Dear Andy, Bill, Jim, and Colby, Please let me say that while I'm not sure exactly how I would fit into some of the more confidential aspects of this project, I'm delighted to offer suggestions or elaborations on general points of theory which don't require knowledge of proprietary internal details. One such interesting general point is Valid Tuning Range (VTR) for what is termed the "schismatic temperament" -- or, as I would say, the schismatic mapping. Here I'll suggest that this mapping, whlle applicable for its original purpose (approximating 5-limit ratios) in the immediate vicinity of Pythagorean intonation or a rounded 702 cents, has a likely VTR from 19-EDO to 17-EDO, or a rounded 695-706 cents. -------------------------------------------- 1. Evaluating VTR for the schismatic mapping -------------------------------------------- As I understand it, VTR pertains not to the absolute sizes of intervals generated through a given mapping, but to their _relative_ sizes, or more precisely the ordering of their sizes. One approach would be to note the ordering of relevant intervals in the schismatic mapping when it is used around 702 cents for its original purpose of obtaining just or near-just 5-limit intervals, and then to test how far in either direction along the continuum that ordering is preserved. Here, for convenience, I will take a rounded 702 cents as the generator for a traditional schismatic mapping, making the arithmetic a bit simpler while arriving at a result virtually identical to that of Pythagorean tuning (701.955 cents). -------------------------------------------------------- Interval Example Cents 5-limit Cents -------------------------------------------------------- Apotome C-C# 114 16:15 112 Diminished 3rd C#-Eb 180 10:9 182 Augmented 2nd Eb-F# 318 6:5 316 Diminished 4th C#-F 384 5:4 386 Augmented 5th F-C# 816 8:5 814 Diminished 7th F#-Eb 882 5:3 884 Augmented 6th Eb-C# 1020 9:5 1018 Diminished 8ve C#-C 1086 15:8 1088 ------------------------------------------------------- As I here define it, the VTR is the range over which these orderings are preserved, with limits at the points where any relevant pair of these intervals have identical sizes, so that any further motion in that direction along the regular diatonic continuum of generators would cause a reversal of sizes, and move us into a different mapping. ---------------------------------------- 1.1. The schismatic mapping at 696 cents ---------------------------------------- Let us first consider two interesting points where the mapping remain valid, but yields very useful interval sizes other than those of the 5-limit. We begin at 696 cents, quite close to Zarlino's 2/7-comma meantone, the first regular temperament to be defined mathematically in terms of fractions of a syntonic comma (1558): -------------------------------------------------------- Interval Example Cents 2-3-7 JI Cents -------------------------------------------------------- Apotome C-C# 72 28:27 63 Diminished 3rd C#-Eb 240 8:7 231 Augmented 2nd Eb-F# 264 7:6 267 Diminished 4th C#-F 432 9:7 435 Augmented 5th F-C# 766 14:9 765 Diminished 7th F#-Eb 936 12:7 933 Augmented 6th Eb-C# 960 7:4 969 Diminished 8ve C#-C 1128 27:14 1137 ------------------------------------------------------- Note that while the interval sizes are quite different than around 702 cents, here approximating 2-3-7 or Archytan JI (named after the diatonic of Archytas, as we'll shortly see) rather than 5-limit JI, the ordering of sizes is the same. Thus we remain within the VTR. Let's quickly consider the classic JI tunings that these two realizations of the schismatic mapping approximation. Around 702 cents our mapping approximates the Syntonic Diatonic of Ptolemy with its prime factors of 2-3-5, later adopted in the 16th century by Fogliano (1529) and Zarlino (1558) as a basic for contemporary vocal polyphony. Here I'll give a version in the Ionian Mode, which Zarlino took as the first of the 12 modes: C D E F G A B C 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1 0 204 386 498 702 884 1088 1200 9:8 10:9 16:15 9:8 10:9 9:8 16:15 204 182 112 204 182 204 112 Here I follow Zarlino in placing the smaller 10:9 tones at D-E and G-A. In practice, of course, singers will dynamically adjust certain notes to preserve concord, for example a 3:2 fifth at D-A. On fixed-pitch instruments, as Zarlino notes, it is much more