----------------------------------------------------------- The e-based tuning (704.607 cents) as a "bifurcated" 17-EDO Amine Beyhom and ideas for a 17-step notation: Part 1 ----------------------------------------------------------- In June 2000, I came upon the idea of a regular tuning where the ratio of logarithmic sizes between the whole tone and diatonic semitone would be equal to Euler's _e_, approximately 2.71828. This tuning has a fifth at about 704.607 cents, a diatonic semitone (e.g. E-F) of 76.965 cents, and a chromatic semitone (e.g. F-F#) of 132.248 cents. At the time, my motivation for choosing the arbitrary mathematical perimeter was simply to explore an interesting region of what I then termed the "neo-Gothic spectrum" between Pythagorean and 22-EDO, and since have come to call "neomedieval" so as to include medieval (and later) Near Eastern styles as well as those of Gothic Europe. My first excited report on the new temperament from July 2000 reflects at once my long experience with medieval European styles and my lack of focus at that time on Near Eastern modal systems and styles: While my first explorations early that summer involved a 12-note tuning, in October 2000 I realized that the region around 704.6 cents did have a special property: a chain of 15 fifths up formed a virtually just 7:4. This serendipitous touch that I had _not_ considered originally in plucking Euler's _e_ out of the air as a mathematical parameter to define a tuning, was nevertheless most felicitous given Leonhard Euler's championship of 7:4 as a concordant ratio which might be used in practiced as a tuning for minor sevenths. Near the end of summer in 2001, a critical turning point came when I found myself in contact with George Secor, whose inestimable guidance and mentorship led my to delve into the realm of Near Eastern music with its rich repertory of Zalzalian or neutral intervals also abounding in the tuning system we were mainly exploring, his 17-tone well-temperament (17-WT) of 1978. Our collaboration also lead to my devising a 24-note system called Peppermint based on two 12-note chains of a regular temperament (704.096 cents) proposed by Keenan Pepper. In terms of the sheer variety of Zalzalian steps and intervals available in "only" 24 notes, and the overall accuracy of prime approximations, I would still recommend Peppermint as an ideal neomedieval temperament. The creative influence of Shaahin Mohajeri leads me to draw a parallel between Peppermint and the varied schemes for tuning the frets of the Persian tar. His skill as a composer and knowledge of the Persian dastgah system plays an important part in my appreciation for this music and the related Near Eastern maqam systems which often show similar patterns of intonation. The e-based system, in contrast, offers the advantages of a completely regular (although noncirculating) temperament: there are fewer types of intervals, but available at more locations with more transpositions easily possible. How did I come, around 2008, to explore this system anew with avidly renewed interest? Here Ozan Yarman lent pivotal impetus by making me more aware of the depth and intricacy of maqam music as actually realized more or less fluently in a given tuning system, theoretical or practical. His compendious 79-MOS for the qanun based on a slightly modified version of 159-EDO at once inspired me to consider the art more deeply, and to renew my interest in the medieval Near Eastern theorists, especially the "Systematists" of the later 13th century such as Safi al-Din al-Urmawi and Qutb al-Din al-Shirazi. Their tuning schemes, sometimes predicated in part on a regular 17-note Pythagorean system, seemed to have a curious modern counterpart in the regular 24-note e-based temperament with its wealth of Zalzalian and septimal intervals. My exploration of the e-based system was further advanced last summer when I studied in more depth some writings of the Lebanese musician and scholar Amine Beyhom, and became aware of a fascinating development in some of his recent articles. In his PhD thesis of 2003 on _Modal Systematics_, he had followed the familiar modern Arab theoretical model of 24 steps per octave, while stressing that this scheme serves only to identify the _type_ of steps in a given maqam or jins (i.e. "genus," for example a trichord, tetrachord, or pentachord), not to specify a precise intonation. Going beyond a caution that these 24 steps are indeed unequal in practice (at least to informed traditional performers!), he suggests the use