of fifths for generating an interval in a given category of interest. A _pental_ (or "5-limit") ratio is generated from prime factors of (2,3,5). The 5:4 major third at 386 cents, and 6:5 minor third at 316 cents, are familiar examples. A _septimal_ ratio is generated from prime factors of (2,3,7). The 9:7 major third at 435 cents, and 7:6 minor third at 267 cents, are again familiar examples. A _pental-septimal_ ratio is generated from prime factors of (2,3,5,7). The simplest ratios for a diminished fifth at 7:5 or 583 cents, and an augmented fourth at 10:7 or 617 cents, are fine examples. In addition to these four basic types of more or less simple ratios, we have two types of ratios known as mediants which can also provide some helpful points of orientation in mapping interval space. The _classic mediant_ offers one helpful reference point within the middle territory between a pental and a septimal ratio -- for example, the small 5:4 and large 9:7 major thirds, at 386 and 435 cents. To find this mediant, we add the numerators and denominators of the two simpler ratios: 5 + 9 14 ----- = -- 4 + 7 11 Thus the ratio for a major third of 14:11 (418 cents) provides one cardinal point of reference within the "middle realm" between the small 5:4 and large 9:7. Another such point of reference in this middle region of major thirds is the Pythagorean 81:64 at 408 cents. Another and more complex type of mediant is the _noble mediant_ formulated by David Keenan and described in a paper which I was honored to coauthor with him: The noble mediant for two ratios such as 5:4 and 9:7 may approximate the region of "metastability" or maximum complexity where an interval is about equally far from the "gravitational influence" of either ratio. It is calculated using the Golden Ratio of Phi, about 1.618034, which is used to "weight" the numerator and denominator of the more complex of the two simple ratios, here 9:7. 5 + 9 Phi --------- = ~422.48 cents 4 + 7 Phi Intervals tuned near a noble mediant, as Keenan suggests, may be called "Nobly Intoned" (NI). This mediant, as a possible point of rough equipose between a pental and a septimal ratio, for example, may be of interest here as an indication of the region where a major third, for example, may be approaching the "sphere of influence" of a septimal ratio such as 9:7. These types of simple and complex ratios, while helpful in mapping the spectrum, are by no means exhaustive. Thus 14:11, the classic mediant between 5:4 and 9:7, can be described generally as one type of "middle major third," but also more specifically as an "undecimal major third" since it brings into play the prime factor of 11. This is the usage of Scala, and nicely supplements the basic concepts of interval region mapping we are about to explore. --------------------------------------------------------- 2. Major and minor thirds and sixths (36:35-plus regions) --------------------------------------------------------- As here conceived, the term "major third" or "minor third" -- and likewise "major sixth" or "minor sixth" -- describes a region of the interval spectrum roughly delineated by the pental and septimal ratios for that category of third or sixth, with at least a few extra cents of leeway on either side. Thus for a major third, the small or pental 5:4 at 386 cents and the large or septimal 9:7 at 435 cents roughly delimit the region. For major intervals, more generally, the pental ratio represents the lower end of the range and the septimal ratio the upper range, with the Pythagorean 81:64 (408 cents) and classic mediant 14:11 (418 cents) ratios typifying the middle range. In practice, as already noted, we will want at least a few extra cents on either side of the region to recognize major thirds slightly smaller than the pental 5:4, or larger than the septimal 9:7. One possible range might be about 372-440 cents, with quite "fuzzy" borders. Similarly, for the minor third, the small or septimal 7:6 at 267 cents and the large or pental 6:5 at 316 cents roughly delimit the region, with the Pythagorean 32:27 at 294 cents and classic mediant 13:11 at 289 cents nicely representing the realm of middle minor thirds. Here, as more generally with minor intervals, the septimal ratio marks the lower end of the range and the pental ratio the upper range. Again, some leeway is needed at either extreme of the range, with something like 260-330 cents as a possible guide for the minor third region, again with "fuzzy" borders. The pental 5:4 and septimal 9:7 ratios for a major third differ by a factor of 36:35, or 48.77 cents -- and likewise 7:6 and 6:5 for a minor third. Thus we may refer to the region of major or minor thirds as a "36:35-plus region" -- the "plus" referring to the extra leeway on either extreme, making these regions actually about 20 cents wider, if the approximate guidelines suggested here are followed. Thus for major or minor thirds we have a rather encompassing region with many shades of small, middle, and large varieties: Major third region (~5:4-9:7) pental Pyth CMed NI septimal 5:4 81:64 14:11 9:7 .......____|---------------|---------|----|----------|___..., 386 408 418 422 435 |--------------------|----------------------|---------------| 372? small 400? middle 423? large 440? Minor third region (~7:6-6:5) septimal NI CMed Pyth pental 7:6 13:11 32:27 6:5 ....____|--------------|----|----|---------------|_____...... 260? 267 284 289 294 316 330? |--------------------|-----------------|--------------------| small 280? middle 300? large While it is easy to identify some characteristic examples of small, middle, and large intervals in these regions, the question of just where a small major or minor third may be approaching a "middle" size, or a middle type a "large" size, is much more fuzzy. Here I have suggested sizes of about 280-300 cents for middle minor thirds, and 400-423 cents for middle major thirds, but this is an approximate and subjective judgment of finely shaded transitional zones. The regions of major and minor sixths may be similarly sketched out, with a mention of two criteria sometimes favored in coordinating these categories with those of the thirds. People may desire either that octave inversion should hold, so that an octave less a major third will produce some kind of minor sixth; or that fourth or fifth complementation should hold, so that a major or minor third plus a fourth yields a major or minor sixth. Since fifths and fourths, although frequently assumed to be at pure ratios of 3:2 and 4:3, are often in practice tempered in either direction, these criteria can have intricate ramifications in fine-tuning a mapping scheme. Here, since the borders are taken to be inherently fuzzy, people should feel free to adjust them according to either of these criteria. Major sixth region (~5:3-12:7) pental Pyth CMed NI septimal 5:3 27:16 17:10 12:7 .......____|---------------|---------|-----|--------|____.... 884 906 918 923 933 |--------------------|-------------------|------------------| 870? small 900? middle 920? large 940? Minor sixth region (~14:9-8:5) NI or septimal CMed Pyth pental 14:9 11:7 128:81 8:5 ....____|-------------|-------|----------------|_____....... 765 782 792 814 |------------------|-----------------|---------------------| 760? small 777? middle 800? large 828? While much effort could be devoted -- and rightly so -- to appreciating some of the fine nuances of these fuzzy borders and transitions, the basic regions and subregions of major and minor thirds and sixths (small, middle, and large) can be helpful in describing the resources of some common tuning systems. Thus a 12-note regular meantone tuning around 1/4-comma or 2/7-comma will have both small and large major thirds -- 386 and 427 cents in 1/4-comma, or 383 and 434 cents in Zarlino's 2/7-comma -- but no middle sizes. Likewise minor thirds are either large or small: in 1/4-comma, 269 or 310 cents; and in 2/7-comma, 262 or 313 cents. In contrast, a 12-note set of the regular Wilson/Pepper temperament with fifths at about 704.096 cents (2.14 cents wide) will have a wealth of middle major thirds around 416 cents, quite close to the classic mediant of 14:11 -- but no small or large sizes. Similarly, middle minor thirds at around 288 cents are very close to the classic mediant between 7:6 and 6:5 at 13:11 or 289 cents, the tridecimal (13-based) minor third -- but we have no large or small minor thirds. George Secor's 17-note well-temperament has middle and large major thirds ranging from 418 to 429 cents, and likewise small and middle minor thirds from 278 to 287 cents -- but no large minor or small major thirds near pental ratios. Many intonational systems, especially larger ones, survey small, middle, and large varieties of major and minor thirds and sixths. This may also happen in a smaller system such as Zest-12, where an irregular 12-note circle based on 2/7-comma meantone with eight narrow and four equally wide fifths produces major thirds at 383, 396, 408, 421, and 435 cents; and minor ones at 274, 287, 300, and 313 cents. Such a system can be attractive if one is drawn to the middle sizes of major and minor thirds, but less so if one strongly prefers that the simplest ratios of 5:4 and 9:7, or 6:5 and 7:6, be approximated at as many locations within the tuning as possible. ------------------------------------------------------------------- 3. Neutral or "interpental" thirds and sixths (25:24-minus regions) ------------------------------------------------------------------- Having considered major and minor thirds and sixths, we may proceed in a natural and intuitive way to neutral thirds and sixths situated somewhere between our minor and major categories. Thus a neutral third should have a size somewhere between that of a large minor third at 6:5 or 316 cents, and a small major third at 5:4 or 386 cents. Since these large minor and small major thirds both have pental ratios, we may also describe neutral thirds as "interpental," or occupying a region between these pental points of demarcation. The ratios of 6:5 and 5:4 differ by 25:24, or 70.67 cents; but, in practice, the neutral region is rather smaller, since these rather simple pental ratios have considerable "spheres of influence" including intervals somewhat larger than 316 cents or smaller than 386 cents. We may thus speak of a "25:24-minus" region of neutral thirds, say around 330-372 cents, with musical context as a factor in shaping perceptions around the fuzzy transitional regions. Within the neutral third region, we may speak of small, middle or central, and large subregions. One possible guideline is that middle or central neutral thirds range from around 39:32 (342 cents) to 16:13 (359 cents) -- or, if we like, a round 360 cents for the upper border, again a fuzzy zone. The simplest neutral third ratio of 11:9 (347 cents), and also Zalzal's famous lute fret as estimated in the tradition of al-Farabi at 27:22 (355 cents), come within this central region. Small neutral thirds of around 330-342 cents are sometimes called supraminor, while large major thirds at around 360-372 cents are also known as submajor. The simplest ratios of 17:14 (336 cents) and 21:17 (366 cents) nicely exemplify these small and large subregions. Neutral third region 63:52 17:14 NI 39:32 11:9 27:22 16:13 21:17 26:21 ..--|------|---|----|-----|-------|-----|-------|-----|--.. 332 336 339 342 347 355 359 366 370 |-------------------|--------------------|---------------| 330? small 342 middle 360 large 372? (supraminor) (central) (submajor) Here it may be noted that the 11:9 third, typical of the central region, is the classic mediant of 5:4 and 6:5. The Noble Intonation (NI) mediant between these ratios is around 339 cents, which might give the upper range of small neutral or supraminor thirds a rather ambiguous or complex quality. For neutral sixths between the pental ratios of 8:5 or 814 cents for a large minor sixth, and 5:3 or 884 cents for a small major sixth, a generally similar situation holds: Neutral sixth region 21:13 NI 34:21 13:8 44:27 18:11 64:39 21:17 104:63 .__|-----|-|---------|-----|-------|-----|------|-----|--... 