Hello, all. With the recent discussion about keyboard and related tunings as a starting point, I would like to share some of my experiences with what I term "regularized" 24-note keyboard tunings, and especially with a just intonation (JI) system of this type focusing primarily on Pythagorean, septimal (prime 2-3-7), and neutral intervals. In contrast to Western European compositional styles of the later 15th-19th centuries, where temperament on keyboard or other fixed-pitch instruments is generally a practical necessity as shown by authors such as Mark Lindley and Claudio Di Veroli, such a JI system addresses a medieval European or Near Eastern setting, and also some more recent Near Eastern styles. Additionally, the 24-note keyboard tuning I describe gives reasonably "user-friendly" access to a fascinating JI system described by Erv Wilson: a "Rast-Bayyati matrix" built from neutral thirds and fifths with a size of 7, 10, or 17 notes. -------------------------------------------- 1. What is a "regularized" 24-note keyboard? -------------------------------------------- The basic idea of a regularized 24-note keyboard is simply an array of two 12-note manuals (e.g. standard MIDI controllers) sharing identical patterns of steps and intervals at some chosen distance apart. This concept may be clearer if we take it step by step. Let us start with a tuning likely common in 14th-century Europe: a 12-note Pythagorean chain of Eb-G#. Such a tuning would fit much of the vocal repertory of the period, as well as the instrumental music of the Robertsbridge Codex noted by Mark Lindley as admirably fitting this choice of accidentals. Using a "regularized" 24-note concept, we would first place one 12-note manual in this Eb-G# Pythagorean tuning, thus arriving at an arrangement likely identical to that of many 14th-century European keyboards, including that of the organ at Halberstadt (as described by Michael Praetorius in 1619). Further following the concept, we then place our second 12-note manual in an identical Eb-G# tuning -- but at some chosen distance from the first manual. For the purposes of the system I here address, that distance is in theory a diesis or thirdtone step of 531:512, or 63.082 cents. For many practical purposes, this may be considered virtually identical to the thirdtone of Archytas at 28:27 or 62.961 cents. The very small difference of 14337:14336 or 0.121 cents is of more theoretical interest than aural consequence. Interestingly, such a 24-note system could be tuned by ear, although I use a TX-802 synthesizer with two 12-note MIDI controllers where one sets a digital tuning table. For example, one could first tune the upper manual using pure fifths or fourths: Eb* - Bb* - F* - C* - G* - D* - A* - E* - B* - F#* - C#* - G#* Note that the asterisk (*) shows a note on the upper manual or chain of fifths. Next, to find a starting point for our lower manual or chain of fifths, we could, for example, tune C at a pure 7:6 minor third below D*, or a pure 7:4 minor seventh below A*: Eb* - Bb* - F* - C* - G* - D* - A* - E* - B* - F#* - C#* - G#* 7:6 7:4 | | C--------------- Having set this note, we then tune the lower manual or chain of fifths in an Eb-G# arrangement identical to that of the upper manual: Eb* - Bb* - F* - C* - G* - D* - A* - E* - B* - F#* - C#* - G#* Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# From one point of view, this regularized 24-note tuning is essentially a JI lattice of primes 2-3-7, with each quadrilateral in the following diagram representing a sonority of 12:14:18:21, with a 7:6 minor third, a 3:2 fifth, and a 7:4 minor seventh above the note at the lower lefthand corner, e.g. Eb-F*-Bb-C*. The diagonal lines represent 7:6 minor thirds (266.871 cents), or aurally equivalent intervals of 4779:4096 (266.992 cents): Eb* - Bb* - F* - C* - G* - D* - A* - E* - B* - F#*- C#*- G#* / / / / / / / / / / Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# Here is a link to a Scala file of this 24-note tuning, and to a listing of the steps in just ratios and cents: ------------------------------------------------------ 2. "Regularization" in practice: Families of intervals ------------------------------------------------------ In discussing tuning systems in general, and JI systems in particular, it is important to consider how the different families of intervals will be used in a given style. Here three families of intervals are of special interest. Regular Pythagorean intervals of a kind common in medieval European and Near Eastern styles are found on either keyboard. While various writers including Lindley have noted how these intervals play a vital role in 13th-14th century European polyphony, they are also a basic ingredient of medieval and modern Near Eastern music. The second family consists of neutral intervals, and explains the choice of a diesis or thirdtone between the two manuals at 531:512 or 63.1 cents, which produces neutral