of plus or minus signs to convey fine points of intonation. In the more recent articles, he intriguingly follows this general approach, giving some actual measurements of interval sizes used by flexible-pitch performers of maqam music, but with a proposal that a conceptual model of 17 steps per octave might better fit many Near Eastern styles of intonation. His ideas seemed beautifully to tie in with some main intonational themes I had been pursuing in the e-based system, a happy cross-pollination, as George Secor might call it, leading to this article. Part I focuses on the structure of the e-based temperament and the three main interval families in a 24-note system, while Part II will explore a slightly modified version of Beyhom's 17-step notation as applied to some medieval and modern Near Eastern tunings as realized in the e-based system. -------------------------------------------------- 1. Overview of the e-based tuning: a 17-step ethos -------------------------------------------------- In its 24-note version, the e-based tuning has a character or ethos largely shaped by three distinctive "colors" or families of intervals met within a 17-note chain of fifths. The choice of a 24-note size serves to make these intervals available at more locations, and also more incidentally to introduce some rarer intervals which, along with some others arising only in a larger tuning set (e.g. 29, 34, 46, 63, or a circulating 109), will be briefly mentioned later in this article. A Scala file follows: ! eb24.scl ! Regular "e-based" tuning, Blackwood's R or tone/limma at e, ~2.71828 24 ! 55.28289 132.24835 187.53125 209.21382 264.49671 286.17928 341.46217 418.42763 473.71052 495.39309 550.67598 627.64145 682.92434 704.60691 759.88980 836.85526 892.13815 913.82072 969.10362 990.78618 1046.06908 1123.03454 1178.31743 2/1 My keyboard layout places this regular chain of 23 fifths on two 12-note manuals each with an accidental range of Eb-G#, indeed the range on the Halberstadt organ which since 1361 has become a byword for this arrangement. Corresponding notes on the two keyboards differ by the diesis (12 fifths up) of 55.283 cents, an interval often serving as an expressive narrow semitone. Since the tuning is entirely regular, it is quite possible to use conventional spellings for all notes, an approach which, as Aaron Johnson has pointed out in the context of 31-EDO, can be useful with some musical software permitting multiple sharps and flats but lacking support for "microtonal" symbols. Here such a notation is shown together with a pragmatic keyboard notation using an asterisk (*) to show a note on the upper keyboard raised by the 55-cent diesis: 187.531 341.462 682.924 892.138 1046.069 C#*/Bx Eb*/D#3 F#*/Ex G#*/F#x Bb*/A# _132.2|77.0_77.0|132.2_ _132.2|77.0_132.2|77.0_77.0|132.2_ C*/B# D*/Cx E*/Dx F*/E# G*/Fx A*/Gx B*/Ax C*/B# 55.283 264.497 473.145 550.676 759.890 969.104 1178.317 1255.283 209.214 209.214 76.965 209.214 209.214 209.214 76.965 -------------------------------------------------------------------------- 132.248 286.179 627.641 836.855 990.786 C# Eb F# G# Bb _132.2|77.0_77.0|132.2_ _132.2|77.0_132.2|77.0_77.0|132.2_ C D E F G A B C 0 209.214 418.428 495.393 704.607 913.821 1123.035 1200 209.214 209.214 76.965 209.214 209.214 209.214 76.965 The 24-note e-based tuning, like 17-note circles including 17-EDO and more accurate unequal temperaments such as Secor's 17-WT, features approximations of primes 2-3-7-11-13; additionally, there some intervals close to ratios of 23. These prime factors, occurring early and often, help shape the three main families of intervals: regular or "accentuated Pythagorean" (1-6 fifths up or down); Zalzalian or neutral (6-11 fifths); and septimal (12-16 fifths). Intervals formed from chains of 1-6 fifths up or down, together with the unison and 2:1 octave, make up the regular diatonic family. The "accentuated Pythagorean" qualities of this family nicely fit much of the polyphonic music of 13th-14th century Europe. Major and minor thirds at 418 and 286 cents (near 14:11 and 13:11 or 33:28), and sixths at 914 and 782 cents (near 22:13 or 56:33 and 11:7) very effectively resolve to stable concords. The limma or diatonic semitone at 76.965 cents, yet more compact and incisive than the Pythagorean step of 256:243 (90.224 cents), facilitates many of these efficient resolutions: it is almost exactly a just 23:22 (76.956 cents), and often represents nearby ratios such as 22:21 (80.537 cents) or 117:112 (75.612 cents). The augmented fourth at 628 cents and diminished