830 833 834 841 845 853 858 864 868 870? |-------------------|--------------------|-----------------| 828? small 840 middle 858 large (supraminor) (central) (submajor) Interestingly, the classic mediant between the pental major and minor sixths at 5:3 and 8:5 is 13:8 (841 cents), located near the lower end of the central neutral sixth range. This "harmonic neutral sixth" or "tridecimal neutral sixth" (since its ratio is 13-based) may have a special attractive force because the upper tone is a harmonic of the fundamental. Not too far from this ratio is the Nobly Intoned or NI small neutral or supraminor third at around 833 cents, defined by the Golden Ratio of Phi itself! More generally, the region offers a wealth of subtle shadings. -------------------------------------------------------- 4. Major seconds and minor sevenths (36:35-plus regions) -------------------------------------------------------- Major seconds and minor sevenths may be convenient to consider next, since they follow much the same pattern as major and minor thirds and sixths. For major seconds, we have three handy ratios of orientation, all of them superparticular in form (n+1:n). These are the small or pental tone at 10:9 or 182 cents; the middle or Pythagorean tone at 9:8 or 204 cents; and the large or septimal tone at 8:7 or 231 cents. As it happens, the classic mediant of the pental 10:9 and septimal 8:7 is also equal to the Pythagorean 9:8, our exemplary "middle tone." As with our regions for major or minor thirds and sixths, the pental and septimal values of 10:9 and 8:7 differ by 36:35 or about 49 cents, but the major third region is actually somewhat larger -- say 180-240 cents. Indeed, intervals somewhat smaller than 180 cents or larger than 240 cents might also sometimes function as "major seconds" -- but these adjoining regions have their own qualities which might best be considered as distinctive, as discussed below. Within the major second region, in addition to the three simplest ratios of the small 10:9, middle 9:8, and large 8:7, there are some other main points of interest. A small 10:9 plus a middle 9:8 tone will form a pure pental 5:4 major third, as will two "mean-tones" each equal to the geometric mean or average of these unequal ratios, approximately 193.16 cents. Likewise, a middle 9:8 plus a large 8:7 tone will form a pure septimal 9:7 major third, as will two "eventones" each equal to the average of 9:8 and 8:7, around 217.54 cents. We may refer to these 193-cent and 218-cent sizes as the "pental meantone" and "septimal eventone" respectively. The classic mediants of 10:9 and 9:8 at 19:17 (193 cents), and 9:8 and 8:7 at 17:15 (217 cents), closely approximate these sizes. Thus we have something like this: Major second region (~10:9-8:7) pental 5:4 Pyth or 9:7 septimal 10:9 meantone CMed 9:8 eventone 8:7 .__|--------|---------|-----------|-------------|____..... 182 193 204 218 231 |-----------------|-------------------|------------------| 180? small 200? middle 220? large 240? One might mention incidentally that the Phi-weighted mediant between 10:9 and 8:7 is around 198.39 cents; but it is not clear whether or how significant this might be for any special quality of "complexity" near this tuning, which is also not too far from the ratio of 9:8. David Keenan (personal correspondence, 19 June 2008) helpfully explains that in fact 10:9 and 8:7 do _not_ form a noble mediant, which would require that a pair of ratios meet an important mathematical test. With two ratios a:b and c:d -- here 10:9 and 8:7 -- the difference between the cross-products of the outer and inner terms, a*d and b*c, or here 10*7 and 9*8, must differ by 1; the difference here is 2 (72 - 70). Thus the "noble mediant" concept is inapplicable to this pair of ratios. For minor sevenths, rather similar pattern holds, except that the simple harmonic ratio of 7:4 (969 cents), the "harmonic minor seventh," may have a great attracting power, thus exerting its influence over a wider portion of the spectrum than with the more complex 9:7 septimal major third, for example. As with the 9:8 middle tone, so with the 16:9 minor seventh (996 cents), the Pythagorean ratio is also the classic mediant between septimal and pental values, here 7:4 and 9:5 (1018 cents). Minor seventh region (~7:4-9:5) CMed or septimal Pyth pental 7:4 16:9 NI 9:5 ...._______|---------------------|-----|-----------|_____.... 969 996 1002 1018 |-----------------------|-------------|---------------------| 960? small 987? middle 1000? large 1025? The Noble Intonation value is around 1001.61 cents, as with the major second not too far from the classic mediant and also Pythagorean value, here 16:9. ---------------------------------------------------------- 5. Minor seconds and major sevenths (405:392-plus regions) ---------------------------------------------------------- One way to seek an approximate range for a minor second region is to take the septimal ratio of 28:27 (63 cents), the favored small semitone or thirdtone of Archytas, as near the lower end; and the pental-septimal ratio of 15:14 (119 cents) as near the upper end. These ratios differ by a factor of 405:392 or 56.48 cents. Allowing, as usual, for some leeway at each end, we thus have a "405:392-plus" region of minor seconds. One rough guideline might be a range of about 60-125 cents. While smaller steps may sometimes function as semitones or minor seconds, they have their own qualities which might fit the label of "diesis," as discussed below. Likewise, while a step as large as 27:25 (133 cents) serves as a minor second in pental just intonation, this size is also a fine example of a small neutral (or supraminor) second, to be considered shortly. Within this range of about 60-125 cents, we have some cardinal ratios of orientation: the small or septimal minor second at 28:27 or 63 cents; the middle Pythagorean semitone at 256:243 or 90 cents; and the large or pental semitone at 16:15 or 112 cents, with 15:14 at 119 cents near the upper end of the region. In addition to the Pythagorean value, the middle region is also represented by the classic mediant between the septimal 28:27 and pental 16:15, 22:21 or 81 cents. Minor second region (~28:27-15:14) pental- septimal CMed Pyth pental septimal 28:27 22:21 256:243 16:15 15:14 ..__|-------------|-------|---------------|-------|___.... 63 81 90 112 119 |----------------|----------------|----------------------| 60? small 80? middle 100? large 125? We may also note that the difference between the middle 9:8 tone and a large or pental 16:15 semitone yields an excellent small semitone at 25:24 or 71 cents; and the difference between 9:8 and the Pythagorean middle semitone at 256:243 yields a large semitone at 2187:2048 or 114 cents, quite close to the pental 16:15. The major seventh region may be regarded as rather similar to this, but with small pental and large septimal ratios, as holds for major categories generally. Major seventh region (~28:15-27:14) pental- septimal pental Pyth CMed septimal 28:15 15:8 243:128 21:11 27:14 ..____|-------|--------------|--------|--------------|___.. 1081 1088 1110 1119 1137 |----------------------|---------------|------------------| 1075? small 1100? middle 1120? large 1140? Small major sevenths are represented by the pental-septimal 28:15 at 1081 cents and pental 15:8 at 1088 cents; middle major sevenths by the Pythagorean 243:128 at 1110 cents and the classic mediant between the pental 15:8 and septimal 27:14 at 21:11 or 1119 cents; and large major sevenths by the septimal 27:14 at 1137 cents. ----------------------------------------------------- 6. Neutral seconds and sevenths (28:27-minus regions) ----------------------------------------------------- One way to approach a neutral second region is to regard it as having a range between the large septimal-pental minor second at 15:14 or 119 cents, and the small or pental tone at 10:9 or 182 cents. These intervals differ by a factor of 28:27 or 63 cents. In practice, however, the range is somewhat smaller, say the 125-170 cents which Hormoz Farhat suggests as the domain of the neutral second in Persian music based on the dastgah system of modality. The handy superparticular ratio of 14:13 (128 cents) favored by Ibn Sina may nicely represent the neighborhood near the lower end of the range, and Ptolemy's 11:10 (165 cents) the neighborhood near the upper end. A middle or central subregion might run from about 135 to 160 cents, with these approximate sizes often preferred according to Farhat in Persian music, and also reported by some Arab musicians to be common in the maqam traditions. Within this central portion of the range we find the superparticular ratios of 13:12 (139 cents) and 12:11 (151 cents), the latter used in Zalzal's scale according to al-Farabi along with the more complex ratio of 88:81 or 143 cents. One general rule is that two neutral seconds should add up to some kind of minor third (Section 2 above). Thus from 14:13 and 13:12 we get the small or septimal 7:6 third; from 13:12 and 12:11, the middle or tridecimal minor third at 13:11; and from 12:11 and 11:10, the large or pental minor third at 6:5. In Near Eastern traditions where many instruments are tuned in pure fifths and fourths, the Pythagorean size of middle minor third at 32:27 or 294 cents may often serve as a guide, with Farhat's rounded 135-cent and 160-cent steps adding up to something around this value -- and Zalzal's 88:81 and 12:11 (143/151 cents) yielding it precisely. Thus we have a situation something like this: Neutral second region 14:13 13:12 88:81 12:11 11:10 ..___|-------------|-----|-------|----------------|-----.. 128 139 143 151 165 |--------------|--------------------------|--------------| 125? small 135? middle 160? large 170? (supraminor) (central) (submajor) From here, it is intuitive to posit a similar situation for neutral sevenths, which may have a range roughly from 20:11 (1035 cents) to 13:7 (1072 cents), with a few cents leeway at either end, say something like 1030-1075 cents: Neutral seventh region 20:11 51:28 11:6 24:13 13:7 ..____|-----|--------------|------------|------------|_____.. 1035 1038 1049 1061 1072 |-----------------|-----------------=------|----------------| 1030? small 1043? middle 1065? large 1075? (supraminor) (central) (submajor) Representative of the small neutral seventh subregion are the 20:11, and also 51:28 (the small 17:14 neutral second plus a 3:2 fifth) at 1038 cents. An interval of special prominence in the middle or central range is 11:6 at 1049 cents, on account of its simple ratio as a zone of harmonic "gravitation." Also within the central subregion is 24:13 (an octave less 13:12) at 1061 cents. In the large neutral seventh subregion, 13:7 at 1072 cents plays a prominent role. The reader may have noticed that there is a gap between the high end of our neutral second region at around 170 cents, or a bit larger than 11:10, and the start of the major second region with the small or pental tone at 182 cents, or a very slightly smaller value around 180 cents. This small intermediate region of around 170-180 cents marks the heart of an "equitable heptatonic" zone where steps have a size at or near 1/7 octave or 171 cents, to be discussed below, and which deserves its own special recognition. ------------------------------------------------------ 7. Perfect fifths and fourths (4096:3959-plus regions) ------------------------------------------------------ It may seem curious to have surveyed an assortment of major, minor, and neutral intervals before coming to those intervals which adorn so many world musics: fifths and fourths. In medieval European and also Balinese or Javanese gamelan polyphony, for example, these intervals serve as stable concords and resting points toward which often complex contrapuntal textures are drawn. Comparing the pure 3:2 fifths and 4:3 fourths of a medieval European Pythagorean tuning with typical gamelan values for the corresponding concordant interval category of _kempyung_ will reveal a variance of often around 10-30 cents between these systems. We thus have a fine illustration of how intonation can vary according to taste, and also timbral conditions. Generally the zone of a perfect fourth or fifth might be taken as extending for about a 64:63 zone, equal to the septimal comma or comma of Archytas at 27.26 cents, on either side of 4:3 or 3:2. This zone would thus have a size equal to twice that of the comma, or of the square of its 64:63 ratio: 4096:3959, or 54.53 cents. In practice, we may find it convenient to round this to an even 60 cents, 30 cents on either side. Thus the zone of perfect fourths would have a range of about 468-528 cents, 30 cents to either side of 4:3 at 498 cents; and likewise the perfect fifth one of about 672-732 cents, centering around 3:2 at 702 cents. Thus gamelan tunings often seem to prefer values around 680 cents or 720 cents, with great variations; indeed, tradition requires that each gamelan ensemble have its own distinctive intonation and not try to copy that of another ensemble. In world musical traditions tending toward an equal or equable heptatonic style of intonation, small fifths around 4/7 octave or 686 cents are often favored; while styles tending toward an equal or equable pentatonic (e.g. the slendro modes of gamelan) may favor a size around 3/5 octave or 720 cents. In world traditions like that of Western Europe, where strongly harmonic timbres (e.g. harpsichord) are favored for polyphonic music, tunings at or quite near 4:3 and 3:2 may prevail; most regular European and related keyboard temperaments stay within about 7 cents of these ratios, whether in the narrow direction of tempering for the fifths and the wide one for the fourths (the historical norm), or in the opposite direction (as in some historical modified meantones for certain fifths and fourths, and regularly in modern neomedieval temperaments). The interval of a fourth narrowed by a septimal comma (21:16, 471 cents), or a fifth enlarged by this comma (32:21, 729 cents) occurs, for example, in just tunings based on the diatonic of Archytas with steps of 9:8, 8:7, and 28:27. Pythagorean tunings with 12 or more notes will produce a narrow fifth at 262144:177147 or 678 cents, smaller than 3:2 by a Pythagorean comma at 531441:524288 or 23.46 cents, and likewise a wide fourth at 177147:131072 or 522 cents. Pental just intonation in classic forms also produces narrow fifths and wide fourths at 40:27 and 27:20 (680 and 520 cents), differing from pure by the syntonic comma of 81:80 or 21.51 cents. If we desire an integer ratio to gauge the neighborhood of the upper end of our region, then 256:189 at 525 cents, a 3:2 fifth plus 64:63, might serve this purpose. Thus we have impressionistic ranges something like this: Perfect fourth region 21:16 2/5 oct 4:3 3/7 oct 256:189 ...