Note to Readers: This letter is written to Ozan Yarman, the outstanding Turkish musician and theorist whose doctoral dissertation has recently become available on the World Wide Web: _79-Tone Tuning and Theory for Turkish Maqam Music As A Solution To The Nonconformance Between Current Model and Practice, Istanbul Technical University, Institute of Social Sciences, Department of Musicology (June 2008). The 79-tone tuning presented in the dissertation is a system which, in simplified terms, is based on 159-EDO, a tuning in which each of the 53 commas of 53-EDO is subdivided into three parts. In the 79-tone system, an octave is built from 78 steps equal to twice the size of a 159-EDO step, or 2/3 comma -- plus one step equal to a full comma, the 46th of the tuning, which has as its upper note a 3:2 fifth above the beginning of the series. Actually the system is a bit more complicated than this, since the 2/3-comma step or yarman, as it is appropriately called, varies minutely in size in order to obtain some pure 3:2 fifths and 4:3 fourths, in contrast to the tiny amount of tempering these intervals receive in 53-EDO. These niceties, however, do not significantly divert the system from its conceptual simplicity as a "79-MOS," that is, a "Moment of Symmetry" in which adjacent tuning steps have only two sizes, here basically either 2/3-comma (the yarman), or a full comma (step 46). In this scheme 33 yarmans form a pure 4:3 fourth, while, as noted 45 of these usual yarmans plus a full comma or approximate 53-EDO step for a pure 3:2 fifth. This letter largely focuses on two medieval traditions involving 17-note Pythagorean systems (also MOS tunings, with adjacent steps of a 90-cent limma or 23-cent comma), and compares these Near Eastern and European tunings with a modern system smaller and less ambitious than the 79-MOS: a 24-note regular temperament with fifths of 704.607 cents, or the same system as implemented on a 1024-EDO synthesizer with small variations in some interval sizes. As observed in the letter, this 24-note system is not itself an MOS, but can be viewed as a union of two 12-MOS systems, and also includes a 17-MOS inviting comparison with the medieval 17-MOS tunings. An appendix proposes a system of interval classification and notation for Near Eastern music based on some familiar Turkish and Syrian concepts combined with certain new categories and symbols, with Ozan's ingenuity and originality a great inspiration in any such effort, whatever the imperfections of my attempt. Margo Schulter 22 September 2008 * * * Dear Ozan, Please let me share with you my great celebration at now having a copy of your dissertation, which I was able to have printed at the local University Library from the PDF file which you have so generously made available at your Web site. Your writing in this most noble document, which I am only beginning to read and digest, together with some of our recent conversations here, moves me to consider one aspect of my more modest 24-note "e-based" temperament for Maqam/Dastgah music, a question relating to the "footing" of this tuning system. The name "e-based," incidentally, refers to a curious mathematical property of the system: the generator of 704.607 cents results in a ratio between the logarithmic sizes (for example as measured in cents) between the regular whole tone and the limma or diatonic semitone equal to Euler's e, approximately 2.71828. ! eb24n.scl ! 1024-tET/EDO version of e-based tuning (Blackwood R=e, 5th 704.607c) 24 ! 55.07813 132.42187 187.50000 209.76563 264.84375 285.93750 341.01562 418.35937 473.43750 495.70312 550.78125 628.12500 683.20312 704.29688 759.37500 836.71875 891.79687 914.06250 969.14063 991.40625 1046.48438 1122.65625 1177.73437 2/1 As you have written, one important goal and feature of the 79-MOS is that it supports, as a subset, a 12-note chromatic cycle; and this is also a feature of Yarman24, which indeed is built upon such a cycle as designed by Rameau, but so artfully augmented as to provide either a 24-note or a 17-note system of the perdeler. For my 24-note system, which seeks to provide a pleasant and reasonable subset of maqam/dastgah colors while at the same time offering a footing in 13th-14th century European styles, a different but somewhat analogous goal might apply: the availability of a 17-note MOS like that of the regular Pythagorean tuning (Gb-A#) described and advocated by Safi al-Din al-Urmavi in the later 13th century, and also by Prosdocimus de Beldemandis (1413) and Ugolino of Orvieto (writing sometime around 1425-1440). Here I will first explore the 17-MOS structure that obtains within this 24-note regular temperament, and then the division brought about when all 24 notes are considered. ----------------------------------------------------------- 1. An important distinction: intervals from 7-10 generators ----------------------------------------------------------- At the outset, I should note an important limitation as well as a possible advantage of the somewhat different structure of the 17-MOS in this temperament than in Pythagorean tuning. While chains of 1-6 generators (fifths or fourths) yield what may be termed "accentuated Pythagoraen intervals" with major intervals somewhat larger and minor ones somewhat smaller, chains of 7-10 generators very significantly yield middle or neutral rather than "schismatic" or 5-limit intervals. When 5-limit flavors are a priority, as certainly in the tuning system and maqamat of Safi al-Din as well as much modern Near Eastern practice, and also in early 15th-century European music where keyboards often have written sharps tuned as Pythagorean flats, the way to acquire these flavors is obviously to choose a different tuning system, with the 79-MOS as one outstanding option. For a subset of maqam/dastgah music where such flavors may not be so indispensable, however, our more modest tempered system has a possible advantage: we can acquire a wealth of neutral as well as "accentuated Pythagorean" intervals with relatively short chains of fifths, so that a 24-note tuning makes them available in diverse locations and patterns. Indeed, although it happens that some intervals have sizes similar or even virtually identical to those of the familiar 53-comma theory, this system might also be seen as a variation on the 17-EDO model of the 17 basic perdeler, since it supports all of the interval types of that model while, like the 53-comma system, providing a smaller and a larger size for each type of middle or neutral interval, as well as some delectable septimal flavors. Of course, as you well observe in your dissertation regarding the 53-comma system, providing only two sizes for each neutral category is more the beginning of wisdom than its full consummation, as may be seen in the subtle variety of shadings realized by musicians in various parts of the Near East, and much more amply sampled in the 79-MOS. Yet I hope that the more imperfect solution of a 24-note regular temperament that at least respects the distinction between a larger and a smaller neutral second step, for example, will offer a system which, although not ideal, is adequate and pleasant. ------------------------------------------------------------------------ 2. Medieval parallels: the Pythagoran 17-MOS in the Near East and Europe ------------------------------------------------------------------------ In showing the 17-MOS within this system, I will take two approaches. The first is that of Prosdocimus and Ugolino, who define their 17-note Pythagoraen tunings as having a range of Gb-A#. If we take C as the 1/1, this means a tuning with 6 fifths down and 10 fifths up. Then we shall compare this with the tuning of Safi al-Din, based as you have explained on a series of 4 fifths up and 12 fifths down. ----------------------------------------------------------- 2.1. Prosdocimus and Ugolino: In search of regular cadences ----------------------------------------------------------- Since the e-based tuning actually uses, by custom, two 12-note keyboards each tuned in a 12-MOS with a range of Eb-G#, with the notes on the upper keyboard raised by the diesis of about 55.28 cents, to get the Prosdocimus/Ugolino arrangement we must pick a note as 1/1 which will accommodate the requisite number of fifths in each direction. Here, for simplicity's sake, I choose the note A. l d l l d l l l d l 77 55 77 77 55 77 77 77 55 77 0 77 132 209 286 341 418 495 572 628 705 A ---- Bb --- A# ---- B ---- C --- B# ---- C# ---- D ---- Eb --- D# ---- E |---------------------|--------------------|-------|---------------------| T T l T 209 209 77 209 l d l l d l l 77 55 77 77 55 77 77 705 782 837 914 991 1046 1123 1200 E ---- F ---- E# ---- F# ---- G --- Fx ---- G# ---- A |---------------------|---------------------|-------| T T l 209 209 77 As this diagram shows, an octave of the 17-MOS may be diatonically divided into five 209-cent tones plus two 77-cent limmas, here placed at A-B-C#-D-E-F#-G#-A. Each of the five tones is subdivided into three steps: a 77-cent limma plus a 