fifth at 572 cents (near 23:16 or 56:39, and 32:23 or 39:28) are also members of this family, and may occur as vertical intervals or melodic outlines in European polyphony around 1200, for example. In many Near Eastern contexts, however, they take on an affinity with the intervals of our next family, and so may conveniently be considered as members of it also. Intervals with chains of 7-11 fifths make up the Zalzalian or neutral family, a basic element of Near Eastern modal and intonational systems. The name Zalzalian, of course, honors Mansur Zalzal, the 'oudist of 8th-century Baghdad famed for adding his _wusta_ or middle finger fret at a neutral third measured by al-Farabi at 27:22 (355 cents) and by Ibn Sina at 39:32 (342 cents). This family includes Zalzalian seconds at 132 and 154 cents (near 14:13 and 12:11 or 128:117); thirds at 342 and 363 cents (near 28:23 or 39:32, and 16:13 or 121:98); sixths at 837 and 859 cents (near 13:8 and 23:14); sevenths at 1046 and 1068 cents (near 11:6 and 13:7); and Zalzalian "tritonic" intervals of 551 and 649 cents, within a cent of the 11:8 "superfourth" and 16:11 "subfifth" as David Keenan has styled them. In many Near Eastern contexts, the 6-fifth intervals of the augmented fourth and diminished fifth at 628 and 572 cents may also be considered as Zalzalian, having a musical kinship with their 11-fifth relatives at 649 and 551 cents respectively, as will be discussed shortly. Intervals from chains of 12-16 fifths make up the septimal family of intervals approximating ratios from the factors of 2-3-7-9. The diesis of 55.283 cents (12 fifths up) serves as a very narrowly tempered variant of the septimal thirdtone at 28:27 (62.961 cents), playing a dramatic and effective role in the resolution of many other intervals in this family. The septimal major third at 440.11 cents is some five cents wide of 9:7, but the minor third at 264.50 cents is only about 2.37 cents narrow of 7:6, and the 969.10-cent minor seventh is a virtually just 7:4 (968.825 cents). Rounding out this family is the colorful narrow fourth at 473.71 cents, not quite three cents wide of a just 21:16. These intervals and their octave complements or inversions occur in various medieval Near Eastern tuning systems or modes, and additionally provide one interpretation for the enhanced cadential intonations of 14th-century European polyphony as described by Marchettus of Padua and practiced by a modern performer such as Christopher Page. In this scheme of three main families there are two areas of overlap. The first, as noted, involves intervals of 6 fifths up or down, which enjoy "dual citizenship" in both the regular diatonic and Zalzalian families, a point influencing the 17-step notation a la Beyhom we shall soon meet. The second, more of a theoretical nicety, involves intervals of 12 fifths up or down at 55 and 1145 cents, which typically represent the septimal ratios of 28:27 or 27:14 (heavily tempered at 7.68 cents narrow and wide respectively), but can also represent the Zalzalian ratios of 33:32 and 64:33 (53 and 1147 cents). A quick way of illustrating both points is to consider two versions of a pentachord called Buzurg (likely Persian) or Buzruk (Arabic). The first, from the "systematists" Safi al-Din and Qutb al-Din, nicely illustrates the use of the 628-cent tritone (6 fifths up) as a Zalzalian interval, here representing 56:39 (626 cents): 1/1 14/13 16/13 4/3 56/39 3/2 0 128 359 498 626 702 14:13 8:7 13:12 14:13 117:112 128 231 139 128 76 F# G* Bb B C* C# 0 132 363 495 628 705 132 231 132 132 77 Here I show both the original just intonation (JI) version and the e-based realization to acknowledge the compromises of a regular temperament: not only do we lose the pure fifths and fourths of the original, but also the subtle distinction between 14:13 (128 cents) and 13:12 (139 cents), both rendered as an identical 132 cents. In both versions, a fourth plus a small Zalzalian second results in an interval of 56:39, or a tempered 628 cents, which itself shares in the pervasive Zalzalian flavor. A variation on this medieval tuning of Buzurg or Buzruk proposed by the Lebanese scholar Nidaa Abou Mrad illustrates how the 55-cent diesis, although naturally placed in our septimal family as a highly tempered equivalent of 28:27, may also represent the Zalzalian "quartertone" at 33:32 (53.273 cents). 