__|------|--------------------|---------------|----------|__.. 471 480 498 514 525 |-------------------------|-------------|-----------------------| 468? small or 491? middle 505? large or 528? narrow wide The situation for perfect fifths may be taken as generally similar: Perfect fifth region 189:128 4/7 oct 3:2 3/5 oct 32:21 ...__|------------|--------------|-------------------|------|__.. 675 686 702 720 729 |-------------------------|-------------|-----------------------| 672? small or 695? middle 709? large or 732? narrow wide These ranges should not be taken as rigid or exhaustive, since, for example, intervals rather larger than 732 cents may sometimes serve musically as perfect fifths -- especially in some inharmonic timbres. However, when treated flexibly, these concepts of general ranges for perfect fourths and fifths nicely set the stage for considering some especially intriguing categories we will meet below. ----------------------------------------------- 8. Tritonic intervals (a 6561:6272-plus region) ----------------------------------------------- As an intuitive approach to defining a region of tritonic intervals (also known as augmented fourths or diminished fifths), we may begin with the concept of a fourth augmented or a fifth diminished by a minor second or semitone of some kind (Section 5). Another way of putting this is that a tritonic interval should have a size somewhere between that of a 4:3 fourth at 498 cents and a 3:2 fifth at 702 cents, while differing from _either_ of these ratios by at least a "semitone" of some kind. If we consider the septimal 28:27 semitone or thirdtone of Archytas at 63 cents as close to the lower end of the minor second range, then the tritonic region would cover a range equal to 9:8 or 204 cents (the space between a 4:3 fourth and 3:2 fifth), less a 28:27 zone at either end of this potential range. To estimate the size of our tritonic region using integer ratios, we might thus take 9:8 at 204 cents and subtract twice the size of a 28:27 semitone at 63 cents, that is 784:729 or about 125.92 cents, leaving a range of 6561:6272 or 77.99 cents. A simpler way of estimating this range is to consider that a 4:3 fourth plus a 28:27 semitone yields an interval of 112:81 or 561 cents; while a 3:2 fifth less 28:27 yields 81:56, or 639 cents. If we round conveniently to 560-640 cents, this gives us a handy gauge for the region. If we strive consistently to apply our earlier concept of a semitone ranging down to about 60 cents, then we might have a slightly wider range of 558-642 cents. Since borders are inherently fuzzy, this is no great matter. From a viewpoint of traditional European theory, an augmented fourth or diminished fifth should more properly be defined as fourth plus a _chromatic_ semitone, or a fifth less such a semitone, rather than by means of a usual or diatonic semitone such as our septimal 28:27. However, we can arrive at the same result as this theory by defining an augmented fourth such as F-B as equal to the fifth (F-C) less a diatonic semitone (B-C); and a diminished fifth such as B-F likewise as a fourth (C-F) plus a diatonic semitone (B-C). Thus the smaller septimal tritonic interval 112:81 at 561 cents may be considered a diminished fifth, and the larger 81:56 at 639 cents an augmented fourth. We could reach the same result more conventionally by using the septimal _chromatic_ semitone, equal to the difference of 9:8 and the small 28:27 semitone, or 243:224 at about 141 cents -- actually by our usual definition a type of neutral second (Section 6)! Defining a septimal augmented fourth as 4:3 plus 243:224 (e.g. F-B as F-Bb plus Bb-B), we arrive at 81:56; and defining a septimal diminished fifth as 3:2 less 243:224 (e.g. B-F as Bb-F less less Bb-B), we get 112:81. This example illustrates how, while a tritonic interval as here defined differs from either 4:3 or 3:2 by _at least_ a "semitone" with a minimum size of 60 cents or so (Section 5), it may sometimes differ from one of these ratios by a small semitone of around 60-75 cents, and from the other by a small to middling neutral second, here 243:224 at about 141 cents. If we take 560-640 cents as a convenient guide to the tritonic region, then the septimal 112:81 and 81:56 at 561 and 639 cents are just within its fuzzy borders. Close by are the simpler tridecimal or 13-based ratios of 18:13 and 13:9 at 563 and 637 cents, which may be produced by the division of a 9:8 tone into a small diatonic semitone at 27:26 or 65 cents, and a chromatic step equal to a 13:12 neutral second at 139 cents. This division yields an augmented fourth equal to 4:3 plus 13:12, or 13:9; and a diminished fifth of 3:2 less 13:12, or 18:13. The augmented fourth and diminished fifth of 17-EDO at about 635 and 565 cents, nicely approximate these values. Pental just intonation also yields some tritonic intervals in this general neighborhood -- but this time with the diminished fifth as the larger interval and the augmented fourth the smaller. Two pure 6:5 minor thirds, for example B-D-F, yield a diminished fifth of 36:25 or 631 cents, which differs from 3:2 by the small chromatic semitone of 25:24 or 71 cents. Tuned in similar fashion, the augmented fourth F-B would exceed 4:3 by the same 25:24 semitone, yielding 25:18 at 569 cents. These tritonic ratios occur in 1/3-comma meantone. In a classic pental just intonation context, the small 25:24 chromatic semitone has its counterpart in the notably large diatonic step of 27:25 at 133 cents -- in many other contexts a small neutral second, as has been noted (Section 5). Together, these steps add up to 9:8. In the tempered system of 1/3-comma meantone, while the tritonic ratios of 36:25 and 25:18 precisely obtain, both of these steps are smaller, at about 64 cents for the small chromatic semitone and 126 cents for the large diatonic semitone -- the latter near our usual border area between a large minor second and a small neutral second. To this point, we have encountered pairs of tritonic ratios where the small form differs from the large by more than 60 cents -- indeed, by almost 78 cents for the septimal pair 112:81 and 81:56. Gradually, however, we have been moving toward the central portion of the tritonic region where these differences become less dramatic. Medieval Near Eastern theory around 1300 features the tritonic ratio of 56:39, or 626 cents, which could be described as an augmented fourth equal to 4:3 plus a small 14:13 neutral second at 128 cents. This ratio differs from 3:2 by a small diatonic semitone of 117:112 or 76 cents. The corresponding diminished fifth, equal to 3:2 less 14:13, is 39:28 at 574 cents. These tritonic ratios, like 13:9 and 18:13, could be described as tridecimal, since they are based on 13 as the highest prime factor. With 56:39 and 39:28, we still have a situation where tritonic ratios are based on the division of a 9:8 tone into a small semitone plus a neutral second, here 14:13. One way to define the "central" tritonic subregion is to specify that intervals in this subregion should differ from either 4:3 or 3:2 by some kind of "semitone" as we have defined it in Section 5: that is, by at least 60 but no more than 125 cents, to take conveniently rounded figures. This criterion suggests a central subregion of about 577-623 cents. In the small tritonic subregion of 560-577 cents, intervals differ from 4:3 by a small semitone but from 3:2 by an interval larger than 125 cents fitting our concept of a neutral second (Section 6). In the large subregion, intervals likewise differ from 3:2 by a small semitone and from 4:3 by a small to middling neutral second. Moving now into the central subregion, we encounter a pair of ratios notable for their simplicity: the pental-septimal diminished fifth at 7:5 or 583 cents, and its complementary augmented fourth at 10:7 or 617 cents. The simplest integer ratio found in the tritonic region, 7:5 is also the classic mediant between 4:3 and 3:2. In a typical just intonation context, a 7:5 diminished fifth (e.g. B-F) resolves to a pure 5:4 major third (e.g. C-E) by stepwise contrary motion, the upper voice descending by a small diatonic semitone of 21:20 (84 cents), and the lower ascending by a usual pental diatonic semitone of 16:15 (112 cents). In the division of a 9:8 tone, the 21:20 semitone is complemented by a larger chromatic semitone of 15:14 or 119 cents. From 3:2 less 15:14 we get the 7:5 diminished fifth; and from 4:3 plus 15:14, the 10:7 augmented fourth. Pythagorean intonation yields rather similar tritonic intervals. The augmented fourth is equal to 4:3 plus a chromatic semitone of 2187:2048 or 114 cents, or 729:512 at 612 cents; and the smaller diminished fifth at 1024:729 to 3:2 less 2187:2048, or 588 cents. In the division of the 9:8 tone, the rather large chromatic semitone of 2187:2048 is complemented by the 256:243 diatonic semitone at 90 cents. This rather compact and incisive semitone makes available a very effective resolution by oblique motion from an augmented fourth at 729:512 to a pure 3:2 fifth, for example G-C# with the lower voice remaining stationary and the upper ascending by a 90-cent semitone to the fifth G-D, as happens in the Montpellier version of Perotin's organum _Alleluia posui adjutorium_ from around 1200. Along with the more polarized tritonic ratios of 36:25 and 25:18 that we have already encountered, pental just intonation also yields the forms within our central subregion of 45:32 and 64:45, respectively a smaller augmented fourth at 590 cents and a larger diminished fifth at 610 cents. These forms are based on the division of a 9:8 tone into a diatonic semitone at 16:15 or 112 cents, and a chromatic semitone at 135:128 or 92 cents. Thus 4:3 plus 135:128 yields the 45:32 augmented fourth, and 3:2 less 135:128 the 64:45 diminished fifth. Two ratios yet more centrally located within the region are 17:12 at 603 cents and 24:17 at 597 cents, based on the near-equal division of the 9:8 tone into semitones of 18:17 at 99 cents and 17:16 at 105 cents. Depending on the context, either 18:17 or 17:16 might be regarded as the diatonic semitone, and the other as chromatic. If we arbitrarily take the smaller 18:17 as diatonic, for example, then the 17:16 chromatic semitone plus 4:3 would yield a 17:12 augmented fourth, and 3:2 less 17:16 a smaller 24:17 diminished fifth. We may refer to these ratios, following Scala, as septendecimal, or 17-based. Some tempered systems have a tritone equal to precisely 600 cents or half of a 2:1 octave, which may be used as either diminished fifth or augmented fourth. This interval will be found in any equal division of the 2:1 octave (EDO) with a number of steps evenly divisible by two, with 12-EDO and 22-EDO as familiar examples. Irregular temperaments such as 12-note modified meantone systems may feature this form of tritone along with other sizes. The following diagram may give an overview of the tritonic region and some of the ratios by which it is populated. Indeed the ratios we have discussed provide a rich enough population that the region is here divided into two mirrorlike halves, with the "axis of symmetry" at 600 cents: Tritonic region (~112:81-81:56) pental- septimal pentalA septimal Pyth pentalB 112:81 18:13 25:18 39:28 7:5 729:512 45:32 24:17 ....|-----|-------|-------|----------|-------|-------|--------|----> 561 563 569 574 583 588 590 597 |------------------------------|-----------------------------------> 560? small 577? middle 600 pental- pentalB Pyth septimal pentalA septimal 17:12 64:45 1024:792 10:7 56:39 36:25 13:9 81:56 <---|--------|----=---|--------|---------|-------|-------|------|.. 