practical to use temperament: 2/7-comma or 1/4-comma meantone for keyboards, and 12-TET for fretted instruments such as the lute. Let us consider an equivalent of this JI tuning in our 702-cent schismatic mapping: B C# Eb E F# Ab Bb B 0 204 384 498 702 882 1086 1200 204 180 114 204 180 204 114 In either the 5-limit JI or schismatic version, there are three types of melodic steps: a larger tone at 9:8 or 204 cents; a smaller tone at 10:9 or 182 cents (180 cents in the schismatic version); and a rather large semitone at 16:15 (112 cents), or 114 cents in the schismatic version. At 696 cents, the schismatic mapping approximates, as we have said, a different kind of JI diatonic: the diatonic of Archytas, adopted by Ptolemy as the "Tonic Diatonic." Here I give this tuning in a permutation which might tend to maximize the degree of sensory consonance in relation to the 1/1; this arrangement might be described as a septimal (i.e. 2-3-7) JI version of the medieval/Renaissance Dorian mode: D E F G A B C D 1/1 9/8 7/6 4/3 3/2 27/16 7/4 2/1 0 204 267 498 702 906 969 1200 9:8 28:27 8:7 9:8 9:8 28:27 8:7 204 63 231 204 284 63 231 We have again three melodic step sizes: a 9:8 tone at 204 cents; a larger 8:7 tone at 231 cents; and a very compact semitone at 28:27 or 63 cents, highly effective melodically, as George Secor has noted. Now let us consider our equivalent for this Archytas or Tonic Diatonic using the schismatic mapping at 696 cents: Ab Bb B Db Eb F F# Ab 0 192 264 504 696 888 960 1200 192 72 240 192 192 72 240 Here the large tone at 240 cents is somewhat wide but not too far from a just 8:7 at 231 cents, with the 72-cent semitone also a bit wider than but fairly close to 28:27 at 63 cents. The biggest difference is the substitution of the regular meantone major second or tone at 192 cents for a just 9:8 at 204 cents in our 2-3-7 JI version. In some musical contexts, this might significantly change the artistic effect. In the JI version, for example, we might have this three-voice cadence with parallel thirds and sixths (and fourths between the upper voices) leading to a stable 2:3:4 sonority: E D E B A B G F E Both G-B-E at a Pythagorean 0-408-906 cents with an 81:64 major third and 27:16 major sixth (64:81:108), and F-A-D at a septimal 0-435-933 cents, with a 9:7 major third and 12:7 major sixth (7:9:12), are rather active and outgoing sonorities, leading nicely to the resolution in a usual 14th-century European fashion. At 696 cents, however, the same progression becomes: Bb Ab Bb F Eb F Db B Bb Here the opening sonority Db-F-Bb is tuned in regular meantone at 0-384-888 cents, very close to a just 12:15:20 with a 5:4 major third and 5:3 major sixth. This contrasts dramatically with the following B-Eb-Ab at 0-432-936 cents, quite close to a just 7:9:12 -- where the third and sixth are 48 cents larger! Such a dramatic difference might be accepted or even relished as "going with the territory," but produces a color distinct from that of the JI Tonic Diatonic version. Interestingly, one element of this 696-cent or more generally meantone schismatic mapping is already used in the Matrix/Thummer system: the meantone augmented sixth (e.g. Ab-F#) as an equivalent of 7:4 in order to approximate a seventh chord at 4:5:6:7 (0-386-702-969 cents) as Ab-C-Eb-F# (here 0-384-696-960 cents, and 0-386-697-966 cents in 1/4=comma). In the Tonic Diatonic or its 696-cent schismatic equivalent, however, a different variety of sonority with a minor seventh is available: 2-3-7 JI version 696-cent schismatic version C F# A Eb F B D Ab 0-267-702-969 cents 0-264-696-960 cents In JI, this chord with a 7:6 minor third, 3:2 fifth, and 7:4 minor seventh has a ratio of 12:14:18:21 (0-267-702-969 cents). Musically, like 4:5:6:7, it is a rather consonant sonority from a sensory standpoint, but may be treated as stable or unstable depending on the style. Just as 4:5:6:7 might serve either as a tuning of the dominant seventh in a major/minor style (as proposed by Euler in 1764), or as the norm of stable concord in Paul Erlich's decatonic system of tonality based on a tuning around 22-EDO, so 12:14:18:21 may either resolve to a yet more concordant sonority such as a 3:2 fifth, or stand as a stable conclusion, as noted by Secor. Typical directed resolutions would be: 2-3-7 JI version 696-cent schismatic version C B F# F A B Eb F F E B Bb D E Ab Bb In these progressions, the 