1/1 12/11 27/22 4/3 16/11 3/2 0 151 355 498 649 702 12:11 9:8 88:81 12:11 33:32 151 204 143 151 53 C* D E F* G G* 0 154 363 495 649 705 154 209 132 154 55 Here Mrad suggests how this pentachord might be realized by a musician using the steps of Zalzal's tuning according to al-Farabi (9:8, 12:11, and 88:81) -- plus a "minimal diesis" of 33:32! The tempered 55-cent diesis nicely approximates this interval. In the absence of intervals supporting a septimal interpretation (e.g. the 8:7 step in the previous example), the 55-cent diesis may take on this special role as a member of the Zalzalian family. A 17-note chain of fifths suffices to produce all the intervals of our three main families. From this perspective, while adding a few rarer intervals to the mix, a 24-note system may serve mainly to make the intervals of these families -- regular, Zalzalian, and septimal -- available at more locations. For example, a 17-note tuning would have our virtually pure 7:4 available at only two locations, while a 24-note tuning expands this to nine locations. Additionally, going to 24 notes brings explicitly into play an interval exerting its influence "behind the scene," as it were, in shaping the structure of a 17-note set. Let us now meet that interval. ---------------------------------------- 1.1. The 17-comma and a 17-step notation ---------------------------------------- In circulating 17-tone systems such as 17-EDO or George Secor's 17-WT, 17 fifths are equivalent to 10 octaves. In the e-based tuning, however, 17 fifths at 704.607 cents each fall short of 10 octaves by about 21.683 cents. This small interval, called the 17-comma, thus makes its appearance when the e-based system is expanded beyond 17 notes, where it results from a chain of 17 fifths down (e.g. E*/Dx-F). While this interval may serve as a direct melodic step or microtonal inflection, or as a vertical sonority (e.g. a "complex unison" in gamelan), its role in the 24-note e-based system is far more pervasive, in practice and theory. The 17-comma spells variety by defining the subtle difference between two intervals of the same general type, such as smaller and larger Zalzalian seconds at 132 and 154 cents (e.g. G#-A*/Gx and G#-Bb), or septimal and regular minor thirds at 264 and 286 cents (e.g. C-D*/Cx, C-Eb). We may note that these "kindred but distinct" pairs of intervals have "17-complementary" chains of fifths: thus 7 fifths up or 10 down for our Zalzalian seconds at 132/154 cents; and 14 fifths up or 3 down for septimal and regular minor thirds at 264/286 cents. This variety often means a special element of artistic choice when two forms of a given step or interval differing by a 17-comma are available from the same location, as often happens in our 24-note system. Consider, for example, how the two interpretations of a Buzurg/Buzruk pentachord given above could both be realized above a final or resting note of G# -- in comparison to the single version with its uniform interval sizes available at this or any other location in 17-EDO. The comparison provides an apt occasion for introducing some symbols for a 17-step notation. Safi al-Din/Qutb al-Din version G# A*/Gx C C# D*/Cx D#/Eb* 0 132 363 495 628 705 0 2s 5L 7 9s 10 132 231 132 132 77 2s 3+ 2s 2s 1 Mrad version G# Bb C C# Eb D#/Eb* 0 154 363 495 649 705 0 2L 5L 7 9L 10 154 209 132 154 55 2L 3 2s 2L 1- 17-EDO (either version) G# Bb/Gx C C# Cx/Eb D# 0 141 353 494 635 706 0 2 5 7 9 10 141 212 141 141 71 2 3 2 2 1 Using both the conventional and pragmatic keyboard spellings for the notes on the upper manual in the e-based versions may make the notation a bit more cluttered, but helps to show how two spellings located a 17-comma apart in this tuning are equivalent in 17-EDO. Thus in our two e-based versions, G#-Gx/G* (7 fifths up) yields a step of 132 cents, and G#-Bb (10 fifths down) a step of 154 cents; in 17-EDO, either spelling yields an identical size of 141 cents. The 17-EDO version is a good place to start in becoming acquainted with the 17-step notation which is a main focus of this article. In this equal temperament, there are only 16 interval sizes other than the unison and 2:1 octave, consisting of from 1 to 16 steps of 1/17 octave. Thus 1 step defines the 71-cent limma or diatonic semitone (also the diesis); 2 steps, the 141-cent Zalzalian second; and 3 steps, the major second or whole tone. We find 5 steps in a 353-cent Zalzalian third, 7 in a 494-cent perfect fourth, 9 in a 636-cent augmented fourth (or fifth-less-diesis), and 10 in a 706-cent perfect fifth. Our two e-based versions, in contrast, show two types of subtle variations each reflected in their 17-step notations. The first type involves the distinction between smaller and