---------------------------------------------------- Pegasus-12: An offshoot of Kraig Grady's Centaur Using prime 59 as aurally equivalent to prime 7 ---------------------------------------------------- Kraig Grady's 12-note Centaur tuning for primes 2-3-5-7 has led me to an offshoot or variation based on primes 2-3-59, with prime 59 producing approximations of 7-based ratios found in the original Centaur varying from these by a factor of 14337:14336 (0.121 cents). At the same time, as suggested by the great philosopher, physician, and music theorist Ibn Sina (980-1037), 59-based ratios of 64/59 and 72/59 reasonably approximate 13/12 and 39/32, for example, the difference being a factor of 768:767 (2.256 cents). The Pegasus-12 tuning, inspired by Centaur and also by Ibn Sina's mention of 72/59, proposed as one possible neutral third fretting for the `oud by Safi al-Din al-Urmawi (1216?-1294), draws on these just intonation "puns" involving prime 59 so as to provide some aurally equivalent versions of septimal (primes 2-3-7) tetrachords and modes like those of Archytas and Ptolemy in the Greek tradition, as well as tetrachords and modes with middle or neutral intervals in the Near Eastern tradition of the Islamic Renaissance (8th-15th centuries). For much fascinating information on Centaur and some of its possible extensions and close relatives, please visit Kraig Grady's web page on this system at . In the Scala scale archive, Centaur is included as grady7.scl. Here is a Scala file for Pegasus 12: ! bamm24b-pegasus12_D.scl ! Offshoot of Kraig Grady's Centaur: Rast/Penchgah plus Archytas-like modes on 1/1 12 ! 531/512 9/8 4779/4096 59/48 4/3 177/128 3/2 1593/1024 27/16 14337/8192 59/32 2/1 ---------------------------------------------- 1. A lattice diagram: one possible perspective ---------------------------------------------- As shown in the lattice diagram below, naming the notes according to their keyboard locations in the Bamm24b tuning with two 12-note Pythagorean chains (Eb-G#) at 531/512 apart (63.082 cents), and with the note located at D on the lower keyboard taken as the 1/1, Pegasus-12 actually consists of only two chains of fifths. There is a five-note or pentatonic including the 1/1 (G-B, or 4/3-27/16), plus a seven-note or heptatonic chain identical to a Pythagorean diatonic (F*-B*, or 59/48-14437/8192). From one practical viewpoint, however, we may find it convenient to regard the lower four notes of this second heptatonic chain as forming septimal ratios with the 1/1 of the kind so prominently featured in Kraig Grady's Centaur (here ~28/27-~7/4). The upper three notes of this second chain form neutral ratios with the 1/1 of 59/48, 59/32, and 177/128, which have their own fine nuances of shading but could be compared, for example, to 16/13, 24/13, and 18/13 at 768:768 larger. The lattice reflects this perspective on Pegasus-12: (this chain continues downward below) ------>---- o | 765.0 267.0 470.9 | A* E* B* | 1593/1024 4779/4096 14337/8192 | ~14/9 ------ 7/6 ------ 7/4 | / / / | / / / v /__________/__________/__________________ | 4/3 1/1 3/2 9/8 27/16 | G D A E B | 498.0 0 702.0 203.9 905.9 | | | | | | | | | | | | | | | | +---------+---------+--------+ <---- o 59/48 59/32 177/128 531/512 ~896/729 ~448/243 ~112/81 ~28/27 F* C* G* D* 357.2 1059.2 561.1 63.1 What Centaur and Pegasus notably share in common is a 1-3-7-9 hexany (or in Pegasus an aurally equivalent structure differing by the factor of 14337:14336), shown in the upper portion of the lattice diagram by slanted lines showing minor thirds at ~7:6 (4779/4096) which form two adjacent slanted quadrangles representing sonorities of ~12:14:18:21. In the lower portion of the diagram, there are three dashed vertical lines showing 59:48 neutral thirds. From the standpoint of our chosen 1/1, the system thus features minor septimal intervals and larger neutral steps and intervals. If we shift our focus to a note such as D* (531/512 or ~28/27) on the upper chain, however, then we encounter many major septimal intervals or virtual equivalents (e.g. D*-G, ~9/7; D*-D, ~27/14) and smaller neutral steps or intervals (e.g. D*-E, 64/59 or ~13/12; D*-B, 96/59 or ~13/8). Often a given mode may focus on either septimal or neutral intervals. Thus this mode from the 1/1 of D could be described as one variation on the Diatonic of Archytas, or Tonic Diatonic of Ptolemy, with steps of 28:27, 9:8, and 8:7, using disjunct tetrachords. Tonic Diatonic 9:8 Tonic Diatonic |-----------------|......|----------------| D D* E* G A A* B* D 1/1 ~28/27 ~7/6 4/3 3/2 ~14/9 ~7/4 2/1 28:27 9:8 8:7 9:8 28:27 9:8 8:7 This mode, with its larger neutral steps and intervals from the same 1/1 of D, is a basic version of a possible shading of a modern disjunct Arab Rast: Rast 9:8 Rast |-----------------|......