7:4 minor seventh contracts to a fifth and the 7:6 minor thirds to unisons, with steps of 9:8 and 28:27 (204 and 63 cents) in JI, and 192 and 72 cents in our 696-cent tuning. These cadences are very effective, and could be considered septimal (i.e. 2-3-7) analogues of the tonal V7-I cadence. Let us now consider the schismatic mapping near the other end of its VTR, where different interval sizes and colors result: 705 cents. ---------------------------------------- 1,2. The schismatic mapping at 705 cents ---------------------------------------- Here the lineup of relevant interval categories for the mapping is as follows: --------------------------------------------------------- Interval Example Cents 2-3-11-13 JI Cents --------------------------------------------------------- Apotome C-C# 135 13:12 139 Diminished 3rd C#-Eb 150 12:11 151 Augmented 2nd Eb-F# 345 11:9 347 Diminished 4th C#-F 360 16:13 359 Augmented 5th F-C# 840 13:8 841 Diminished 7th F#-Eb 855 18:11 853 Augmented 6th Eb-C# 1050 11:6 1049 Diminished 8ve C#-C 1065 24:13 1061 --------------------------------------------------------- Again, while the interval sizes have changed, the ordering remains the same, thus keeping us within the VTR for the schismatic mapping. Using this mapping, we can approximate another traditional JI tuning, this one however coming from the early medieval rather than ancient Greek era: the `ud or lute tuning of Mansur Zalzal used in 8th-century Baghdad, as reported by al-Farabi (?-950): 1/1 9/8 27/22 4/3 3/2 18/11 16/9 2/1 0 204 355 498 702 853 996 1200 9:8 12:11 88:81 9:8 12:11 88:81 9:8 204 151 143 204 151 143 204 As with the Syntonic Diatonic and the Archytan or Tonic Diatonic, there are three step sizes: a 9:8 tone (common to all three tunings), a larger neutral second at 12:11 or 151 cents, and a smaller neutral second at 88:81 or 143 cents. In a 705-cent schismatic mapping, there is a subtle change of interval sizes and colors which may most notably affect the smaller neutral second step, here reduced from 88:81 at 143 cents to 135 cents, or slightly smaller than a just 13:12 at 139 cents. Thus some ratios of 13, while not an element of Zalzal's scale, are closely approximated at some locations in this tempered version: F# G# Bb B C# Eb E F# 0 210 360 495 705 855 990 1200 210 150 135 210 150 135 210 Small variations of this kind may be typical of medieval and modern Near Eastern tuning practices: thus Scott Marcus reports that modern performers of traditional Arab music in Egypt may favor in certain contexts a smaller neutral second around 135-145 cents, with our 705-cent schismatic version of Zalzal's tuning illustrating the lower end of this range, and al-Farabi's original version approaching the higher end. George Secor describes a different variety of 7-note tuning which features a 13/8 "harmonic" neutral sixth above the 1/1: 1/1 13/12 11/9 4/3 3/2 13/8 11/6 2/1 0 139 347 498 702 841 1049 1200 9:8 44:39 12:11 9:8 13:12 44:39 12:11 204 209 151 204 139 209 151 Here, as in the older JI tunings we have considered so far, there are three step sizes: a tone at 44:39 or 209 cents, slightly larger than the classic 9:8 at 204 cents; and neutral second steps at 12:11 (151 cents) and 13:12 (139 cents). Our schismatic mapping at 705 cents can quite closely approximate these sizes (here 210, 150, and 135 cents): Bb B C# Eb F F# G# Bb 0 135 345 495 705 840 1050 1200 135 210 150 210 135 210 150 The versatile schismatic mapping can approximate 5-limit intervals at 702 cents; septimal (2-3-7 JI) intervals at 696 cents; or neutral intervals (2-3-11-13 JI) at 705 cents. Let us now map the outer boundaries of the VTR for this mapping. ------------------------------------------- 2. The limits of the VTR: 19-TET and 17-TET ------------------------------------------- We found that the schismatic mapping is valid at 696, 702, and 705 cents -- and now wish to locate the limits of the VTR. Let us first consider the set of relevant intervals in 19-TET, with a generator of 694.74 cents: -------------------------------------------------------- Interval Example Cents JI Cents -------------------------------------------------------- Apotome C-C# 63 28:27 63 Diminished 3rd C#-Eb 252 15:13 248 Augmented 2nd Eb-F# 252 15:13 248 Diminished 4th C#-F 442 9:7 435 Augmented 5th F-C# 758 14:9 765 Diminished 7th F#-Eb 948 26:15 952 Augmented 6th Eb-C# 948 26:15 