larger forms of Zalzalian intervals, here most notably the seconds at 132 or 154 cents. These may be considered as "small" and "large" counterparts of the 2-step interval in 17-EDO at 141 cents, or 2s and 2L for short. Our first e-based version uses the smaller 132-cent step (2s) only, representing either 14:13 or 13:12, while the second version uses the 154-cent step (2L) as the first and fourth melodic intervals of the pentachord. The penultimate note of the pentachord stands at an interval above the final G# of 628 cents in the first version (9s), and 649 cents (9L) in the second, representing 56:39 and 16:11 in the original JI versions; in any 17-EDO version, this 9-step interval is a uniform 635 cents, a fine representation of the intermediately sized 13:9. The second refinement of the 17-step notation uses a plus/minus modification (+/-) to distinguish a regular interval, shown simply by a number of steps without any sign, from a kindred interval of the septimal family at a 17-comma wider (+) or narrower (-). In the first version, the second melodic step of the pentachord is a septimal 8:7 tone shown as "3+"; in the second version, it is a regular 9:8 tone at a tempered 209 cents, shown simply as "3". Likewise, the last melodic interval in the first version of the pentachord is a regular limma or diatonic semitone at 77 cents (simply "1"); but in the second it is the narrower diesis at 55 cents, which may represent a septimal 28:27 or, as here, a Zalzalian 33:32 (marked as "1-"). Generally, as illustrated in the 17-step notations of these two e-based versions, regular intervals are shown as a simple number of steps without any marking; Zalzalian intervals are shown as small or large (s/L); and septimal intervals are shown as a comma narrower or wider (-/+) than their regular counterparts. For the purposes of this notation, intervals of 628 and 572 cents (6 fifths up or down) are treated as Zalzalian and grouped with the related forms derived from a chain of 11 fifths: thus 551/572 cents as 8s/8L; and 628/649 cents as 9s/9L. Intervals from 12 fifths up or down at 55 and 1145 cents, whether in a septimal context (28:27 and 27:14) or a Zalzalian one (33:32 and 64:33), take the septimal symbols: 1-/16+. Another helpful point is that regular major intervals (seconds, thirds, sixths, and sevenths) have wide (+) septimal forms, as does the perfect fifth; while regular minor intervals have narrow (-) septimal forms, as does the perfect fourth. This and other points may become clearer as we explore the 17-step notation in more detail. ---------------------------------------------------------- 1.2. 17-EDO "bifurcation": Interval catalogue and overview ---------------------------------------------------------- In 17-EDO, there are 16 categories of intervals other than the unison and octave, each with a single uniform size throughout the tuning: for example, the 5-step Zalzalian or neutral third at 352.942 cents. In a 17-note e-based tuning, each of these categories is in effect "bifurcated" into a pair of kindred but distinct intervals with sizes differing by the 17-comma of 21.683 cents. For example, we have smaller and larger Zalzalian thirds at 341.462 and 363.145 cents, shown as "5s" and "5L" in our 17-step notation. From this "bifurcation" of the 16 categories in 17-EDO, we arrive in a 17-note e-based system at 32 intervals other than the unison and octave, grouped into 16 kindred pairs. The following table shows each of the 16 category in 17-EDO in order of increasing size, and the corresponding pair of e-based intervals. In reading the table, two points may be helpful. First, all 17-EDO intervals are generated from a 17-note circle of fifths at 10/17 octave or 705.882 cents, and have sizes which are multiples of the 1-step minor second (also the diesis) at 1/17 octave or 70.588 cents. Secondly, given that 17-EDO consists of a 17-note circle of fifths, a given interval can be generated by moving along the circle in either an upward (sharpward) or downward (flatward) direction: our 5-step Zalzalian third at 353 cents, for example, from either 8 fifths down or 9 fifths up (-8/+9 fifths). More generally, moving by n fifths in one direction or (17-n) in the other produces the same interval. In contrast, in the e-based system, such "17-complementary" chains as 8 fifths down or 9 fifths up will produce a kindred pair of intervals differing by a 17-comma: here respectively our 5L/5s thirds at 363/341 cents. We might describe this 17-comma as the engine of