603 610 612 617 626 631 637 639 <-------------------------------------|---------------------------| 600 middle 623? large 640? Starting at the left of the first portion at 560 cents, and at the right of the second portion at 640 cents, one encounters the pairs of tritonic intervals considered above in the order we have discussed them: first the small 112:81 and large 81:56, then 18:13 and 13:9, and so on until we reach 24:17 and 17:12 near 600 cents. This is only a sampling of the many shades found in just and tempered systems. ------------------------------------------------ 9. Interseptimal intervals (49:48-minus regions) ------------------------------------------------ We now come to the intriguing categories of intervals which may be called _interseptimal_, since they occupy regions intermediate between two septimal ratios such as 8:7 and 7:6, or 12:7 and 7:4. We will begin with these two regions, with ranges of about 240-260 cents and 940-960 cents. Intervals of these sizes routinely and beautifully adorn many world musics, for example in the slendro tunings of gamelan. From the perspective of at least one world musical tradition, the Western European, however, they may have more of an unconventional beauty which captures the imagination by exploring the space intervening between two more familiar interval categories. ----------------------------------------------------------- 9.1. Regions between large major and small minor categories ----------------------------------------------------------- Thus the region of 240-260 cents intervenes between the large or septimal major second at 8:7 (231 cents), and the small or septimal minor third at 7:6 (267 cents). The ratios 8:7 and 7:6 differ by a factor of 49:48, or 36 cents, which is the potential size of this interseptimal region. However, since we wish to allow 8:7 some "sphere of influence" including slightly larger intervals, and likewise with 7:6 in regard to slightly smaller intervals, a rounded 240-260 cents may roughly represent the range with an "interseptimal" quality somewhat distinct from either of these simpler ratios. Similarly, we find an interseptimal region between the large 12:7 major sixth at 933 cents and the small 7:4 minor seventh at 969 cents; a rounded range of 940-960 cents might represent this distinct region. These regions are of a type we might call "major-minor," since they are situated in the space between a large major and small minor category -- the space between 8:7 and 7:6, or 12:7 and 7:4. The other type, which we will explore in the next subsection, is found between a large major and a perfect category, or a perfect and small minor one. Looking first at the 240-260 cent region between 8:7 and 7:6, we find that the complex integer ratio of 147:128 at a rounded 240 cents is one possible guide to the lower border area. This is equal to 8:7 plus a small interval of 1029:1024 or 8.43 cents. The 147:128 ratio is very close to an even 1/5 octave or 240 cents, one possible point of departure for the subtly unequal division of the octave typical of slendro. From a neomedieval European-oriented perspective, a 240-cent interval is large enough that, while still often serving as a large melodic or vertical major second, it can also serve in some contexts as a very small and intriguing kind of "minor third," for example contracting to a unison with one voice moving by a small tone of about 180-192 cents and the other by a step of about 50-60 cents which serves as a semitone, although it may be rather smaller than 60 cents we have set as the minimum size for a "usual" semitone. One theoretical point of interest is the Nobly Intoned or NI mediant between 8:7 and 7:6 at around 344 cents. The complex integer of 1152:1001 at 243 cents, found in the Zephyr-24 tuning based on Erv Wilson's eikosany concept (with factors of 1-3-7-9-11-13), nicely approximates this NI value, as does the 242-cent interval found in extended versions of Zarlino's 2/7-comma meantone (regular 192-cent tone plus 50-cent enharmonic diesis). The classic mediant of 8:7 and 7:6 is 15:13 at 248 cents, very closely approximated in 29-EDO. As we continue our journey through the region, another of the simpler ratios we encounter is 22:19 at 254 cents. Not too far from the upper border area of the region around 260 cents is 297:256 at 257 cents. An interval of around this size seems to me largely interchangeable with 7:6, and yet may have its own distinct charm and shading -- as also with sizes in the more immediate neighborhood of 250 cents. Here is an overview of the region showing these landmarks: 1/5 oct or CMed 8:7-7:6 147:128 1152:1001 NI 15:13 22:19 297:256 ..|----------|-----|------------|---------------|-------|-------|.. 240? 243 244 248 255 257 260? Over much of this region, an interval in a vertical context may act as _either_ a very large or "ultramajor" second or a very small or "ultraminor" third; toward the upper end, intervals may lean more in a "thirdlike" direction. Often the musical impression may depend on the context: an interseptimal interval around 250 cents may sound like a major second if it expands stepwise to a fourth, and like a minor third if it contracts stepwise to a unison. This creative ambiguity is one of the delights of neomedieval style. The region of 940-960 cents between 12:7 and 7:4 is in many ways similar: here, in a neomedieval style based largely on 13th-14th century European polyphony, an interval may often act either as an "ultramajor sixth" expanding to an octave, or an "ultraminor seventh" contracting to a fifth. Indeed, some interpretations of Marchettus of Padua (1318), who calls for a very large major sixth in directed progressions expanding from this interval to the octave, suggest a size of around 950 cents; an alternate reading calls for something like 12:7 (933 cents), for example. With the region between 8:7 and 7:6, the Noble Intonation value at about 244 cents was located closer to the more complex 8:7. Here it is likewise, at about 943 cents, located closer to the more complex 12:7 (933 cents) than to the simpler 7:4 (969 cents). This value is very nicely approximated in 14-EDO. The classic mediant of these septimal ratios is 19:11, at 946 cents. Another ratio of interest is 26:15, at 952 cents, or its near-just equivalent in 29-EDO, also close to the geometric mean of 3:2 and 2:1 at around 951 cents. It is a curious question just how far the "sphere of influence" of 7:4 might reach out to embrace somewhat smaller intervals. Thus the fine 958-cent interval of Zarlino's 2/7-comma (e.g. Bb-G#) comes within our interseptimal region, although it could also be taken as a rather heavily tempered representation of 7:4. Here 960 cents, or 4/5 octave, is arbitrary taken as around the upper border of the region. CMed 2/7-comma 12:7-7:4 meantone NI 19:11 26:15 (e.g. Bb-G#) 4/5 oct ..|----------|--------|-------------|-------------|-----------|.. 940? 943 946 952 958 960? This region, like that of around 240-260 cents, is very typical of the slendro tunings of gamelan. An alternate term for an interval of around 250 cents would be a "hemifourth," equal to about half of a fourth of some kind as defined in Section 7 (around 468-528 cents). Thus the 940-960 cent region could be described as that of "an octave less hemifourth," or possibly of "a fifth plus hemifourth." Such terms might be attractive, for example, in describing the Soft Diatonic of Aristoxenos, which as interpreted by John Chalmers includes a step of half a fourth or 250 cents. Whereas neutral or "interpental" intervals (Sections 3, 6) occur in regions between a large minor and small major category as typified by pental ratios -- for example, neutral thirds between 6:5 and 5:4 -- interseptimal intervals of the two regions we have just considered occur in the space intervening between a large major and small minor category as typified by septimal ratios: 8:7 and 7:6, or 12:7 and 7:4. An advantage of distinguishing between "neutral" intervals and "interseptimal" ones is that some tuning systems may feature intervals of one type but not the other. Thus George Secor's 17-tone well-temperament offers a wealth of neutral intervals, but no interseptimal ones around 250 or 950 cents (or 450 and 750 cents, as we are about to consider). In contrast, his 29-tone High Tolerance Temperament (HTT-29) offers an abundance of intervals in all of these categories. ------------------------------------------- 9.2. Major-perfect or perfect-minor regions ------------------------------------------- A related type of interseptimal region occurs at around 440-468 cents, between the large or septimal major third at 9:7 (435 cents) and the narrow fourth at 21:16 (471 cents); and likewise at around 732-760 cents, between the wide fifth at 32:21 (729 cents) and the small or septimal minor sixth at 14:9 (765 cents). These regions, like those around 240-260 and 940-960 cents, are very commonly represented in gamelan music, for example. They are also represented as premiere resources in a tuning system such as 24-EDO or 29-EDO which features sizes at or around 250, 450, 750, and 950 cents. Especially with certain types of strongly harmonic or customized inharmonic timbres, the regions of 440-468 and 728-760 cents can have distinctive qualities because of their rather close proximity to the simple ratio of a 4:3 fourth or 3:2 fifth. This attraction is most evident for the portions of these regions nearest to 4:3 or 3:2, say 458-468 cents and 732-745 cents. Depending on the choice of timbre and style, the musical consequences may be undesirable or highly desirable. Thus in a 1/4-comma meantone tuning used in a 16th-century European fashion, the interseptimal intervals at around 462 cents (Eb-G#) and 738 cents (G#-Eb) are mostly regarded as "Wolf" fourths or fifths, "howling" and musically "mistuned" counterparts of usual fourths and fifths close to 4:3 and 3:2. Here both the intense beating which occurs in a strongly harmonic timbre (e.g. organ or harpsichord), and the categorical "strangeness" of an interval somewhere between a major third and perfect fourth, or perfect fifth and minor sixth, may have contributed to a cognitively as well as acoustically dissonant impression. In contrast, the very similar interval sizes available in 13-EDO at around 462 and 738 cents can in certain customized inharmonic timbres serve in a neomedieval context as stable "fourths" and "fifths" -- and the 462-cent "fourth" also as a very large cadential major third expanding to a 738-cent fifth! This practice might be taken as a tribute to the large gravitational spheres of 4:3 and 3:2. Conversely, intervals in the lower portion of the 9:7-21:16 region or the upper portion of the 32:21-14:9 region, say 440-455 and 745-760 cents, may have qualities rather like those of large septimal major thirds around 9:7 or small septimal minor sixths around 14:9, with the former in a neomedieval context often expanding stepwise to a fifth and the latter to an octave, for example. Looking first at the region between the large 9:7 major third and the narrow 21:16 fourth, we find that this interseptimal zone has a size equal to the 49:48 difference between these two ratios, or 36 cents, less a few cents at either end to allow these ratios their "spheres of influence": thus about 440-468 cents. Here a rounded 440 cents is taken as marking a fuzzy border area where we are moving from widely tempered variants of 9:7 to intervals with a more distinct character. Thus the 442-cent interval of 19-EDO, or the 443-cent interval of 46-EDO, might be regarded either as a rather imprecise version of 9:7, or as having its own interseptimal flavor. By around 22:17 or 446 cents, we may have entered more distinctively interseptimal space. While 9:7 and 21:16 are the septimal ratios bounding this region, it may be best as a guide to Noble Intonation (NI) to seek a Phi-weighted mediant between 9:7 and the simpler ratio of 4:3, yielding an NI value of around 448 cents, or not far from 22:17. This is in theory one measure of the area of maximum complexity between 9:7 and 4:3, and it is a very pleasant and effective one in neomedieval contexts. The classic mediant of 9:7 and 4:3 is 13:10 at 454 cents, an interval available in virtually just form in 29-EDO, and also in Zarlino's 2/7-comma meantone, for example at Eb-G#. It has been proposed that the style of vocal tuning described by Marchettus of Padua in 1318 calls for an enlarged major third of around this size in a directed progression expanding to a fifth -- although a more moderate interpretation prefers something around the septimal 9:7. In any event, an interval at 13:10 or so acting as an "ultramajor third" expanding to a fifth can be musically at once striking and beautifully effective. Much larger than this, and we move more into the peripheral sphere of influence of the fourth. The classic mediant between 9:7 and 21:16 at 30:23, or about 460 cents, is one possible guide to this transition. At 462 cents we encounter the interval of 4:3 less a 49:48 diesis (as we shall term many intervals in the range of about 33-60 cents), or 64:49, with the augmented third or small fourth of 1/4-comma meantone (e.g. Eb-G#), or the small fourth of 13-EDO, very close to this. At 17:13 or around 464 cents we are getting closer to the upper border of the region, and the 467-cent interval found in 72-EDO and the version of George Secor's Miracle temperament based on it might be considered as a variant on the 21:16 fourth. The following diagram may give an overview of some of these points, with the reader's patience asked for the multiple labels attached to certain rounded values in cents: 2/7-comma/ CMed 1/4-comma/ 19- 46- 9:7-4:3 9:7-4:3 13-EDO/ EDO EDO 22:17 NI 13:10 30:23 64:49 72-EDO ..