952 Diminished 8ve C#-C 1137 27:14 1137 ------------------------------------------------------- Here we have reached a boundary where the diminished third at C#-Eb has a size identical to that of the augmented second at Eb-F# at 252 cents. If we seek a JI equivalent involving prime factors no higher than 13, then a good choice would be 15:13 at 248 cents; 22:19 at 254 cents would by somewhat closer. Similarly, the diminished seventh F#-Eb at 948 cents is identical to the augmented sixth Eb-C#. Here 26:15 (952 cents) is a reasonable JI equivalent, although the 19-prime 19:11 (946 cents) would be closer. These intervals are what I have termed "interseptimal," located between the two simple septimal ratios of 8:7 (231 cents) and 7:6 (267 cents) at 252 cents; or between 12:7 (933 cents) and 7:4 (969 cents) at 948 cents. See An interesting musical property of this 19-TET mapping is that the 252-cent interval might represent either a very wide major second or a very small minor third; and likewise 948 cents might be used as a very large major sixth or a very narrow minor seventh. These two alternative resolutions illustrate this creative ambiguity: C# C C# E Bb C Bb B Eb F Eb E In the first alternative, the outer interval Eb-C# at 948 cents acts like a narrow minor seventh, contracting to a fifth, with steps of a regular meantone major second at 189 cents (Eb-F) and a small semitone C#-C (63 cents). At the same time, the upper interval Bb-C# at 252 cents acts like a small minor third contracting to a unison via the same melodic step sizes (Bb-C, C#-C). In the second resolution, Eb-C# instead acts as a wide major sixth, expanding to the octave of the 2:3:4 sonority E-B-E, while the upper Bb-C# acts like a wide major second expanding to a fourth. These progressions, as in the first resolution, involve steps of 189 and 63 cents. It is perhaps psychologically convenient that 19-TET should be the lower limit of the VTR for the schismatic mapping, since it might also be considered the lower limit of the meantone range proper, if we regard meantone temperament as primarily a strategy for optimizing JI factors 2-3-5 among regular diatonic intervals. Around 19-TET, or the virtually identical 1/3-comma meantone (694.79 cents), we get a pure 6:5 minor third at 316 cents. Tempering the fifth more than in 1/3-comma or 19-TET (where the minutely greater temperament produces precise mathematical symmetry, although 1/3-comma will circulate without any problem, as will 1/4-comma in 31 notes vis-a-vis 31-TET) moves the fifth, major third, and minor third alike away from their values in the 5-limit, and thus moves into another realm where different motivations prevail. Let us now consider the upper limit of the VTR for the schismatic mapping: 17-TET, with a generator at 705.88 cents. -------------------------------------------------------- Interval Example Cents JI Cents -------------------------------------------------------- Apotome C-C# 141 13:12 139 Diminished 3rd C#-Eb 141 13:12 139 Augmented 2nd Eb-F# 353 27:22 355 Diminished 4th C#-F 353 27:22 355 Augmented 5th F-C# 847 44:27 845 Diminished 7th F#-Eb 847 44:27 845 Augmented 6th Eb-C# 1059 24:13 1059 Diminished 8ve C#-C 1059 24:13 1059 ------------------------------------------------------- Here we may in brief describe what has happened by saying the neutral second, third, sixth, and seventh have single values at 141, 353, 847, and 1059 cent. The apotome is identical to the diminished third; the augmented second to the diminished fourth; the augmented fifth to the diminished seventh; and the augmented sixth to the diminished octave. In the "neomedieval" portion of the continuum which runs from around 702 to 709 cents, or Pythgorean to 22-TET, 17-TET marks not only the upper limit of the VTR for the schismatic mapping, but also a divide between two approaches to approximating septimal (2-3-7) intervals where they are available in reasonably representative versions within this part of the spectrum. In the range from 702 to 706 cents (or more precisely 17-TET), septimal approximations, where available, involve rather long chains of fifths. Optimal spots for such purposes are right around or very slightly above Pythagorean, where chains of 14, 15, or 16 generators approximate 7:4, 7:6, and 9:7. Exactly at Pythagorean with a 3:2 fifth, these intervals will vary from just by the "septimal schisma" of