bifurcation. ---------------------------------------------------------------------- Table of "Bifurcated" 17-EDO Counterparts in 704.607-cent tuning ---------------------------------------------------------------------- 17-EDO steps 5ths type cents 704.607: symbol 5ths cents ~JI ---------------------------------------------------------------------- 1- +12 55.28 32:31 1 -5/+12 min2 70.59 1 -5 76.97 23:22 ---------------------------------------------------------------------- 2s +7 132.25 14:13 2 +7/-10 Zal2 141.18 2L -10 153.93 12:11 ---------------------------------------------------------------------- 3 +2 209.21 44:39 3 +2/-15 Maj2 211.76 3+ -15 230.90 8:7 ---------------------------------------------------------------------- 4- +14 264.50 7:6 4 -3/+14 min3 282.35 4 -3 286.18 13:11 ---------------------------------------------------------------------- 5s +9 341.46 28:23 5 -8/+9 Zal3 352.94 5L -8 363.14 21:17 ---------------------------------------------------------------------- 6 +4 418.43 14:11 6 +4/-13 Maj3 423.53 6+ -13 440.11 9:7 ---------------------------------------------------------------------- 7- +16 473.71 21:16 7 -1/+16 4 494.12 7 -1 495.39 4:3 ---------------------------------------------------------------------- 8s +11 550.68 11:8 8 -6/+11 dim5 564.71 8L -6 572.36 32:23 ---------------------------------------------------------------------- 9s +6 627.64 23:16 9 +6/-11 Aug4 635.29 9L -11 649.32 16:11 ---------------------------------------------------------------------- 10 +1 704.61 3:2 10 +1/-16 5 705.88 10+ -16 726.29 32:21 ---------------------------------------------------------------------- 11- +13 759.89 14:9 11 -4/+13 min6 776.47 11 -4 781.57 11:7 ---------------------------------------------------------------------- 12s +8 836.86 13:8 12 +8/-9 Zal6 847.06 12L -9 858.54 23:14 ---------------------------------------------------------------------- 13 +3 913.82 22:13 13 +3/-14 Maj6 917.65 13+ -14 935.50 12:7 ---------------------------------------------------------------------- 14- +15 969.14 7:4 14 -2/+15 min7 988.24 14 -2 990.79 39:22 ---------------------------------------------------------------------- 15s +10 1046.07 11:6 15 -7/+10 Zal7 1058.82 15L -7 1067.75 13:7 ---------------------------------------------------------------------- 16 +5 1123.03 44:23 16 +5/-12 Maj7 1129.41 16+ -12 1144.72 31:16 ---------------------------------------------------------------------- In many ways 17-EDO is an admirably compact system: all intervals are available from any location in the system, something not true in a 17-note or even a 24-note e-based temperament. The circle neatly provides a full complement of regular diatonic intervals, including the augmented fourth and diminished fifth at 635 and 565 cents with their frequently Zalzalian character, plus a Zalzalian second, third, sixth, and seventh, making up an attractive set for neomedieval music in an "accentuated Pythagorean" style a la 13th-14th century Europe or a Near Eastern style. From a Near Eastern perspective, a main weakness of this solution is that we are restricted to one Zalzalian interval in each category, a limitation shared in common with 24-EDO when literally applied. However, as Beyhom and others have shown, we can take a conceptual map of either 17 or 24 steps per octave as a convenient basis for a flexible model of intonation where the steps and intervals are open to nuances and shadings. The e-based tuning is one possibly way to realize certain aspects of such an approach on a fixed-pitch instrument, with a 24-note tuning having the advantage of making available at more locations the intervals found within a 17-note chain of fifths and shown in the above table. Having catalogued these intervals, we can get an overview of the order in which they are generated by a chain of fifths in the following diagram showing the three main families we earlier surveyed: small/large (s/L) Zalzalian (Neutral) {------------------------}... +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 705 209 914 418 1123 628 132 837 341 1046 551 55 760 264 969 474 10 3 13 6 16 9s 2s 12s 5s 15s 8s 1- 11- 4- 14- 7- -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16 495 991 286 782 77 572 1068 363 859 154 649 1145 440 936 231 726 7 14 4 11 1 8L 15L 5L 12L 2L 9L 16+ 6+ 13+ 3+ 10+ {------------------}... {-------------------} Regular diatonic Septimal No modifier plus/minor (+/-) Intervals of +/- 6 fifths use Zalzalian s/L, and may also be regular Intervals of +/- 12 fifths use septimal +/-, and may also be Zalzalian Here the upper half of the table shows