|-----|---|-------|------|---------|--------|------|------|---|.. 440? 442 443 446 448 454 460 462 467 468? The region of 732-760 cents between 32:21 and 14:9 is rather similar, with the especially strong "gravitational attraction" of the 3:2 fifth often exerting its sway over the lower portion of the region. At about 738 cents we encounter the ratio of 49:32, equal to 3:2 plus the 49:48 diesis. This interval can have a very striking character in certain septimal tunings, including one of Robert Walker's, where the 49:48 step actually serves as a kind of very small semitone or "ultraminor second." In such a setting, it might be perceived as a very small "minor sixth" contrasting with a 3:2 fifth. In the different context of 13-EDO, as mentioned, an interval of almost this identical size can serve as a "xentonal" equivalent of 3:2 in a neomedieval setting. This is also the size of an interval such as G#-Eb in 1/4-comma meantone. In gamelan, interval sizes in the neighborhood of 740 cents may occur in both slendro and pelog tunings, with much variability in size: intervals which would generally be described here as wide fifths, small minor sixths, or denizens of the intervening interseptimal region seem all congenial to the style, with the choice between these varying from ensemble to ensemble. By 20:13, or 746 cents, we may be moving more into the domain of a "sixthlike" impression, with an interval such as G#-Eb in Zarlino's 2/7-comma meantone at almost precisely this size. Such an "ultraminor sixth," much like a more conventional septimal minor sixth at 14:9, might in a neomedieval context expand stepwise to an octave, or resolve by oblique motion to a fifth with the upper voice descending by a small step of about 50 cents (in context, a kind of "ultraminor second"). The classic mediant of 3:2 and 14:9 is 17:11, or 754 cents, an intriguing form of very small "minor sixth": from here on we more and more closely approach the upper border of the region, or the lower border of the 14:9 neighborhood, here placed at a rounded 760 cents. In this upper portion of the region are found the 46-EDO interval of 757 cents and the 19-EDO interval of 758 cents. 1/4-comma/ 2/7-comma/ CMed 13-EDO/ 29-EDO 3:2-14:9 46- 19- 49:32 20:13 17:11 EDO EDO ..|-----------|-----------|-----------------|---------|---|-----|.. 732? 738 746 754 757 758? 760? In various musical styles, these interseptimal regions typified by intervals of around 450 or 750 cents happily combine with those of around 250 and 950 cents that we have considered. Thus in gamelan, all of these ranges are common. In neomedieval styles, also, interseptimal intervals of the "major-minor" type (240-260 or 940-960 cents) nicely complement those of the "major-perfect" or "perfect-minor" type (440-468 or 732-760 cents). For example, using the rounded values of 24-EDO for convenience, a sonority of 0-700-950 cents is rather ambiguous: the outer 950-cent interval might be taken as an "ultramajor sixth" seeking expansion to the octave or an "ultraminor seventh" inviting contraction to a fifth. The upper 250-cent interval, likewise, might be an "ultramajor second" expanding to a fourth or an "ultraminor third" contracting to a unison. While precisely this creative ambiguity may be the desired effect, with either resolution open, adding a fourth voice can resolve the ambiguity. Thus 0-450-700-950 cents, with the 450-cent "ultramajor third" adding both excitement and definition, suggests the interpretation of 950 cents as a very large major sixth and 250 cents as a very large major second, with all three intervals resolving by stepwise expansion to the fifth, octave, and fourth respectively. In contrast, 0-250-700-950 cents, with the added 250-cent interval above the lowest voice as an "ultraminor third," suggests that the identical interval between the highest voices should likewise be taken as a very small minor third, and the outer 950 cents as an "ultraminor seventh," with these thirds contracting to unisons and the seventh to a fifth. One purpose of the term "interseptimal" is to describe these intervals in a relatively objective way based on their location along the spectrum, whether they represent the routinely flowing intonational currents of a gamelan ensemble, for example, or the less familiar effects of some other world musical settings where superlatives like "ultraminor" or "ultramajor" might be more in order. While giving these interseptimal regions a choice place on the continuum, we should also remember that in various world musical traditions they may blend quite unassumingly with what are here considered distinct categories, as in the free gamelan use of intervals we might here consider "fifths," "minor sixths," or somewhere in between. ---------------------------------------------------- 10. Superfourths and subfifths (49:48-minus regions) ---------------------------------------------------- We now come to the regions of superfourths and subfifths, most versatile and intriguing intervals for whose names I am much indebted to David Keenan. As here defined, superfourths are found in the region between a wide perfect fourth at around 256:189 or 525 cents and a small septimal tritonic interval at 112:81 or 561 cents -- say around 528-560 cents. Likewise subfifths populate a region situated between the large septimal tritone at 81:56 or 639 cents and the narrow fifth at 189:128 or 675 cents; allowing our usual leeway at each end of the range, around 640-672 cents. Especially familiar examples are the 11:8 superfourth at 551 cents, and the 16:11 subfifth at 649 cents, with many other shadings also in evidence, as we shall see. Superfourths and subfifths have affinities to either neutral or interseptimal types of intervals, and might be considered as an instance of either, but are perhaps best considered in their own special category. In gamelan music, they are often favored along with interseptimal and neutral intervals alike, the latter especially in pelog. In various Near Eastern music, neutral steps and intervals interact with others such as the 9:8 tone to generate superfourth or subfifth relationships, as with the 11:8 or 16:11 intervals found within Zalzal's tuning according to al-Farabi. From a certain theoretical viewpoint, the affinity between superfourths or subfifths and interseptimal categories is especially compelling. If we take a 4:3 fourth and subtract an interval up to around a 64:63 comma at 27 cents or a tad more, say around 30 cents, we have some kind of narrow fourth, ranging down to 21:16 (471 cents) or a bit smaller. If we subtract a small semitone of around 28:27, then we have a large major third at around 9:7 (435 cents). If we instead subtract an interval somewhere in the range of 30-60 cents, a _diesis_ as we shall term it, then an interseptimal interval somewhere in the range between 9:7 and 21:16 results. Similarly, if we begin again with 4:3 and add some kind of comma up to 64:63 or so, we have a wide fourth at up to around 256:189 (525 cents) or a bit larger. If we add a small semitone around 28:27, we have a small septimal tritone or diminished fifth at around 112:81 (561 cents). Should we add a diesis somewhere in the 30-60 cent range, however, then we have a superfourth of some flavor. The same logic would apply for subfifths. Adding a diesis of 30-60 cents to a 3:2 fifth produces an interseptimal flavor somewhere between the wide 32:21 fifth and the small 14:9 minor sixth. Subtracting 30-60 cents from 3:2 produces some flavor of subfifth siituated between the large septimal tritone or augmented fourth at 81:56 (639 cents) and the narrow fifth at 189:128 (675 cents). A certain affinity between these categories is also suggested by the intonational practices of gamelan. Just as either slendro or pelog often favors intervals at around 445-480 cents, which using our terminology would be either narrow fourths or interseptimals in the 9:7-21:16 range, so pelog especially often favors intervals around 515-545 cents, which might be styled either wide fourths or superfourths. This association might lead one to think of the two categories as kindred -- and likewise with intervals around 15-50 cents from 3:2 in either the wide or narrow direction. While this last example illustrates a pleasant association between these categories, it can also serve as a useful caution on the limitations of any generalized interval mapping scheme. Following the scheme presented here, we might classify a pelog interval of 515 cents as a "wide fourth" very close to 3/7 octave, but 535 cents as a "small superfourth." What is not clear, however, is that a Balinese or Javanese musician would take these sizes as representing two distinct categories, rather than shadings of a single basic category. Any generalized scheme, including this one, should always supplement rather than substitute for the categories and patterns of a specific musical tradition. Douglas Leedy has made a very cogent case that superfourths and subfifths as here styled (following Keenan) might be considered members of the same family as neutral thirds and sixths, for example. If we take an 11:9 neutral third at 347 cents and add a 9:8 tone, we arrive at an 11:8 superfourth at 551 cents. Similarly, one might add, subtracting a 9:8 tone from an 18:11 neutral sixth at 853 cents yields a 16:11 subfifth at 649 cents. Situations of this kind often arise in Near Eastern maqam and dastgah traditions, and would also occur in certain ancient Greek modes based, for example, on the Equable Diatonic tetrachord of Ptolemy with its division of the 4:3 fourth into string ratios of 12:11:10:9. Regarding superfourths and subfifths as belonging to their own category leaves open an appreciation of their co-occurence with neutral and/or interseptimal intervals (as here described) in various world musics, and also some conceptual affinities between these categories. As here defined, the region of superfourths or of subfifths spans a range similar to that of an interseptimal region, for reasons already touched upon in the comparison above. To derive a superfourth from a 4:3 fourth, for example, it must be widened by somewhat more than a septimal comma at 64:63 or 27 cents, say 30 cents, but less than a small or septimal semitone at 28:27 or 63 cents, which would yield a small septimal tritone at 112:81 or 561 cents. The difference between 28:27 and 64:63 is 49:48 or 36 cents. In practice, the range is a bit smaller, say 528-560 cents, or about 32 cents. Starting from 528 cents, we soon encounter the interesting ratio of 49:36 or 533 cents, with 36-EDO and multiples providing a very close approximation. This very small superfourth is equal to 4:3 plus a septimal diesis at 49:48, or 3:2 less the very large septimal chromatic step of 54:49 at 168 cents, near the top of our neutral second range (125-170 cents), and equal to the difference between a small 7:6 minor third and a large 9:7 major third. Possibly this small superfourth is large enough to have a degree of "independence" from the region of perfect fourths because it differs from 4:3 by 49:48, a diesis step more melodically discrete than a smaller interval of the comma type (up to about 30-33 cents). Another small superfourth is 15:11 at 537 cents, equal to 3:2 less Ptolemy's large neutral second at 11:10 or 165 cents, or 4:3 plus a small 45:44 diesis at 39 cents. As the studies of John Chalmers have suggested, 15:11 would arise in some ancient Greek modes based on the Equable Diatonic of Ptolemy with its 11:10 steps. This superfourth is very closely approximated by the 538-cent equivalent in 29-EDO. Possibly one of the first superfourths written in the Western European tradition of composition occurs in Fabio Colonna's treatise of 1618 on his harpsichord with 31 notes per octave, _La Sambuca Lincea_, where the upper voice starts at a fourth from a middle voice and then rises a diesis, about a fifth of a tone, to form a superfourth. If the instrument is tuned in 1/4-comma meantone, and we take to diesis to be the regular one of 128:125 or 41 cents, then this interval would be equal to the slightly wide meantone fourth at 503 cents plus the diesis, or 544 cents. If the octave were divided into 31 precisely equal parts, then the fourth would remain at around 503 cents but the diesis be slightly smaller, at around 39 cents, for a 542-cent superfourth. At 545 cents, we encounter a superfourth which, in systemic terms, is the regular diminished fifth of 22-EDO (e.g. B-F), equal to the somewhat wide fifth at 709 cents less the large chromatic step at 164 cents, very close to Ptolemy's 11:10. This superfourth is also equal to the 22-EDO fourth at 491 cents plus the regular diatonic semitone (or literally diatonic quartertone, since four make a major second) at not quite 55 cents. Although considerably smaller than a usual tritone as defined above (starting around 560 cents), it can quite happily in this context be called both a regular 22-EDO diminished fifth and a superfourth. At 551 cents, we have the cardinal landmark of 11:8, the simplest ratio in the region and also, interestingly, the classic mediant between 4:3 and the 7:5 diminished fifth. it is equal to 3:2 less a 12:11 middle neutral second, and thus arises both in ancient Greek modes based on Ptolemy's Equable Diatonic (12:11:10:9 tetrachord), and in Zalzal's scale as described by al-Farabi with its 12:11 steps. The 11:8 superfourth is also equal to a 4:3 fourth plus the large diesis of 33:32 or 53 cents, a large enough step to serve under some circumstances as a kind of small semitone. The 550-cent superfourth of 24-EDO and multiples provides an excellent approximation. Extended versions of Zarlino's 2/7-comma meantone also offer a good approximation of 11:8 by adding the meantone diesis of about 50 cents to the somewhat widened fourth at 504 cents, yielding a 554-cent superfourth. In 13-EDO, we also find an interval of this rounded size. At around 558 cents, near the upper border of the region, we have another form of superfourth (or small tritone) occurring in Zalzal's scale: 243:176, equal to 3:2 less the smaller 88:81 middle neutral second of this tuning at 143 cents. If we choose to regard the minimal size for a usual "tritone" as 4:3 plus a semitone at the rounded size of 60 cents, then 243:176 would be considered a very small tritone, since it differs from 4:3 by a large diesis or small semitone of 729:704, or 60.41 cents. If we prefer a rounded minimal size of 560 cents, then it is a very large superfourth. Since borders are fuzzy, the question seems somewhat moot. CMed 2/7- 36-EDO/ 29- 31- 1/4- 22- 24- 4:3-7:5 comma 49:36 15:11 EDO EDO comma EDO EDO 11:8 13-EDO 243:176 ..