about 3.80 cents. By making the fifth minutely wider, we can temper out this small schisma: at 702.21 cents, with the fifth larger than pure by 1/15 of this schisma, 7:6 is pure, and 7:4 and 9:7 impure by the same tiny amount as the fifth, about 0.25 cent. A second "sweet spot" is found at around 704.6 to 705 cents, where chains of 13, 14, or 15 generators produce 9:7, 7:6, and 7:4. Here the overall accuracy cannot be quite so great as in the niche just above Pythagorean, where the fifth is virtually just, since the tempering of the fifth at around 3 cents wide of pure must make some of the septimal ratios impure by at least this amount. An optimal strategy might be setting 7:6 pure, using a generator of 704.78 cents, with 9:7 and 7:4 impure by the same amount as the fifth, about 2.82 cents. Starting at around 17-TET, however, another strategy prevails: simply tempering out the septimal comma so that regular intervals approximate septimal (2-3-7) ratios, much as the syntonic comma is tempered out in meantone to approximate 5-limit intervals. At 17-TET, we are on the threshold of this strategy: the minor and major thirds and minor seventh at 282, 423, and 988 cents have a rather different musical quality than 7:6, 9:7, and 7:4 at 267, 435, and 969 cents, one which may be more evocative of the "middle" territory between Pythagorean and septimal flavors, a territory also characteristic of neomedieval style. At around 707 cents, these regular intervals become more convincingly "septimal," as in the nearer transpositions of George Secor's 17-note well-temperament (where the larger fifths are 707.22 cents). In the territory around 708-709 cents, these approximations become more accurate. At 22-TET (709.09 cents), major thirds at 436 cents are very slightly wide of a just 9:7; minor thirds at 273 cents about 6 cents wide of 7:6; and minor sevenths at 982 cents about 13 cents wide of 7:4. Above 17-TET, the schismatic mapping is no longer valid, which is to say that it is replaced by a related by different mapping we will briefly consider below. ---------------------------------------------------------------- 3. An overview, the schismatic and "reverse schismatic" mappings ---------------------------------------------------------------- Having taken "snapshots" of the schismatic mapping as it operates at 696, 702, and 705 cents, and ascertained its limits of validity as 19-TET and 17-TET, we may get an overview of this mapping by sampling the values of our relevant intervals at generator sizes located along the VTR for this mapping. Here we start at 19-TET and conclude at 17-TET, sampling intermediate generators at reference points about one cent apart: ------------------------------------------------------------------------------ Intvl Example 19-TET 696 697 698 699 700 701 702 703 704 705 17-TET ------------------------------------------------------------------------------ Apot C-C# 63 72 79 86 93 100 107 114 121 128 135 141 dim3 C#-Eb 252 240 230 220 210 200 190 180 170 160 150 141 Aug2 Eb-F# 252 264 273 282 291 300 309 318 327 336 345 353 dim4 C#-F 442 432 424 416 408 400 392 384 376 368 360 353 Aug5 F-C# 758 766 776 784 792 800 808 816 824 832 840 847 dim7 F#-Eb 948 936 927 918 909 900 891 882 873 864 855 847 Aug6 Eb-C# 948 960 970 980 990 1000 1010 1020 1030 1040 1050 1059 dim8 C#-C 1137 1128 1121 1114 1107 1100 1093 1086 1079 1072 1065 1059 ------------------------------------------------------------------------------ This overview of the schismatic mapping suggests some general regions along the VTR for approximating different flavors of intervals. At 19-TET or around 695 cents, we generally get interseptimal flavors, although the apotome or chromatic semitone at 63 cents provides a virtually just 28:27, and likewise the diminished octave a virtually just 27:14. Around 696-697 cents, or in the general zone of 2/7-comma, 1/4-comma, and 31-TET, we get the best septimal (2-3-7) approximations for this mapping, with 9:7 virtually just around 2/7-comma, and 7:4 around 31-TET. At 697-699 cents, we get intervals in the intermediate range between Pythagorean and septimal -- also a staple of regular neomedieval tunings around 702-706 cents, where intervals of this type are the usual diatonic sizes. At 700 cents, we have a special situation where schismatic intervals are identical to regular diatonic ones (e.g. the augmented second to a regular