intervals generated from upward chains of fifths, while the lower half focuses on downward chains. Each half has an upper line showing the number of fifths up (+) or down (-); a middle line showing interval sizes in cents; and a lower line showing the 17-step notation for each interval. The scope of each of the three families is also shown, with dotted lines showing that a given family extends to an interval also belonging to the next family and using the 17-step modification symbols of that family: thus the "regular but also Zalzalian" intervals of +/- 6 fifths with their s/L modifications; and the "septimal but also Zalzalian" intervals of +/- 12 fifths with their +/- notations. ------------------------------------------------------- 2. More remote intervals: beyond 17 fifths, or 24 notes ------------------------------------------------------- In addition to the three main families we have surveyed, all found within a 17-note version of the e-based temperament, a 24-note tuning gives rise to some more remote intervals which here, as promised, will be briefly surveyed. A tuning set larger than 24 would generate yet more sizes of intervals, some of which might be of special interest to a musician such as Ozan Yarman. We have already encountered the 17-comma of 21.68 cents generated by a chain of 17 fifths down -- with 17 fifths up producing an octave less this comma, or 1178.32 cents. In a septimal context, these intervals might be interpreted as 64:63 and 63:32 at 27.26 and 1172.74 cents. While a special comma sign such as "k" might be desirable, we could show these intervals within our 17-step system of notation as respectively a "0+" to show a regular unison or "0-step" interval plus a comma, and "17-" to show a usual 17-step interval (the octave) less a comma. The remaining "remote" intervals generated in our 24-note system from chains of 18-23 fifths have the common property of being regular diatonic intervals (here including the augmented fourth or diminished fifth) altered by a 17-comma in a direction _opposite_ to that of the septimal family (12-16 fifths). In other words, the perfect fifth, major intervals, and also the augmented fourth are in these remote variations a comma narrower; while the perfect fourth, minor intervals, and diminished fifth are a comma wider. These intervals can thus nicely be expressed by the same +/- notation we use for the septimal family, but with opposite signs for septimal and remote variations on the same regular interval. Starting from an interval in many ways more pervasive than "remote," the 17-comma itself, we thus have the following intervals: ---------------------------------------------------------------------- fifths cents 17-step symbol type ---------------------------------------------------------------------- +17 1178.32 0+ octave less 17-comma -17 21.68 17- 17-comma ---------------------------------------------------------------------- +18 682.92 10- narrow fifth -18 517.08 7+ wide fourth ---------------------------------------------------------------------- +19 187.53 3- narrow major second -19 1012.47 14+ wide minor seventh ---------------------------------------------------------------------- +20 892.14 13- narrow major sixth -20 307.86 4+ wide minor third ---------------------------------------------------------------------- +21 396.75 6- wide major third -21 803.25 11+ wide minor sixth ----------------------------------------------------------------------- +22 1101.35 16- narrow major seventh (~17:9) -22 98.65 1+ wide minor second (~18:17) ----------------------------------------------------------------------- +23 605.96 9- narrow augmented fourth -23 594.04 8+ wide diminished fifth ----------------------------------------------------------------------- The notation for the intervals at +/- 23 fifths at 606 and 594 cents follows a convention that when a usually Zalzalian number of steps normally taking the s/L modifications is instead follows by +/- sign, this notation means the larger Zalzalian size plus a comma or the smaller size minus a comma. Thus 9- means 9s (a 628-cent augmented fourth) less a 22-cent comma; while 8+ means 8L (a 572-cent diminished fifth) plus a comma, yielding 606 and 594 cents respectively. Here I have taken note of possible just ratios only for the intervals of +/- 22 fifths, which yield virtually just versions 17:9 and 18:17, the latter a ratio famously present in some lute fretting schemes. The wide minor and narrow major thirds from -20 and +21 fifths, at 308 and 397 cents, are not too far from the