|------------------| D E F* G A B C* D 1/1 9/8 59/48 4/3 3/2 27/16 59/32 2/1 9:8 59:54 64:59 9:8 9:8 59:54 64:59 ----------------------------------------------------- 2. Pegasus as a Centaur offshoot: interval categories ----------------------------------------------------- A quick and general overview might note that Centaur and the Pegasus offshoot share in common a set of Pythagorean and septimal intervals generated from primes 2-3-7 -- or, in Pegasus, "virtual septimal equivalents" from ratios of 59. Additionally, Centaur has a variety of ratios of 2-3-5 or 2-3-5-7, where in Pegasus a corresponding number of scale steps often produce neutral ratios of 59. While both 12-note systems, according to Scala, are strictly proper as well as constant structures, this contrast between 5-based and neutral flavors of intervals correlates some differences between the two tunings in the organization of their interval types and sizes. In a very effective graphical overview of Centaur's 50 interval sizes , Kraig Grady presents two concepts helpful for such a comparison: "span" and "gap." If we group the intervals of a tuning into categories based on the number of scale steps subtended, for example 3-step minor thirds and 4-step major thirds in Centaur, then the "span" of a given category is equal to the range of sizes within it. The "gap" between two categories is equal to the difference or "jump" between the largest 3-step and smallest 4-step interval, for example. In Centaur, while there is considerable variation as we move through the groups of intervals formed by from 1 to 11 scale steps found between the 1:1 unison and 2:1 octave, we find that these spans and gaps are often comparable, with spans for categories other than 6-step tritones at 48.8-56.5 cents, and gaps at 35.7-63.0 cents. (Tritones or 6-step intervals have a somewhat wider overall span of 78.0 cents.) A general impression might be one of balance between spans and gaps. In Pegasus, this equation is considerably altered by the presence of smaller and larger neutral categories which often result in much smaller and more subtle gaps between adjacent categories. The following table, modelled after Kraig's also less elegant, shows some of these patterns: -------------------------------------------------------------------- Pegasus-12 interval categories and sizes -------------------------------------------------------------------- 1-step intervals 11-step intervals (minor and small neutral 2nds) (large neutral and major 7ths) 531:512 63.1 1024:531 1136.1 256:243 90.2 span: 77.7 243:128 1109.8 64:59 140.8 59:32 1059.2 ____________________________________________________________________ gap: 12.5 -------------------------------------------------------------------- 2-step intervals 10-step intervals (large neutral and major 2nds) (minor and small neutral 7ths) 59:54 153.3 108:59 1046.7 9:8 203.9 span: 77.7 16:9 996.1 16384:14337 231.1 14437:8192 968.9 ___________________________________________________________________ gap: 35.9 ------------------------------------------------------------------- 3-step intervals 9-step intervals (minor and small neutral 3rds) (large neutral and major 6ths) 4779:4096 267.0 8192:4772 933.0 32:27 294.1 span: 77.7 27:16 905.9 72:59 344.7 59:36 855.3 ____________________________________________________________________ gap: 12.5 -------------------------------------------------------------------- 4-step intervals 8-step intervals (large neutral and major 3rds) (minor and small neutral 6ths) 59:48 357.2 96:59 842.8 81:64 407.8 span: 77.7 128:81 792.2 2048:1593 435.0 1593:1024 765.0 ___________________________________________________________________ gap: 35.9 ------------------------------------------------------------------- 5-step intervals 7-step intervals (fourths and superfourths) (subfifths and fifths) 43011:32768 470.9 65536:32768 729.1 4:3 498.0 span: 77.7 3:2 702.0 81:59 548.6 118:81 651.4 ____________________________________________________________________ gap: 12.5 -------------------------------------------------------------------- 6-step intervals (tritonic) 177:128 561.1 256:177 638.9 1024:729 588.3 span: 77.7 729:512 611.7 ____________________________________________________________________ -------------------------------------------------------------------- A curious characteristic of Pegasus-12 is that every category has a span of 77.7 cents. This range of 32768:31329 (77.747 cents) is equal to the difference, for example, between 531:512 (63.1 cents) and 64:49 (140.8 cents) as complements adding up to a 9:8 tone. The smaller gaps at 12.5 cents are equal to the difference, for example, between smaller and larger neutral second steps at 64:59 (140.8 cents) and 59:54 (153.3 cents), complements adding up to a 32:27 minor