minor third at 300 cents, and the diminished fourth to a regular major third at 400 cents). These regular and schismatic intervals might be taken either as somewhat subdued approximations of Pythagorean intonation, or rather inaccurate 5-limit approximations (more persuasive in less assertively harmonic timbres, for example a 16th-century lute or 20th-century piano). Around 701-703 cents we have the "schismatic" mapping in its familiar 5-limit sense, with an optimal niche in the zone at or just below Pythagorean if it is these 5-limit intervals which are to be optimized (e.g. 53-TET). Starting around 29-TET (703.45 cents), we enter a zone where the schismatic mapping shifts from the 5-limit to the neutral region -- a zone extending to 17-TET, the upper limit of the VTR. Note in 29-TET, the augmented second at 331 cents and diminished fourth at 372 cents might be considered either rather inaccurate 5-limit approximations, or as reasonable approximations of small and large neutral thirds, for example 63:52 (332 cents) and 26:21 (370 cents). By 704 cents, the neutral character of these "schismatic" thirds at 336 and 368 cents is more pronounced, and by 705 cents were are into the "central" neutral region of thirds from around 39:32 (342 cents) to 16:13 (359 cents). At 17-TET, the 706-cent fifth is divided into two precisely equal neutral thirds at 353 cents -- or, more precisely, 705.88 cents and 352.94 cents. In fact, 17-TET marks the end not of the great region where augmented and diminished intervals yield neutral sizes, but simply of the portion where the augmented second is smaller than the diminished third. Here it may be instructive to see what happens between 702 and 709 cents (or 22-TET at 709.09 cents), and a bit beyond that. ------------------------------------------------------------------------------ Intvl Example 702 703 704 705 17-TET 707 708 22-TET 710 711 712 ------------------------------------------------------------------------------ Apot C-C# 114 121 128 135 141 149 156 164 170 177 184 dim3 C#-Eb 180 170 160 150 141 130 120 109 100 90 80 Aug2 Eb-F# 318 327 336 345 353 363 372 382 390 399 408 dim4 C#-F 384 376 368 360 353 344 336 327 320 312 304 Aug5 F-C# 816 824 832 840 847 856 864 873 880 888 896 dim7 F#-Eb 882 873 864 855 847 837 828 818 810 801 792 Aug6 Eb-C# 1020 1030 1040 1050 1059 1070 1080 1091 1100 1110 1120 dim8 C#-C 1086 1079 1072 1065 1059 1051 1044 1036 1030 1023 1016 ------------------------------------------------------------------------------ At 17-TET, we have not only the upper limit of the schismatic mapping but also the lower limit for the VTR of mapping we might describe as "reverse schismatic," with the order of certain pairs of intervals reversed. Thus the diminished third is now larger than the apotome; the augmented second than the diminished fourth; the augmented fifth than the diminished seventh; and the augmented sixth than the diminished octave. From 17-TET to around 708 cents, this mapping produces neutral intervals of various shadings -- a kind of "mirror reversal" of the schismatic mapping around 703.45-705.88 cents (or 29-TET to 17-TET). As we approach 709 cents or 22-TET, we enter a zone where the reverse schismatic mapping -- like the schismatic mapping around 702 cents -- produces 5-limit approximations. Indeed the 382-cent augmented second of 22-TET is very close to the 384-cent diminished fourth at 702 cents, with the 22-TET augmented sixth at 1091 cents likewise comparable to the 702-cent diminished octave at 1086 cents. The diminished fourth of 22-TET at 327 cents, however, is considerably larger and further from a just 6:5 (316 cents) than the Pythagorean schismatic equivalent at 318 cents, approaching the lower end of the neutral zone at around 330 cents. As the fifth is tempered more and more heavily in the wide direction, here about 7.14 cents (comparable to 19-TET at 7.22 cents in the opposite direction), such divergences among at least some intervals of a given "family" of approximations from JI values (here 5-limit) become inevitably greater and greater. By conventional standards of historical European keyboard temperaments, about 7 cents may be the maximum desirable tempering of the fifth in a regular system, with 1/3-comma meantone or 19-TET as a 16th-century precedent. Moving further along the continuum is to enter a "xentonal" region where other