|------|------|----|-----|---|-----|------|-----|------|-------|.... 528? 533 537 538 542 544 545 550 551 554 558 560? The realm of subfifths is similarly diverse in its shadings, ranging from about 640 to 672 cents. In the Persian dastgah tradition, for example, it is very common in a dastgah or family of related modes such as the popular Shur for the fifth step of a mode to have two forms: a regular form at around 3:2, plus an inflected form lowered by a _koron_, a small interval which may range from about 30 to 70 cents. If values in the range of 30-60 cents were subtracted from a 3:2 fifth, then some flavor of subfifth would result. Indeed the Persian tuning of the fretted tar as suggested by Dariush Anooshfar based on often complex integer ratios, and available in the Scala scale archive (persian.scl), has some subfifths at 3200/2187 or 659 cents. The tuning described by Hormoz Farhat has subfifths at 655 and 660 cents. Gamelan tunings abound with intervals in this range, as also with superfourths. They may also occur in a tuning style such as that of the xylophone tuning favored by the Chopi people of southern Mozambique, where fifths may average around 4/7 octave or 686 cents, but with considerable variation in both directions, so that some are small enough to come within our subfifth region. This variety of "equable heptatonic" tuning, with melodic steps tending toward an average size of about 1/7 octave or 171 cents, will be considered at more length below. Here is a diagram showing some landmarks of this subfifth region. 13- 24- 22- 9-EDO 352:243 EDO 16:11 EDO EDO 22:15 72:49 ..----|--------|-----|----|---------|------------|---------|----.. 640? 642 646 649 650 655 663 667 672? If we take the region as starting around 640 cents, then just wider than this is the 352:243 subfifth at 642 cents found in Zalzal's scale. We quickly move into the nearer neighborhood of 16:11 at 649 cents, with either the 13-EDO interval at 646 cents or 24-EDO at 650 cents, for example, providing a fine approximation of this subfifth with the simplest integer ratio. In a neomedieval style, subfifths around 16:11 are often substituted for regular fifths close to 3:2 in minor seventh sonorities such as 1:1-13:11-16:11-39:22 (0-289-649-991 cents) or tempered variations. Here it is common for the lower two voices as usual to contract from minor third to unison, and the outer ones from minor seventh to fifth; if the lower voice rises by a semitone, for example, with the one immediately above it and the highest descending by a tone to bring about these resolutions, then the voice at 16:11 ascends by a neutral second to arrive at a unison with the highest. Here the subfifth is closely allied with neutral intervals, both vertical and melodic. In 22-EDO, the 655-cent subfifth is the regular tritone or augmented fourth (e.g. F-B), equal to a 709-cent fifth less the regular semitone of almost 55 cents. This interval lends itself to a very effective neomedieval resolution by oblique motion, for example from F-B to F-C, with the lower voice stationery and the upper ascending by the very incisive 55-cent step B-C. This is a kind of modern variation on the classic idiom of Perotin's era around 1200 mentioned in the discussion of tritonic intervals (Section 8). In 11-EDO, a subset of 22-EDO, this same 655-cent interval is used in certain "xentonal" styles as a kind of curious equivalent for a 3:2 fifth, with customized timbres designed to maximize a "fifthlike" impression. Given that this "xenofifth" is about 47 cents narrow of a pure 3:2, the resemblance is necessarily partial, with one listener hearing a kind of synthesis between a large tritone and a fifth. The 22:15 subfifth at 663 cents is notable for its use in a just gamelan tuning of Lou Harrison, with intervals of around 660-670 cents often reported in Javanese or Balinese pelog tunings, as well as others in nearby portions of the spectrum. The 667-cent subfifth found in 9-EDO and its multiples is reported sometimes used in certain timbres as a very small perfect fifth, likely somewhat more firmly than its 11-EDO counterpart at 655 cents. By this point we are approaching the region of small fifths, here estimated to have its fuzzy lower border, also the fuzzy upper border for subfifths, at around 672 cents. Superfourths and subfifths thus occur in various world musical traditions and tuning systems. They may be found in association with a rich use of neutral intervals, as in medieval and current Near Eastern traditions; or with the use of interseptimal and sometimes also neutral intervals, as in certain pelog tunings of gamelan. While superfourths and subfifths are available in some tuning systems such as extended meantones as supplementary or "special effects" intervals, in others such as 22-EDO when used as a regular neomedieval temperament and some 17-note circulating systems they occur as usual diminished fifths or augmented fourths, for example. In certain 17-note circles, we might diminished fifths of around 550-575 cents and augmented fourths of around 625-650 cents, thus spanning a range from the superfourth or subfifth region to that of small or large tritonic intervals as defined above (Section 8). Since these diverse sizes at different locations in the circle are regarded as musically interchangeable, this is an instructive example of how theoretical divisions of the interval spectrum should not trump the more flexible realities of musical practice which often disregard such borders. ------------------------------------------------------------- 11. Small intervals: commas, dieses, and "ultraminor seconds" ------------------------------------------------------------- To this point, while we have often brought intervals smaller than a usual "semitone" into play when defining or exploring other categories, we have not yet made these types of intervals our main focus. We shall do so now, considering the many-featured terrain to be found and surveyed between a pure unison and what we have termed a small usual semitone at the septimal 28:27 (63 cents) or the slightly narrower size of a rounded 60 cents. By tradition, this territory may be divided into two general categories: "commas" or the like ranging from the smallest discernible intervals to around 30-33 cents or so; and "dieses" here taken as ranging from around 30-33 cents to our small form of "usual" semitone at 60 cents. One imperfect but helpful concept is that notes at a comma of some kind apart may often be perceived as two "versions" of the same note or step; while notes connected by the step of a diesis are heard as more clearly "different" or "discrete" in their identity, whether that difference is perceived as a kind of small semitone, or as something quite distinct from a semitone. The first situation of a diesis used as a very small semitone is common in neomedieval styles, where steps of around 50-60 cents are routinely used in forming regular modes as well as in special cadential progressions. The second, of a diesis used as "something else again" and quite distinct from a semitone, occurs par excellence in Nicola Vicentino's 16th-century enharmonic style where meantone dieses of somewhere around 35-41 cents are used as special melodic "fifthtone" steps. Here we shall begin with the smaller intervals or commas, then moving to the larger dieses and "ultraminor seconds," while emphasizing that this is a very partial and incomplete survey not meant to substitute for various more comprehensive lists and discussions. Pyth 352:351 pental 531441: septimal 41- 364:363 896:891 144:143 72-EDO 81:80 524288 64:63 EDO |------|-------|-------|--------|--------|------|---------|-----|...| 0 5 10 12 17 22 23 27 29 30? A small comma often coming into play in neomedieval tunings is 352:351 (about 4.925 cents), the difference between the middle minor third at the classic mediant of 7:6 and 6:5, 13:11 (289 cents), and the slightly larger 32:27 Pythagorean minor third (294 cents) from three pure fifths or fourths. This comma is also the difference between a Pythagorean major third at 81:64 or 408 cents and a slightly larger 33:26 third at 413 cents, the latter plus 13:11 yielding a pure 3:2. A related comma minutely smaller is 364:363 (4.763 cents), the difference between a 33:26 major third and the classic mediant of 5:4 and 9:7 at 14:11 (418 cents), the simplest ratio for a middle major third. This 364:363 comma is also the difference between 13:11 and the slightly smaller 33:28 minor third at 284 cents, the fifth complement of 14:11. Adding these two commas yields 896:891 (9.688 cents), the difference between the Pythagorean 81:64 and 14:11, or likewise 32:27 and 33:28. In regular neomedieval temperaments where fifths are about two cents wide of a pure 3:2, a chain of four such fifths will come close to dispersing the full 896:891 comma, so that usual major thirds are quite close to 14:11. A chain of three fifths up or fourths down disperses an amount slightly in excess of the 352:351 comma, so that usual minor thirds are likewise close to 13:11. A comma of 144:143 or 12.044 cents defines the difference between the neutral seconds at 12:11 or 151 cents and 13:12 or 139 cents, and interestingly is available as a direct melodic step in tuning systems such as Zephyr-24 based on Erv Wilson's 1-3-7-9-11-13 eikosany. This small comma thus also marks, for example, the difference between the large tritone or augmented fourth at 13:9 (637 cents) and the 16:11 subfifth (649 cents), equal respectively to 4:3 plus 13:12 or 12:11. This fascinating comma also seems to approximate the difference between a Nobly Intoned (NI) major third at around 422-423 cents, the evident region of maximum complexity between 5:4 and 9:7, and the pure septimal ratio of 9:7 at 435 cents. In tuning systems like Zephyr-24, both of these flavors of active major thirds are thus available. The step of 1/72 octave or 16-2/3 cents available in 72-EDO is another interesting size of "comma" which in context can represent various integer ratios. It is also used by contemporary composers such as Julia Werntz as a melodic "microinterval" in its own right, without any deliberate regard to approximating intervals based on small integers, for example. We now come to the "big three" among the commas, which might be known by this phrase both because they are relatively large, and because they are the ones likely most often mentioned. The pental or syntonic comma of 81:80 (21.506 cents) is equal to difference between a small or pental major third at 5:4 (386 cents) and a Pythagorean third at 81:64; and likewise between a Pythagorean minor third at 32:27 and a large or pental minor third at 6:5. In meantone temperaments, this is the comma dispersed by chains of somewhat narrowed fifths (or widened fourths). Thus 1/4-comma produces pure 5:4 major thirds; 1/3-comma, pure 6:5 minor thirds; and 2/7-comma, equally impure approximations of 5:4 and 6:5 alike (both at 1/7 comma narrow, or about 3.072 cents). The syntonic or pental comma is also known as the comma of Didymis, named after a Greek musician describing the use of the 5:4 third. The Pythagorean comma at 531441:524288 or 23.460 cents, is equal to the difference between 12 pure 3:2 fifths and 7 octaves at 2:1. This comma must be dispersed in 12-note circulating temperaments, such as well-temperaments or modified meantone systems, where all 12 fifths are "reasonably close" to 3:2 (typically within about 7 cents for Western European styles often realized in strongly harmonic