55-cent diesis plus a 77-cent limma, here abbreviated l-d-l. The 77-55-77 cent division of the tone resembles 17-EDO with its equal division at a rounded 71-71-71 cents (actually about 70.588 cents for each equal step) -- in contrast to Pythagorean tuning with its division of the 9:8 tone into limma-comma-limma at a conveniently if not so accurately rounded 90-24-90 cents (actually with a 256:243 limma at 90.225 cents and a comma at 531441:524288 or 23.460 cents). In the classic Pythagorean division, the ratio of step sizes is close to 4-1-4, as precisely holds in 53-EDO with 9 Holdrian commas per tone; in the e-based temperament, the ratio is very close to 109-EDO, or 7-5-7. As in either Pythagorean intonation of 53-EDO, however, there is a small interval or comma which marks the difference in size between certain steps, here the 77-cent limma and 55-cent diesis: 23.60 cents in Pythagorean tuning, 22.64 cents in 53-EDO (the Holdrian comma), and 21.68 cents in a strictly regular version of the e-based system. In our 17-MOS, this comma does not actually appear as a direct step, but makes its presence known, for example, if we compare the different sizes of intervals formed from two adjacent MOS steps. A limma plus a diesis or "l + d" (e.g. A-Bb-A#) forms an apotome or chromatic semitone such as A-A# at 132 cents, or roughly 68:63, between 14:13 (128 cents) and 13:12 (139 cents) and a bit closer to the smaller of these superparticular neutral seconds. The combination of two limmas or "l + l" (e.g. A#-B-C) forms a diminished third at 154 cents, or twice the size of the limma, a larger neutral second a bit wider than 12:11 (151 cents), and very close to Safi al-Din's ratio of 59:54 (153 cents). The smaller and larger middle or neutral seconds at 132 and 154 cents -- differing by the 22-cent comma -- are not too far from intervals of respectively 6 and 7 Holdrian commas in the 53-EDO system, with rounded values of 136 and 158 cents. Here both steps are somewhat narrower than their Pythagorean or 53-EDO counterparts, as is the regular minor third formed from their sum (e.g. A-A#-C) -- rather smaller at 286 cents than the Pythagorean 32:27 (294 cents); and quite close to 13:11 (289 cents), and yet more so to 33:28 (284 cents). Thus in this 17-MOS of the 704.607-cent temperament, as in 17-EDO (with fifths of 705.882 cents), any two adjacent steps will form a neutral or middle second, more generally yielding neutral intervals quite efficiently with a relatively small number of generators. In addressing the Pythagorean 17-MOS system of Prosdocimus and Ugolino, we should show how this tempered counterpart can satisfy the stylistic goals that led these theorists to champion a 17-note tuning. Unlike Safi al-Din, who obviously sought after schismatic or approximate 5-limit intervals in ajnas and maqamat such as his Rast (equivalent to a modern Turkish 5-limit Rast), these medieval European theorists were interested in obtaining regular Pythagorean intervals such as major and minor thirds and sixths for directed polyphonic progressions. In these progressions, unstable or "imperfect" intervals such as major or minor thirds and sixths seek resolution to "perfect" or stable concords: as in the Near Eastern theory of the Mutazilah Era, and in the early Greek theory shared as a source for both of these medieval traditions, the Pythagorean concords of the unison (1:1), octave (2:1), fifth (3:2), and fourth (4:3). Specifically, it is axiomatic in the standard 14th-century European style that major thirds often expand stepwise to fifths, major sixths to octaves, and major tenths to twelfths; while minor thirds often likewise contract to unisons, with one voice of the interval moving by a 9:8 tone (204 cents) and the other by a 256:243 limma (90 cents). Prosdocimus and Ugolino are concerned that when accidentals are required to obtain a major third before a fifth, or a major sixth before an octave, these thirds and sixths have their "fully perfected" size of 81:64 or 408 cents, and 27:16 or 906 cents, so that can efficiently expand to the fifths and octaves toward which they "strive." Likewise, when a third contracting to a unison must be made minor by an accidental, it is important in Ugolino's language that the third be fully "colored" or reduced in size to the regular 32:27 (294 cents), so that it likewise may efficiently reach its goal through motions of a 204-cent tone and 90-cent limma. The Pythagorean 17-MOS of these two theorists is specifically designed to support cadences with