assumptions, aesthetics, and timbral conditions may often prevail. Between 22-TET (709.09 cents) and 712 cents, various intervals move through a 5-limit region, and some are still in this range at 712 cents. The apotome has grown from 164 cents in 22-TET, very close to Ptolemy's large neutral second at 11:10 (165 cents), to a near-just 10:9 at 184 cents, actually a bit larger than the just ratio (182 cents). The diminished fourth at 304 cents is still not too far from 6:5 (316 cents), actually somewhat closer than in 12-TET. However, the augmented second at 408 cents is a virtually just Pythagorean major third at 81:64, while the augmented sixth at 1120 cents, very close to a just 21:11 (1119 cents), is about ten cents larger than the Pythagorean major seventh at 243:128 or 1110 cents. The tempering of the fifth at a full 10 cents wider than pure helps to account for some of these colorful variations between related categories of intervals in the reverse schismatic mapping. What might be the upper limit of the VTR for the reverse schismatic mapping? Starting from 17-TET, we can survey its values in order to seek an answer to this question. Here interval categories are ordered so that the smaller category of a relevant pair appears first in the table: ------------------------------------------------------------------------------> Intvl Example 17-TET 707 708 22-TET 710 711 712 713 714 715 716 717 ------------------------------------------------------------------------------> dim3 C#-Eb 141 130 120 109 100 90 80 70 60 50 40 30 Apot C-C# 141 149 156 164 170 177 184 191 198 205 212 219 dim4 C#-F 353 344 336 327 320 312 304 296 288 280 272 264 Aug2 Eb-F# 353 363 372 382 390 399 408 417 426 435 444 453 dim7 F#-Eb 847 837 828 818 810 801 792 783 774 765 756 747 Aug5 F-C# 847 856 864 873 880 888 896 904 912 920 928 936 dim8 C#-C 1059 1051 1044 1036 1030 1023 1016 1009 1002 995 988 981 Aug6 Eb-C# 1059 1070 1080 1091 1100 1110 1120 1130 1140 1150 1160 1170 ------------------------------------------------------------------------------> ------------------------------- Intvl Example 718 719 720 ------------------------------- dim3 C#-Eb 20 10 0 Apot C-C# 226 233 240 dim4 C#-F 256 248 240 Aug2 Eb-F# 462 471 480 dim7 F#-Eb 738 729 728 Aug5 F-C# 944 952 960 dim8 C#-C 974 967 960 Aug6 Eb-C# 1180 1190 1280 ------------------------------- Since the ordering of intervals remains valid to the upper regular diatonic limit of 5-TET or 720 cents, we can conclude that the reverse schismatic mapping has a VTR running from 17-TET to 5-TET. Over this imposing range of the spectrum we find neutral intervals from 17-TET or 706 cents to about 708 cents; 5-limit approximations around and somewhat beyond 22-EDO or 709 cents; and approximations in the range from Pythagorean to septimal, for the most part, between about 712 and 715 cents. The latter value yields an apotome at 205 cents or a near-just 9:8 tone, and an augmented second at 435 cents or a virtually just 9:7. At 716-717 cents, the diminished fourth provides very nice approximations of a 7:6 minor third, while some other intervals move into the interseptimal region -- specifically the augmented second and diminished seventh, respectively 444 and 756 cents with a 716-cent generator; and 453 and 747 cents at 717 cents, ratios very close to 13:10 and 20:13 (454 and 746 cents). The diminished third, as we move beyond 715 cents, shrinks to less than 50 cents -- about the minimum value for a routine "semitone" in neomedieval styles where about 50-90 cents may be the norm. At we more and more closely approach 720 cents, the impending convergence on 5-TET values becomes more clear. Interestingly, 719 cents offers an augmented second at a virtually just 21:16 at 471 cents, and a diminished octave at 967 cents, very close to 7:4 at 969 cents. Curiously, the literal tritone in this tuning at 709 cents is almost identical to the 22-TET fifth! At the limiting value of 720 cents or 5-TET, the diminished third becomes a unison, and the augmented sixth a 1200-cent octave. Thus the reverse schismatic mapping between 17-TET and 5-TET provides a spectrum of intervals ranging from neutral to 5-limit to Pythagorean and septimal to interseptimal. At 5-tET, the intervals of 240 and 960 cents embody the interseptimal region, with the first having qualities of either a very wide major second or a very narrow minor third, and the second