third. This difference of 3481:3456 (12.478 cents) makes for a gentle and subtle transition. A clearer transition occurs between categories such as 2-step and 3-step intervals, for example, with 16384:14337 (231.1 cents) as the largest interval of the first category, and 4779:4096 (267.0 cents) as the smallest of the second, these sizes being almost identical to the more familiar 8:7 (231.2 cents) and 7:6 (266.7 cents) of Centaur. With these simpler ratios, the "gap" would be 49:48 or 35.697 cents; since the Pegasus ratios involve a very slightly smaller ~8:7 tone and larger ~7:6 minor third, the difference is 68516523:67108864 (35.938 cents), a minutely larger gap. To note these ~49:48 gaps in either Centaur-12 or Pegasus-12 is another way of saying that in these areas of transition neither tuning set happens to include what might be termed "interseptimal" intervals, for example hemifourths significantly larger than 8:7 but smaller than 7:6. However, extensions of either system might well add such intervals. One rather arbitrary although sometimes possibly useful distinction made in the above table is between "superfourths" or "subfifths" formed by adding to a fourth around 4:3, or subtracting from a fifth around 3:2 an intervals somewhat larger than a septimal comma (64:63, 27.264 cents) but smaller than a usual "semitone" of around 28:27 (62.961 cents) or larger; and "tritonic" intervals involving the addition or subtraction of such semitones. By this distinction, for example, the 5-step interval of 81:59 (548.6 cents), equal to a 4:3 fourth plus a diesis at 243:236 (50.6 cents) could be considered a "superfourth" (a term for which I thank Dave Keenan, who takes 11:8 as an example), while the 6-step interval of 177:128 (561.1 cents), a 4:3 fourth plus a semitone or thirdtone at 531:512 (63.1 cents) could be considered a more usual "tritonic" interval. More generally, we should note that Centaur offers a greater variety of interval sizes, 50 as opposed to 34 in Pegasus; and that Centaur has four sizes of adjacent steps, all superparticular (28:27, 21:20, 16;15, and 15:14), while Pegasus as three sizes, none of them superparticular (531:512, 256:243, 64:59). While it would generally be informative to say that ratios of 2-3-5 or 2-3-5-7 in Centaur often correspond to neutral ratios of 59 in Pegasus, this does not fully convey the greater range of interval sizes in the former tuning. For example, Centaur has a 4-step interval of 80:63 (413.6 cents) which would nicely add to the shadings of major thirds in Pegasus in the Pythagorean-to-septimal range (81:64-9:7 or so). Another way of expressing this last point is to say that while Centaur sometimes has large "gaps" between categories (e.g. from a 15:14 semitone as the largest 1-step interval to a 10:9 tone as the smallest 2-step interval, a difference of 28:27 or 63.0 cents), Pegasus consistently has its largest "jumps" between interval sizes occurring _within_ categories. These jumps of 243:236 or 50.6 cents occur, for example, between the usual Pythagorean limma at 256:243 or 90.2 cents and the small neutral second at 64:59 or 140.6 cents -- both 1-step intervals. Further, Pegasus consistently has its smallest "jumps" or differences of 12.5 cents between interval sizes located as gaps _between_ categories. Within any given category, the smallest jump is found only among the 6-step or tritonic intervals, between the Pythagorean diminished fifth at 1024:729 (588.3 cents) and augmented fourth at 729:512 (611.7 cents), a difference equal to the Pythagorean comma at 531441:524288 at 23.460 cents. Otherwise, all differences within a category are equal either to 131072:129033 (27.143 cents), virtually equivalent to the familiar 64:63 septimal comma (27.264 cents); or to 243:236 at 50.6 cents as mentioned in the previous paragraph. In contrast, intracategory differences in Centaur range from 225:224 (7.712 cents, e.g. 16:15 at 111.7 cents and 15:14 at 119.4 cents) to the septimal or Archytan comma at 64:63. These often subtle gradations within a given category show one very sophisticated side of just intonation, especially in so compact a system. A lesson of these contrasts is that simply referring to a given tuning as "strictly proper" or a "constant structure" leaves opens many details and nuances of intervallic patterns and spacing along the continuum. ---------------------------------------------------------------- 3. Pegasus and the 729:728 connection between ratios of 7 and 13 ---------------------------------------------------------------- While from one perspective, as reflected in the lattice diagram at the beginning of this article, Pegasus-12 can be seen as featuring a mixture of Pythagorean, quasi-septimal, and neutral intervals, the latter two categories are in