timbres.) The septimal comma or comma of Archytas at 64:63 or 27.264 cents marks the difference between the Pythagorean 81:64 middle major third and the large septimal major third at 9:7 (408 and 435 cents); and likewise between the Pythagorean 32:27 minor third at 294 cents and the small 7:6 minor third at 267 cents. A slightly larger interval, near the upper end of the "comma" range, is the 41-EDO step of 29.268 cents. Exactly where the line should be drawn between a "comma" and a larger "diesis" is unclear, and in any event it is a fuzzy border; but somewhere around 30-33 cents seems about right. To sum up on our sampling of commas, we might note that the smaller ones like 352:351, 364:363, and their sum of 896:891 serve to mark subtle differences of shading, as between middle major thirds at the Pythagorean 81:64 or slightly larger 33:26 or 14:11; and likewise middle minor thirds at the Pythagorean 32:27 or the slightly smaller 13:11 or 33:28 popular in neomedieval tuning systems. These commas can be dispersed in regular temperaments by a gentle widening of the fifth on the order of 2 cents, bringing about some new colors without compromising too seriously the medieval idea of pure fifths and fourths. The syntonic or Didymic comma at 81:80 or 22 cents, and the septimal or Archytan comma at 64:63 or 27 cents, are considerably larger, and thus call for a considerably greater compromise of the fifths if they are to be dispersed in a regular system such as a Renaissance European meantone for 81:80; or 22-EDO for 64:63, where a regular major third is very close to the large septimal ratio of 9:7. Some modified meantone systems seek a curious balance betwen the syntonic, septimal, and Pythagorean commas. Most of the fifths are tempered narrow by around 1/4 or 2/7 syntonic comma, but a chain of a few in the remote portion of the 12-note circle are tuned about equally wide, thus producing pental-flavor intervals in the nearer portion of the circle, a few septimal-flavor intervals in the most remote portion, and at the same time balancing out the Pythagorean comma so that all 12 fifths are within about 7 cents or pure and the system circulates. While even the largest commas, approaching the range of 30-33 cents, may often be heard more as two "versions" of "the same step," there are special situations where these intervals can take on more discretely melodic properties. For example, taking a given note and the pure 3:2 fifth above it as a sustained drone, let us add two notes at a 21:16 narrow fourth, and at 63:32 or a 3:2 fifth above 21:16 (471 and 1173 cents) -- with these added notes then moving to a simple 4:3 fourth and 2:1 octave (498 and 1200 cents), and thus making the 64:63 or 27-cent step more tangible. Above our drone we could also have two moveable voices starting at 21:16 and 7:4 (471 and 969 cents), and moving up 64:63 to 4:3 and 16:9 (498 and 996 cents). LaMonte Young and his colleagues are renowned for this type of melodic use of the septimal comma, which can also be approximated in temperaments with reasonable equivalents of these ratios. Moving now to large intervals or "dieses," the lower zone of demarcation remains uncertain, but as suggested might be placed around 30-33 cents. 2/7- smaller larger comma 1/4- 1/4-comma Pyth 24- 22- 36-EDO comma 49:48 31-EDO 29-EDO diesis 36:35 EDO 33:32 EDO 91:88 |....|------|-----|-----|-------|---------|------|----|----|---|----|..| 30 33 35 36 39 41 45 49 50 53 55 58 60? The 36-EDO step of 33-1/3 cents might be considered a very large comma or a very small diesis, depending on the musical context; this interval may be near the transitional zone between the two categories. An interval that nicely epitomizes a small diesis is the 34.990-cent step found in 1/4-comma meantone, which together with the larger 128:125 diesis at about 41 cents divides a 76-cent small or chromatic semitone (e.g. C-C#) into two nearly equal melodic intervals. These small steps play a premier role in the enharmonic or "fifthtone" music of such Manneristic composers as Nicola Vicentino and Fabio Colonna. Their harpsichord or organ tunings of 31 fifthtone steps to the octave, if realized in 1/4-comma, would thus produce diesis steps of 35 and 41 cents, a tuning quite successful in practice although not the only possibility. The 49:48 diesis at 35.697 cents, which we have met frequently in this exploration of the interval spectrum, represents the different for example between the large major second at 8:7 and the small minor third at 7:6; or the large major sixth at 12:7 and the small minor seventh at 7:4. In tunings such as that of Robert Walker having steps above the modal final or note or repose such as 8:7 and 7:6, or 12:7 and 7:4, this 49:48 or 36-cent step between the major second and minor third degrees, or major sixth and minor seventh degrees, can indeed play the role of a very small "semitone" or "ultraminor second." The 31-EDO diesis at 38.710 cents divides the small or chromatic semitone into precisely steps, and is another possibility for the interpretation of fifthtone music by Manneristic composers such as Vicentino and Colonna. In 1666, at any rate, Lemme Rossi demonstrated the very small mathematical distinction between this tuning and 1/4-comma, and suggested that Vicentino's tuning favors the former arrangement, championed by Christian Huygens in 1691. In 1/4-comma meantone, as already mentioned, a large diesis or fifthtone at 128:125 or 41.059 cents complements the smaller 35-cent fifthtone in making up a 76-cent chromatic semitone. Interestingly, 29-EDO has an almost identical diesis at 41.379 cents. While the 1/4-comma or 31-EDO dieses as typically used in a Manneristic enharmonic or "fifthtone" style have an effect quite distinct from that of a "semitone," the 41-cent fifthtone of 29-EDO is sometimes used in effect as a very small semitone in certain neomedieval cadential progressions involving interseptimal intervals. Thus a sonority with the 29-EDO "ultramajor third" at 455 cents and "ultramajor sixth" at 952 cents (close to 13:10 and 26:15) may resolve with these intervals expanding respectively to fifth and octave, the lower voice descending by a usual 207-cent tone and the upper ones ascending by 41-cent dieses. The Pythagorean diesis of about 45.112 cents is equal to half of a usual 256:243 semitone at around 90 cents; the term "diesis" is also sometimes used for this semitone itself, and "diaschisma" for this division of it into two equal parts. Note that "diaschisma" can also refer to a ratio of 2048:2025 or about 19.553 cents, a later usage. This type of division was likely used in some ancient Greek tunings in the enharmonic genus where a semitone or some kind is divided into two more or less equal parts, and is also mentioned as an added feature for a sophisticated organ by Ugolino of Orvieto writing around 1425-1440. After advocating a Pythagorean tuning with 17 notes per octave (Gb-A#), he adds that the diatonic semitones E-F and B-C (at 256:243 or 90 cents) might be divided into two approximately equal parts after the manner of the ancients. The 36:35 diesis at 48.770 cents is found in ancient Greek enharmonic tunings, and marks the difference between a small pental major third at 5:4 and its large septimal counterpart at 9:7 (386 and 435 cents); and likewise a small septimal minor third at 7:6 and its large pental counterpart at 6:5 (267 and 316 cents), etc. At around 50 cents, we enter a range where dieses may more readily or often serve as "ultraminor seconds" in a variety of tuning systems; Ivor Darreg has suggested that around 50-55 cents might mark a transition from "microtonal" intervals to intervals with a more familiar "semitone" quality. Two tuning systems notable for steps of around 50 cents are 24-EDO, where precisely this size of quartertone obtains, and 2/7-comma at 50.276 cents. In a regular form of extended 2/7-comma, or a modified form such as Zest-24, the versatile 50-cent step can be used now as a Vicentino-like meantone diesis (although other features of a 1/4-comma or 31-EDO division do _not_ obtain), now as a small neomedieval cadential semitone in cadences involving interseptimal intervals -- and also, in Zest-24, as a very "narrowly tempered" variation on a usual 28:27 semitone at 63 cents in certain septimal modes. The 33:32 diesis at 53.273 cents represents the difference between a 12:11 neutral second at 151 cents and a 9:8 tone at 204 cents; or a 4:3 fourth and a superfourth at 11:8 (498 and 551 cents). I recall the opinion of one Turkish music theorist that this is the smallest recognized "semitone" in some Turkish styles, which might reflect a perception similar to that of Ivor Darreg. It is narrower than the small 28:27 semitone by 896:891 or about 10 cents, so that in a tuning system like Zephyr-24 both 28:27 and 33:32 occur and may be used more or less interchangeably. In 22-EDO, the regular diatonic semitone (or literally quartertone) at 54.545 cents serves in effect as a tempered form of 28:27 in a septimal context, for example as the difference between the regular near-9:7 major third at 436 cents and the perfect fourth at 491 cents; or the large major sixth at 927 cents and the smallish minor seventh at 982 cents, the latter about 13 cents wide of 7:4 but still close enough to be nicely within its "septimal suburbs." Some people find this small a regular diatonic semitone quite pleasant, and others more problematic. A large diesis closely approaching the realm of usual small semitones is 91:88 at 58.036 cents, narrower than the septimal 28:27 by only a comma of 352:351 or about 5 cents. This step occurs in Zephyr-24, for example, and is almost exactly approximated by the interval of 58.090 cents in George Secor's HTT-29 and some derivative variants. The "quasi-diesis" of the Peppermint-24 system, based on the regular Wilson/Pepper temperament with fifths at about 704.096 cents, is a very comparable 58.680 cents. In a neomedieval style, these 58-59 cent steps invite use much like the slightly larger 28:27. By this point we are near the border area between "dieses" and more usual small semitones, possibly located somewhere around 60 cents. George Secor has suggested that one very attractive range for melodical optimal semitones may be found around 60-80 cents, with the 28:27 semitone or thirdtone of Archytas as a fine example. Having quickly surveyed the interval region of 0-60 cents, we should note that especially with inharmonic timbres such as those used in Javanese or Balinese gamelan, the smaller or "comma-like" intervals of this region can very effectively play the role of complex and shimmering unisons. Thus in Zephyr-24, the intervals of 144:143 or 12 cents, 78:77 or 22 cents, and 64:63 or 27 cents (the first and last considered above) very nicely serve this purpose in a gamelan style. ----------------------------------------------------- 11.1. Ultramajor sevenths and the "perioctave" region ----------------------------------------------------- Our exploration of the territory between a pure 1:1 unison and a usual small semitone at about 60 cents or large suggests a similar survey of the region between a large major seventh at around 1140 cents and a 2:1 octave. Here 1140 cents or so, a slightly wide tempering of the septimal major seventh at 27:14 or 1137 cents, seems a likely place for the lower border of this region. For the sake of symmetry, the region of 1140-1200 cents is shown in the first portion, and the region of 1200-1250 cents on the other side of 2:1 in the second portion. 88- 22- cET EDO 64:33 25-EDO 96:49 63:32 2/7-comma 143:72 2:1 |----|----|-----|------|-----------|------|-------|--------|-------|> 1140? 1144 1145 1147 1152 1164 1173 1180 1188 1200 2:1 288:143 128:63 88-cET <|--------|------------------|---------|.........................