these desired intervals and steps of a classic 14th-century technique, with either ascending or descending semitonal motion, on each of the six "fixed" steps or perdeler, as we might also say, of the standard medieval gamut (C, D, E, F, G, A -- with B/Bb regarded as a fluid degree with both forms as elements of the regular gamut, or _musica recta_). For example, a piece centered on a finalis of D, in what might be considered the mode of Protus or Dorian, will typicallly conclude in the 14th-century with an intensive cadence on this step calling for the accidentals C# and G#; and feature as part of its seyir or development remissive cadences on the second degree E or fifth degree A, the latter involving the "soft" (_molle_) or flat form of the flexible degree B/Bb, or Bb, for example. These accidental inflections in routinely occurring cadences are thus a feature of the " polyphonic seyir" of this mode, much as customary inflections are part of the seyir of a maqam such as Ushshaq (the Arab Bayyati) with its flexible sixth degree above the final, or Buselik where the degree at a tone below the finalis is often raised when ascending to the finalis. Here are examples of these progressions and inflections using the very popular cadence where a major third expands to a fifth while a major sixth expands to an octave, arriving at a complete 2:3:4 sonority (e.g. D-A-D, E-B-E, or A-E-A) which generally defines the standard of complete stable harmony in 13th-14th century European music. Final cadence on D Internal cadences on E or A C# D D E | G A G# A A B D E E D F E Bb A Indeed, the contrast between polyphonic cadences with ascending or descending semitones -- here, following one medieval usage, referred to as "intensive" and "remissive" -- is based to the musical development or seyir of 14th-century compositions and forms. An intensive cadence with ascending semitones typically signals a more conclusive feeling than a remissive cadence with descending semitones, which often is used as an internal cadence on a step other than the finalis. In French, one speaks in a 14th-century context of an _ouvert_ or "open" cadence, often remissive; and a _clos_, that is a "close" or final cadence, generally intensive. These contrasting cadences, and the accidentals they frequently require, shape both the short-range and long-range organization of forms such as the French ballade or virelai and the Italian ballata, guiding the expectations of an attuned listener, much like the seyir of a maqam. Scholars such as Richard Crocker and Sarah Fuller have elucidated these structural aspects and subtleties of 14th-century European music, just as others such as Ali Jihad Racy have illuminated features of the maqam system which are likewise implicit to attuned musicians and listeners. While many of the typical patterns of the 14th century, as exemplified by composers such as Machaut in France and Landini in Italy, can be realized on a 12-note keyboard with a likely common accidental range of Eb-G#, Prosdocimus and Ugolino wanted to accommodate the more adventurous composers or improvisers: around 1400, for example, Solage in his famous _Fumeux fume_ uses a 15-note accidental range (Gb-G#). Here the use of A as our 1/1 instead of C results in a transposed basic medieval European gamut of A-B-C#-D-E-F# plus the fluid degree G/G#, so that to meet the purposes of Prosdocimus and Ugolino, our 17-MOS should support both intensive and remissive progressions on the first six of these degrees while keeping all simultaneous intervals in their usual sizes (e.g. 286-cent minor thirds, 418-cent major thirds, 914-cent major sixths), and permitting all melodic steps in these cadences to be either regular 209-cent tones or 77-cent limmas. Our examples again feature the very characteristic cadence where a major third above the lowest voice expands to a fifth while a major sixth expands to an octave. For each degree, the intensive form of this cadence is shown first, and then the remissive form. Cadences on A Cadences on B Cadences on C# Int Rem Int Rem Int Rem G# A | G A A# B | A B B# C# | B C# D# E D E E# F# E F# Fx G# F# G# B A Bb A C# B C B D# C# D C# Cadences on D Cadences on E Cadences on F# Int Rem Int Rem Int Rem C# D | C D D# E | D E E# F# | E F# G# A G A A# B A B B# C# B C# E D Eb D F# E F E G# F# G F# Here it might duly be added that while Prosdocimus and Ugolino were motivated in part by a desire to facilitate these familiar 14th-century cadences in remote locations, their close analysis of intonation may