likewise of either a very wide major sixth or a very narrow minor seventh. In larger tunings such as 20-TET, these possible identities may be defined in contrapuntal terms, as when a 240-cent interval acts as a "quasi-third" by contracting to a unison, one voice moving by 180 cents and the other by a 60-cent semitone; or when by a similar motion a 960-cent "quasi-sixth" expands to an octave. We have now accounted for the schismatic mapping with its VTR from 19-TET to 17-TET; and the "reverse schismatic" mapping with a VTR from 17-TET to 5-TET. This raises the question: what of the territory between 7-TET and 19-TET? ------------------------------------------------- 4. Inframeantone: a mapping below the schismatic? ------------------------------------------------- We have seen that 19-TET is the lower limit of the VTR for the schismatic mapping, the point at which a diminished third and an augmented second (e.g. C#-Eb and Eb-F#) have identical sizes of about 252 cents. In the schismatic mapping, a diminished third is smaller than an augmented second -- for example, 240 and 264 cents at a generator of 696 cents, and 141 and 353 cents at the mapping's upper limit of 17-TET. What happens below 19-TET, the lower limit for this mapping and also for what might be called "meantone proper" where the fifth is narrowed in order to optimize tempered ratios within the 5-limit such as 3:2, 5:4, and 6:5? Here, again, it may be instructive to sample some interval sizes from 7-TET (685.71 cents) to 19-TET (694.74 cents): ---------------------------------------------------------------------- Intvl Example 7-TET 687 688 689 690 691 692 693 694 19-TET ---------------------------------------------------------------------- Apot C-C# 0 9 16 23 38 37 44 51 58 63 Aug2 Eb-F# 171 183 192 201 210 219 228 237 246 252 dim3 C#-Eb 343 330 320 310 300 290 280 270 260 252 dim4 C#-F 514 504 496 488 480 472 464 456 448 442 Aug5 F-C# 686 696 704 712 720 728 736 744 752 758 Aug6 Eb-C# 857 870 880 890 900 910 920 930 940 948 dim7 F#-Eb 1029 1017 1008 999 990 981 972 963 954 948 dim8 C#-C 1200 1191 1184 1177 1170 1163 1156 1149 1142 1137 ---------------------------------------------------------------------- Here it appears that we have a mapping with an ordering valid from 7-TET to 19-TET -- might it be called "infraschismatic"? In an overview like this, interval values sweep through wide swaths of the spectrum. The diminished third, for example, a neutral third at 7-TET with a size of 343 cents (very close to 39:32 at 342 cents), shrinks at 687 cents to 330 cents, at the lower end of the neutral third region; it then moves through the 5-limit region, by 690 cents reaching 300 cents, more on the outskirts of the Pythagorean zone near a just 32:27 (294 cents). At 690-693 cents we then move through this Pythagorean zone and the territory between Pythagorean and septimal, by 693 cents reaching a size of 270 cents, very close to 7:6 (267 cents) -- and by 19-TET, the fine interseptimal 252 cents, identical in size to the augmented second, a convergence defining the upper limit of the VTR. -------------------------------------------- 5. Conclusions: mappings for 7-10 generators -------------------------------------------- Possibly the best way to describe the infraschismatic mapping (7-TEt to 19-TET), the schismatic mapping (19-TET to 17-TET), and the reverse schismatic or ultraschismatic mapping (17-TET to 5-TET) as is mappings of "7-10 generators," since the relevant intervals are generated from chains of this length. Given the diversity of interval regions generated within the territory of each mapping, a term like "schismatic" may be more of an historical curiosity than a comprehensive description of what these mappings are about. We get neutral intervals, major and minor intervals of various colors, and interseptimal intervals -- as well as, sometimes, what we might call perfect fourths and fifths at or near a just 4:3 or 3:2, as with the 496-cent diminished fourth and 704-cent augmented fifth at a generator size of 688 cents. While the range from around 695 to 709 cents, or from 19-TET to 22-TET, might be considered familiar, the regions below and above deserve their own thorough exploration in their own terms. A realization that each interval or generator region has its own charm is a vital element in this process. Most appreciatively, Margo Schulter 10 September 2008