fact realized alike by steps forming a single heptatonic chain of fifths. This structure raises an interesting question: if Pegasus includes 59-based ratios virtually equivalent to more familiar ratios of 7, couldn't an aurally equivalent version of Pegasus be realized simply by using pure ratios of 7? The quick answer is yes! For purposes of tuning by ear, in fact, Pegasus could be defined in precisely such terms. Thus, in theory -- considerations of the safety of strings could modify the order of tuning in practice, as explained below! -- one might first tune the pentatonic chain including the 1/1: 4/3 1/1 3/2 9/8 27/16 G D A E B 498.0 0 702.0 203.9 905.9 The next step might be to tune either a 7/6 or a 7/4 step, for example, thus setting the first note of the second chain of fifths, which from there would be completed in usual Pythagorean fashion: 896/729 448/243 112/81 28/27 14/9 7/6 7/4 F* C* G* D* A* E* B* 357,1 1059.1 561.0 63.0 764.9 266.9 968.8 | | 4/3 1/1 3/2 9/8 27/16 G D A E B 498.0 0 702.0 203.9 905.9 In practice, a safer procedure for stringed instruments such as the harpsichord might be to tune the _upper_ heptatonic chain first at pitch level known to involve safe levels of tension, and then, for example, taking E* as a reference, to tune D at a 7:6 minor third _below_ this note, or G at a 7:4 minor seventh below it! Here the complex septimal ratios of 896/729, 448/243, and 112/81 are virtually identical to 59/48, 59/32, and 177/128 in Pegasus, which differ by only 14337:14336. These 59-based ratios are founded in the convenient arithmetic division on a monochord or `oud of a 32:27 minor third into neutral seconds of 64:59:54 (140.8-153.3 cents). However, as Ibn Sina's original suggestion in the 11th century that such a division might reasonably approximate his favored division of this minor third into steps of 13:12 and 128:117 (138.6-155.6 cents) may move us to consider, in either a 59-based or pure septimal version of Pegasus, these intervals can be taken as "justly tempered" variations on ratios of 13. Our lattice in the 7-based version could thus be read: (~16/13) (~24/13) (~18/13) (~27/26) 896/729 448/243 112/81 28/27 14/9 7/6 7/4 F* C* G* D* A* E* B* 357,1 1059.1 561.0 63.0 764.9 266.9 968.8 | | 4/3 1/1 3/2 9/8 27/16 G D A E B 498.0 0 702.0 203.9 905.9 With 7-based ratios, the difference between these complex septimal forms and their pure 13-based counterparts is equal to that between thirdtone steps of 28/27 and 27/26: 729:728 (2.376 cents). Between 59-based and pure 13-based counterparts such as 72/59 and 39/32, it is a very slightly smaller 768:767 (2.256 cents). And the yet smaller difference between these differences is 14337:14336 (0.121 cents). A scenario to illustrate the 729:728 has us start with a 9/7 septimal major third at 435.1 cents. Now we form a characteristic type of septimal augmented fourth by adding a 9:8 tone, arriving at 81/56 or 639.0 cents (an interval present in Centaur and virtually identical in the 59-based version of Pegasus at 256:177 or 638.9 cents). Either the 7-based or 59-based version of this interval nicely approximates a pure 13/9 (636.618 cents). Now if we move up by a 3:2 fifth and subtract a 2:1 octave, we have arrived at 243:224 or 140.9 cents, a step famous for its inclusion in the Chromatic of Archytas, where together with a 28/27 step it forms a 9/8 tone (1/1-28/27-9/8-4/3 or 0-63.0-203.9-498.0 cents). This interval is larger by only 729:728 than a 13:12 neutral second, to which it may serve as a good approximation, as may 64:59. Now if we move up by another 3:2 fifth, we arrive at 729:448 or 842.9 cents, wider by only a 729:728 than a just 13.8 at 840.5 cents. In a system like Centaur, while the chain of septimal intervals goes far enough to produce ratios of 112:81 (~18:13) and 81:56 (~13:9), it does not reach quite far enough to bring about neutral seconds, thirds, sixths, or sevenths. Rather, this 12-note system complements its septimal (i.e. 2-3-7) intervals with others based on factors of 2-3-5 or 2-3-5-7. In Pegasus, we might say that the septimal or quasi-septimal chain is sufficiently lengthened to bring about these neutral intervals, so that all intervals are based on ratios of 2-3-59 in the standard version, or 2-3-7 in the modified version based on tuning a pure interval of 7/6 or 7/4 to locate the first note of the second or heptatonic chain of fifths. While Ibn Sina recognized the kinship between ratios of 13 and 59 (differing by 768:767), and the 729:728 relationship between ratios of 7 and 13 has also been recognized, I am not sure how widely the almost perfect coincidence of 7 and 59 and their 14337:14336 relationship has been realized, a recognition celebrated by the Pegasus tuning. Most appreciatively, Margo Schulter mschulter@calweb.com