seconds at a smaller 64:59 (140.828 cents) and a larger 59:54 (153.307 cents) as would obtain in tunings described by Ibn Sina in the earlier 11th century and Safi al-Din al-Urmawi in the 13th century. These tunings result from the arithmetic division of a 32:27 minor third (294.135 cents) into neutral seconds of 64:59:54. For Ibn Sina, this is an optional approximation of his favored division based on steps of 13:12 (138.573 cents) and 128:117 (155.562 cents); for Safi al-Din, it is a common fretting for the `oud where neutral intervals are preferred. For example, here is a location for Ibn Sina's mode of Nawa as interpreted by Cris Forster in his _Musical Mathematics: On the Art and Science of Acoustic Instruments_ (p. 685), or the very similar modern Iranian Dastgah-e Nava, a parallel over the time spane of about a millennium which Forster rightly emphasizes: 702.0 996.1 0 203.9 294.1 498.0 702.0 792.2 996.1 1200.0 3/4 8/9 1/1 9/8 32/27 4/3 3/2 128/81 16/9 2/1 A* C* D* E* F* G* A* Bb* C D B 48/59 842.8 And here is a realization of what could be the "textbook" form of either the Arab Maqam Bayyati or the Daramad (introductory theme) of the Iranian Dastgah-e Shur: 702.0 996.1 0 294.1 498.0 702.0 792.2 996.1 1200.0 3/4 8/9 1/1 32/27 4/3 3/2 128/81 16/9 2/1 A* C* D* F* G* A* Bb* C D B E 48/59 64/59 842.8 140.8 One feature common to these modalities is the contrast between a neutral third step below the final or note of repose (the 1/1), and a minor sixth step above it, the latter sometimes inflected to a neutral sixth. In these modalities, a 32:27 minor third is characteristically divided into a smaller 64:59 step preceding a larger 59:54 step. The converse order prevails in an Arab Rast of a shading like that documented by al-Farabi (c. 870-950) and still practiced today in many parts of the Arab world, with many fine gradations of intonation: 357.2 855.3 59/48 59/36 C* F* A B D E G A 1/1 9/8 4/3 3/2 16/9 2/1 0 203.9 498.0 702.0 996.1 1200.0 In addition to regular Pythagorean and neutral intervals, some maqamat (plural or Arab _maqam_ or modality) or dastgah-ha (plural of Persian _dastgah_ or modal family) may also bring septimal intervals into play, as with this version of an Arab Maqam Hijazkar: 357.2 1059.2 59/48 59/32 G* D* E F A B C E 1/1 256/243 4/3 3/2 128/81 2/1 0 90.2 498.0 702.0 792.2 1200.0 In this flavor of Hijazkar, which the Lebanese theorist Amine Beyhom associates with the name Zirkula, we have tetrachords E-F-G*-A and B-C-D*-E consisting of a regular diatonic semitone at 256:243, a small minor third at 7:6 (or, in theory, the minutely larger 4779:4096 at 0.121 cents wider), and a smaller neutral second at 64:59 (90.2-267.0-140.8 cents). Septinal intervals also play an important role in classic Greek tunings. For example, here is a common version of the Diatonic of Archytas, with steps of 28:27, 8:7, and 9:8 (63.0-231.2-203.9 cents), known by Ptolemy as the Tonic Diatonic. A cautionary "~" symbol indicates that the virtually just septimal ratios are, at least in theory, actually wider by 14337:14336 or 0.121 cents. 63.1 765.0 ~28/27 ~14/9 E* B* E G A B D E 1/1 32/27 4/3 3/2 16/9 2/1 0 294.1 498.0 702.0 996.1 1200.0 An important caution about ancient Greek music is that while we have some information about tetrachords and modes and certain instrumental tunings, we have only fragmentary evidence on how these tunings may have been applied. However, the tunings are beautiful and invite exploration in many directions. ---------------------------------------------------- 3. Septimal intervals in a medieval European setting ---------------------------------------------------- While the performance of 14th-century European polyphony will generally emphasize the Pythagorean intervals available on a single Eb-G# keyboard, the "accentuated Pythagorean" ethos of Marchettus de Padua (1318) invites the use of certain septimal steps and intervals also as one possible realization of his expressive intonation recommended to singers for directed progressions involving sharps. Here it is important to emphasize that we have no evidence that 14th-century _keyboards_ were tuned in a manner that would permit the emulation of Marchettus's inflections described and recommended for _vocal_ music. Rather, we have a 21st-century keyboard emulation of a possible 14th-century shade of vocal intonation. For Marchettus, as for many 14th-century theorists, the most important directed progressions often involve the expansion of a major third to a fifth, a major sixth to an octave, or a major tenth to a twelfh by stepwise contrary motion. If natural degrees or flats are involved, and not sharps, then a usual Pythagorean intaontion seems indicated, available on either 12-note keyboard: G A E F D E B C Bb A or G F These progressions have regular Pythagorean major thirds at 81:64 and major sixths at 27:16 (407.820 and 905.865 cents) expanding to pure 3:2 fifths and 2:1 octaves, arriving at