related ratios of 81:68 (the wusta or middle finger placement known as _Fars_ or "Persian," at 302.86 cents); and 34:27 at 399.09 cents, or 4:3 less 18:17. The longest chains in this tuning, at +/- 23 fifths, produce intervals rather close to 17:12 and 24:17 at a virtually precise 603 and 597 cents. The remote thirds and sixths in this family could also be described as having a "meantone" quality, allowing one to produce, at a few locations, plausible approximations of Near Eastern interpretations of ratios of 5. This liberty is to me most pleasing, and most in keeping with the spirit of a 24-note tuning, when other and often more characteristic intervals are also present to give the intonation a special quality. For example, Amine Beyhom describes a striking Turkish intonation he measured for a Hijaz tetrachord -- about 130-265-90 cents or 0-130-395-485 cents -- which I found myself emulating as follows: Bb B C#* D* 0 132 397 474 132 264 77 The narrow fourth, close to 21:16 in this version as opposed to the somewhat more nuanced 485 cents in the Turkish version, and the middle step at 7:6, contribute to my fascination with this emulation. The 397-cent third might represent 5:4 (not unlikely in a Turkish tradition), 34:27, or simply "the 395 cents that the performer used." One application for the virtually just 18:17 step (-22 fifths) was inspired by a suggestion of the Syrian musician and theorist Tawfiq al-Sabbagh, as translated and explained by the Lebanese performer and teacher Ali Jihad Racy, that a Maqam called Sikah Arabi should be played in the manner of "Turkish Huzam," with steps measured in Pythagorean of 53-EDO commas at 6 9 5 12 5 9 7. What I arrived at was this, with steps shown in approximate 53-commas and 17-step notation: Bb* B* C#* Eb F* F#* G#* Bb* 0 132 341 440 705 837 1046 1200 0 6 15 19 31 37 46 53 0 2s 5s 6L 10 12s 15s 17 132 209 99 264 132 209 154 6 9 4 12 6 9 7 2s 3 1+ 4- 2s 3 2L Here I will offer no analysis except to say that it is interesting all the melodic steps except the 99-cent or 18:17 step at C#*-Eb come from the three main families; and that I only became aware of some of the other remote intervals (here from 20-23 fifths) present when I tried "SHOW INTERVALS" in Scala. Also, I have used some quite inaccurate rounded numbers of 53-commas: 99 cents is close to 4.5 commas, and likewise 440 cents to 19.5 commas. My purpose, perhaps a la Beyhom, is to convey the type of interval rather than offer an exact measurement. Finally, it may be noted that going beyond 24 notes would immediately introduce a new interval: a double diesis (24 fourths up) equal to 110.566 cents, closely approximating a just 16:15 (111.731 cents). A chain of 25 fifths down, subtracting this interval from a regular fourth, would result in a major third at 384.827 cents, about 1.49 cents narrow of 5:4. A chain of 26 fifths up would yield a minor third at 319.780 cents, about 4.14 cents wide. For a musician who seeks a fixed-pitch system of intonation with a versatile choice of intervals including accurate ratios of 5, how would a 46-note or larger e-based system rate? Judging by my dialogues with Ozan Yarman, and his eloquent writings, two vital aspects of his 79-MOS based on a subtly modified 159-EDO for the qanun and other instruments would not be present. First, the system would not fully circulate until one reached 109 notes, at which point it would almost identical to a complete set of 109-EDO. Secondly, while a 24-note e-based system is intended to focus mainly on the families of regular, Zalzalian, and septimal intervals generated within the first 17 fifths, plus the subtle 17-comma as the instrument of artful discretion, a performer seeking 5-based intervals as primary ingredients might find the decidedly non-meantone structure of the e-based tuning as a weighty impediment. Both considerations are central to the design of Ozan Yarman's 79-MOS: not only are many 5-based intervals present, but they are often available from meantone-like chains where no fifth is impure by more than 1/3 Holdrian comma (about 7.55 cents). Thus a 17-step notation for the 24-note e-based system not only follows up a brilliant suggestion by Beyhom, however modestly and inexpertly, but fits the primary purpose of the tuning as it has developed over the last 10 years: a "bifurcated" variation on 17-EDO realizing the admirable ethos of that system while ornamenting it with diversity and variety. Most respectfully, Margo Schulter mschulter@calweb.com Most appreciatively, Margo Schulter mschulter@calweb.com