| 1200 1212 1227 1232 1250? Starting at 1140 cents, we soon a very interesting xentonal interval: the 1144 cents of Gary Morrison's 88-cET, where steps of 88 cents are used in an equal temperament without repetition at the 2:1 octave. This interval, while it might play the role of a very large and active major seventh in a more conventional setting, has been optimized for an aurally "consonant" or blending quality in certain timbres designed by William Sethares, who showcases Morrison's tuning as part of his _Xentonality_ CD. In 22-EDO, we find a regular major seventh at 1245 cents -- an octave less the small minor or "ultraminor" second at 55 cents -- which lends itself, for example, to a very effective oblique resolution ascending by this small semitone to the 2:1 octave, a la Leonin or Perotin in the style of Gothic Europe using the Pythagorean major seventh and semitone at 1110 cents and 90 cents. The interval of 64:33 at 1147 cents, equal to a 2:1 octave less the diesis or very small semitone at 33:32, can likewise serve as an "ultramajor seventh," more or less interchangeable with the septimal 27:14, as may happen in Zephyr-24 (which includes all these ratios). In 25-EDO, the ultramajor seventh available at 1152 cents is an especially striking example of this neomedieval effect, resolving to an octave via the step of 1/25 octave or 48 cents, an "ultraminor second" indeed! This seems to me a distinctively 21st-century sound with 12th-13th century roots. The ratio of 96:49, or 1164 cents, is equal to an octave less a septimal diesis at 49:48. It occurs, for example, in Robert Walker's earlier noted tuning if one plays the interval formed by two notes located at a 7:6 minor third and 16:7 major ninth (8:7 plus a 2:1 octave) above the modal center. I find that this interval can indeed have the effect of an "ultramajor seventh" if it resolves obliquely to the octave via a 49:48 or 36-step step in the upper voice; and has a very curious impression, possibly something between a major seventh and a kind of tempered octave, when it contracts conjunctly to a fifth, each voice moving by an 8:7 step. Moving further along the spectrum, at around 1267-1270 cents we enter the domain of intervals which may be defined as an octave less some comma of up to 30-33 cents. The ratio of 63:32 (an octave less 64:63) or 1173 cents can, as mentioned above in the discussion of that septimal comma, be used very tellingly to make 64:63 at 27 cents sound as a discrete melodic step, although somewhat different from a "semitone." In certain gamelan-like timbres, for example, an interval of around 63:32 can be used as a complex octave with a pleasantly shimmering sound. This kind of treatment is also attractive for ratios closer to 2:1 such as 143:72 at 1188 cents in Zephyr-24, a small octave narrower than 1200 cents by the characteristic comma of 144:143 or 12 cents discussed above. The second portion of the diagram above, starting at 1200 cents, shows a few possible ratios for wide octaves in gamelan-like situations, for example. Thus, in Zephyr-24, 288:143 at 1212 cents is wider than 3:2 by 144:143; and 128:63 at 1227 cents by the septimal comma at 64:63. A memorable instance of a wide octave is the 1232-cent interval found in Gary Morrison's 88-cET, and optimized for aural consonance by William Sethares as part of his artful exploration of this system. Indeed, Sethares has designed timbres where a ratio as large as 21:10, or 1284 cents (a middle 21:20 or pental-septimal semitone plus 2:1), has "octave-like" qualities. Jacky Ligon has suggested that narrow or wide octaves treated musically in a role analogous to 2:1, more or less, might typically range about 50 cents on either side of 1200 cents. Here, as the octave variations favored for instruments such as xylophones or metallophones in many world musics, and also the computerized timbres of Sethares, suggest, it is difficult to draw boundaries but most edifying even briefly, as here, to survey some of the terrain. -------------------------------------------- 12. The equable heptatonic and its heartland -------------------------------------------- To conclude our survey of the interval spectrum and its regions, we now come to a special kind of region which has a fairly clear heartland located at around 160-182 cents, but no clear borders: what is here termed the "equable heptatonic." The basic pattern of an equable heptatonic style of intonation is to divide the octave into seven not-too-unequal steps which thus average at around 1/7 octave or 171 cents. In one variety of equable heptatonic tuning reported as common in Thailand and in some African traditions, for example, these steps are often consistently within a relatively few cents of 171 cents, or a precise 7-EDO. In other varieties, however, along with some steps in this "heartland" of around 160-182 cents, say, there are others which may range from middle neutral seconds on the lower end to around a 9:8 tone on the upper end, say around 140-200 cents. Tunings of both varieties may be found in the Scala scale archive, for example. Thus one Burmese tuning reported by von Hornbostel (burma3.scl) has a range of about 163-182 cents, all within the equable heptatonic "heartland"; and a different tuning (burma.scl) a range of about 154-193 cents, also encompassing what we might call the realms of middle-to-large neutral seconds and small tones. Likewise, while the xylophone tunings, for example, of some African traditions may have steps consistently in the heartland range, South African composer Kevin Volans reports a scale (volans.scl) with steps ranging from 140 to 200 cents. A Chopi xylophone tuning from southern Mozambique as reported by Andrew Tracy similarly appears to have steps with a range of around 144-199 cents. Since in Section 4 we defined the range of the major second region as about 180-240 cents, and in Section 6 a neutral second region of around 125-170 cents, our heartland of the equable heptatonic nicely situates the intermediate range of 170-180 cents in a rightful home of its own. Here is one possible map of this heartland: CMed 11:10-10:9 55- 256: 65536: 11:10 54:49 7-EDO 21:19 EDO 231 59049 10:9 <|------------|--------|---------|--------|------|-----|-----|-----|> 160 165 168 171 173 175 178 180 182 The lower portion of this range coincides with the domain of large neutral seconds (around 160-170 cents). Thus starting from 160 cents, we soon encounter Ptolemy's 11:10 at 165 cents, and then the septimal chromatic step at 54:49 or 168 cents (the difference between a small 7:6 minor third and large 9:7 major third, at 267 and 435 cents, for example). At 171 cents, or more precisely 171.429 cents (171-3/7 cents), we have the 7-EDO step dividing the octave into 7 equal parts. A ratio quite close to this is 21:19 at 173 cents, the classic mediant of the 11:10 neutral second and the small 10:9 tone. Interestingly, 55-EDO, a tuning system close to 1/6-comma meantone sometimes proposed in 18th-century Europe as a useful model for intonation, has a 175-cent step which makes available a fine equable heptatonic division of the octave with six steps of 175 cents and one smaller step of 153 cents. The ratio of 256:231 or 178 cents occurs in Zephyr-24. Slightly larger is the complex Pythagorean ratio of 65536:59049, equal to twice the diatonic semitone of 256:243 or 90 cents, or 180 cents, and more precisely to around 180.450 cents. This interval is located at around our lower border of 180 cents for a small major second or tone, a category exemplified by the pental ratio of 10:9 or 182 cents. While equable heptatonic tuning systems have sometimes been referred to above rather freely as "scales," they are often used as gamuts or collections of notes from which pentatonic modes, for example, may be selected. In Southeast Asian traditions, pieces may shift within this gamut from one modal pattern or center to another, a process known by the ancient Greek term of _metabole_ orginally used to describe an analogous change from one genus or modal center to another. In defining a region of minor sevenths at around 960-1025 cents (Section 4), and of neutral sevenths at around 1030-1075 cents, we likewise left a small gap at 1025-1030 cents which can now find its own home in an equitable heptatonic heartland region centered at the reference point of 6/7 octave or 1029 cents. 59049: 55- 9:5 32768 EDO 38:21 7-EDO 49:27 20:11 51:28 31:17 <|-------|-------------|-------|------|-------|------|------|------|> 1018 1020 1025 1027 1029 1032 1035 1038 1040 Here we begin with the large or pental minor seventh at 9:5 or 1018 cents, with the slightly larger Pythagorean interval of 59049:32768 or 1020 cents (an augmented sixth, e.g. Eb-C#, equal precisely to five 9:8 or middle tones) in close proximity. At 1025 cents, we encounter the 55-EDO interval of 1025 cents. With 38:21 at 1027 cents, we are in the immediate vicinity of 6/7 octave at 1029 cents, Moving to intervals slightly larger than this 7-EDO size, we have 49:27 at 1032 cents, a complex septimal ratio which plays a very prominent role in the "aaron" scale designed by Gene Ward Smith in honor of composer and performer Aaron K. Johnson (available in the Scala archive as pipedum_12k.scl). By around 20:11, we are entering a portion of the equable heptatonic territory which is also a typical domain for small neutral or supraminor sevenths. The ratio of 51:28 or 1038 cents is equal to the small 17:14 neutral third, emblematic of its subregion (Section 3), plus a 3:2 fifth. Our diagram concludes at 1040 cents, with 31:17 an integer ratio of almost precisely this size (1040.080 cents). As with the equable heptatonic "heartland" around 171 cents, so with that around 1029 cents, actual tunings in this style in different world musical traditions may either remain within these precincts or also include considerably smaller or larger intervals. The picture becomes yet more rich and intricate when we consider that octaves somewhat smaller, or larger, than 2:1 at 1200 cents may be used in some of these traditions. Some tuning frequencies for a xylophone of the Chopi people of Southern Mozambique, as reported by Andrew Tracy, show octaves with a range of around 1174-1223 cents. Intervals of a single step have a range of about 144-199 cents, and those in the general vicinity of 6/7 octave a range of around 1003-1056 cents. Such a brief discussion of the equable heptatonic can only begin to suggest the diversity of world musical traditions and tunings fitting this general and often quite flexible approach. If this section has done so, highlighting "heartland" ranges of the interval spectrum especially characteristic of this approach while emphasizing how adjoining regions of the spectrum are often embraced in equable heptatonic styles of intonation, then it has served its purpose. -------------------------------------------------------- 13. Summary of suggested interval regions and subregions -------------------------------------------------------- The following table attempts to summarize the interval categories proposed above by surveying the spectrum from a 1:1 unison to a 2:1 octave. Section numbers are included to show where each category receives its main discussion. Note that this table presents a crude although convenient overview because, for "the sake of simplicity," it overlooks some basic and vital points presented in the text. Thus a "comma" of up to 30 cents or so might often serve as a kind of complex unison in a style such as gamelan, and likewise an interval in the range of 1170-1200 cents, for example, as a complex octave. As developed in the text (Section 11), the line between a "comma" and a "diesis" might be drawn at a fuzzy border region somewhere around 30-33 cents; this table uses the rounded value of 30 cents. Pure unison (1:1) 0 cents Commas 0-30 cents (Section 11) Dieses 30-60 cents (Section 11) Minor seconds 60-125 cents (Section 5) small 60-80 cents middle 80-100 cents large 100-125 cents Neutral seconds 125-170 cents (Section 6) small 125-135 cents middle 135-160 cents large 160-170 cents Equable heptatonic (heartland range) 160-182 cents (Section 12) Major seconds (or tones) 180-240 cents (Section 4) small 180-200 cents middle 200-220 cents large 220-240 cents Interseptimal (Maj2-min3) 240-260 cents (Section 9) Minor thirds 260-330 cents (Section 2) small 260-280 cents middle 280-300 cents large 300-330 cents Neutral thirds 330-372 cents (Section 3) small 330-342 cents middle 342-360 cents large 360-372 cents Major thirds 372-440 cents (Section 2) small 372-400 cents middle 400-423 cents large 423-440 cents Interseptimal 440-468 cents (Section 9) (Maj3-4) Perfect fourths 468-528 cents (Section 7) small 468-491 cents middle 491-505 cents large 505-523 cents Superfourths 528-560 cents (Section 10) Tritonic region 560-640 cents (Section 8) small 560-577 cents middle 577-623 cents large 623-640 cents Subfifths 640-672 cents (Section 10) Perfect fifths 672-732 cents (Section 7) small 672-695 cents middle 695-709 cents large 709-732 cents Interseptimal 732-760 cents (Section 9) (5-min6) Minor sixths 760-828 cents (Section 2) small 760-777 cents middle 777-800 cents large 800-828 cents Neutral sixths 828-870 cents (Section 3) small 828-840 cents middle 840-858 cents large 858-870 cents Major sixths 870-940 cents (Section 2) small 870-900 cents middle 900-920 cents large 920-940 cents Interseptimal 940-960 cents (Section 9) (Maj6-min7) Minor sevenths 960-1025 cents (Section 4) small 960-987 cents middle 987-1000 cents large 1000-1025 cents Equable heptatonic 1018-1040 cents (Section 12) (heartland range) Neutral sevenths 1030-1075 cents (Section 6) small 1030-1043 cents middle 1043-1065 cents large 1065-1075 cents Major sevenths 1075-1140 cents (Section 5) small 1075-1100 cents middle 1100-1120 cents large 1120-1140 cents Octave less diesis 1140-1170 cents (Section 11) Octave less comma 1170-1200 cents (Section 11) Pure octave (2:1) 1200 cents -------------- 14. Conclusion -------------- This paper has sought to sketch out an approach to interval spectrum regions which recognizes the musical beauty and value of each region or subregion, and encourages the further exploration of this spectrum whose multicolored aural hues provide many diverse palettes for the musics of the world. Margo Schulter Sacramento, California, USA mschulter@calweb.com 26 July 2010