a complete 2:3:4 sonority (A-E-A or F-C-F). Voices move by regular melodic steps of either a 9:8 tone or a 256:243 semitone. On a typical 14th-century keyboard, progressions involving sharps would be similarly intoned, for example: F# G C# D C# D G# A A G or E D On our 24-note regularized keyboard, however, we have the additional option of "accentuated" progressions of a kind described by Marchettus with wider major thirds and sixths, and narrower semitone (or diesis) steps, although the exact sizes of these intervals are subject to various interpretations. Christopher Page indeed favors a vocal performance free of constraining fixed-pitch instruments, where singers can tune each passage as seems best. In this keyboard realization, we could play the last two progressions in an accentuated style as follows: G G* D D* D D* A A* A* G* E* D* In these examples, the unstable major third and sixth are at a just or virtually just 9:7 and 12:7 (435.084 and 933.129 cents if pure, and 0.121 cents narrower in our theoretical system based on prime 59); the lower voice descends by the usual 9:8 while the upper voices each ascend by a diesis or thirdtone of 28:27 or a minutely larger 531:512. Whether 14th-century singers pursuing this kind of intonational style -- as Page suggests they often may have in performing the music of a French composer such as Guillaume de Machaut, as well as the early 14th-century Italian styles of Marchettus -- would find a 7:9:12 intonation of the unstable sonorities in these cadences as a point of attraction remains an open question, but both George Secor and I have found it very pleasing. It seems safest to say that it is one possible historical interpretation. ------------------------------------------------ 3. Just Intonation and Pythagorean compatibility ------------------------------------------------ A common theme in these ancient Greek, medieval European, and medieval and later Near Eastern contexts is the free use and acceptability of regular Pythagorean intervals such as thirds and sizths, as well as of intervals from the neutral and septimal families. These latter two families, which on our 24-note regularized keyboard involve mixing notes from both chains of fifths, thus supplement rather than replace their usual Pythagorean counterparts. Where Pythagorean thirds are freely acceptable, many just tuning structures become practical and attractive which would be quite alien to a Renaissance-Romantic European setting. One example is Erv Wilson's "Rast-Bayyati matrix," mentioned above, with the link repeated for convenience: Wilson's matrix presented in this paper is based on al-farabi's neutral thirds at 27:22 (354.547 cents) and 11:9 (347.408 cents), with our regularized 24-note tuning giving slightly more contrasting sizes at 59:48 (357.217 cents) and 72:59 (344.738 cents). The matrix, built from a chain alternating these two sizes of neutral thirds, with every two generators forming a pure fifth, is easy to find on our 24-note keyboard, as here shown for a 10-note set: (#1) (#3) (#5) (#7) (#9) 357.2 1059.2 561.1 63.1 765.8 59/48 59/32 177/128 531/512 1593/1024 F* C* G* D* A* D A E B F# 1/1 3/2 9/8 27/16 81/64 0.0 702.0 203.9 905.9 407.8 (#0) (#2) (#4) (#6) (#8) Such a set yields a rich assortment of familiar and not-so-familiar modes of a Near Eastern variety. Both Erv Wilson's Rast-Bayyati matrix, and our 24-note regularized JI tuning with two 12-note Pythagorean chains at a 531:512 or the almost identical 28:27 apart, illustrate how just intonation systems can feature a variety of complex as well as simple intervals. More specifically, the 24-note regularized tuning I have presented tends to invite either primarily melodic idioms such as those of traditional Near Eastern practice, or styles of polyphony contrasting stable 3:2 fifths and 4:3 fourths with various types of complex and partially concordant intervals such as thirds and sixths: Pythagorean, neutral, or septimal. Such styles are quite different from those of late 15th-19th century European music, where the need for regular thirds and sixths reasonably close to 5-based ratios strongly favors temperament of one variety or another on fixed-pitch instruments. From an ergonomic standpoint, the regularized 24-note keyboard may be seen as an approach, albeit less elegant than that of a generalized keyboard, for making intonational complexity more accessible and manageable in real time. The fact that each 12-note manual itself offers a continuous chain of 11 fifths, here the Eb-G# tuning likely popular in 14th-century Europe, helps to provide a secure basis for approaching the wealth of neutral and septimal intervals to be found by mixing notes from the two chains. My purpose here is not to propose one definitive solution, but to suggest the wide range of world musics, styles, and tuning systems to be considered in these discussions. Most appreciatively, Margo Schulter mschulter@calweb.com