also reflect another factor: the evidently widespread popularity in the early 15th century of Pythagorean keyboard tunings in which some or all of the usual written sharps (F#, C#, and G# in a usual 12-note tuning) were in fact realized as Pythagorean flats (Gb, Db, and Ab). Such tunings served to produce schismatic or near-5-limit thirds in sonorities and cadences involving sharps -- for example, the widely favored 12-note tuning of Gb-B. Any Turkish or other Near Eastern musician sitting down at such a European keyboard could have found Safi al-Din's Rast, or the same 5-limit flavor as often favored in Turkey today, by the use of a few accidentals (here shown along with modern perde names): rast dugah segah chargah neva huseyni evdj gerdaniye D E Gb G A B Db D 1/1 9/8 8192/6561 4/3 3/2 27/16 4096/2187 2/1 0 204 384 498 702 906 1086 1200 204 180 114 204 204 180 114 commas: 0 9 17 22 31 40 48 53 9 8 5 9 9 8 5 In an early 15th-century European style, however, this kind of transposition placing schismatic or 5-limit intervals on the regular steps of a mode is not the norm, however. Rather, there is a contrast of "modal color," as Mark Lindley has termed it, between sonorities with regular Pythagorean thirds or sixths spelled with diatonic notes or flats only (e.g. F-A-D, Bb-D-G), and those with 5-limit flavors of these intervals spelled with written sharps: e.g. E-G#-C# before D-A-D, played at E-Ab-Db at 0-384-882 cents in place of the traditional 14th-century intonation at 0-408-906 cents with the major third and sixth "fully perfected." While Prosdocimus and Ugolino expressed a preference for the traditional intonation, their version of the 17-MOS would more generally offer performers a choice between the regular Pythagorean and schismatic or 5-limit flavors at many locations in sonoorities involving accidentals: it includes all the notes of the traditional 12-note Eb-G# tuning as well as the newer tunings such as Gb-B. By the mid-15th century, in at least some parts of Europe, there was evidently a preference not merely to have an artful mixture of these flavors, but to favor a 5-limit flavor wherever possible -- a desire fulfilled around this epoch by the advent of meantone temperament, an ingredient of the 79-MOS which makes possible a 5-limit Maqam Rast with an unbroken chain of fifths. The tempered 17-MOS with a fifth at 704.607 cents looks at once back to the love of Prosdocimus and Ugolino for wide major intervals, narrow minor intervals, and efficient cadential resolutions based on fifths and fourths as the favored stable intervals -- and beyond the realm of historical European music to a special treasure of the Near Eastern tradition: the plethora of middle or neutral intervals, and of maqamat or dastgah-ha in which they are featured as routine elements of a transcendent art. Many of these maqamat or dastgah families may be relished in this 17-MOS or the full 24-note system. Further, the system supports an approach to polyphonic realizations in maqam-related or dastgah-related styles based on stable fifths and fourths, already an element of some medieval and modern performance traditions (e.g. Ibn Sina's _tarqib_ or "composite sound" of simultaneous intervals, with fourths especially preferred, a term also used to describe an especially intricate maqam compounded of others), and drawing on some European aspects of practice and theory in the Mutazilah Era in order to mix and contrast these stable concords pleasingly with the full range of intervals arising in maqam/dastgah music -- major, minor, and middle or neutral. Before considering the question of maqam/dastgah polyphony, however, we must be sure that a system offers agreeable realizations of a reasonable range or subset of maqamat or dastgah-ha as pure melody, with enough scope for the seyir to unfold without undo intonational obstacles or impediments. This is a worthy and indeed precious goal in its own right, as well as the necessary foundation for any polyphonic texture aptly ornamenting and adorning the traditional melodies and intricate modulations. --------------------------------------------------------------- 2.2. Safi al-Din al-Urmavi and Nasir Dede: A tempered variation --------------------------------------------------------------- Considering the Pythagorean 17-MOS of Safi al-Din based on a tuning from the 1/1 of 12 fifths down and 4 fifths up provides an introduction to the somewhat different range of maqam/dastgah colors offered by the corresponding 17-MOS in our 704.607-cent temperament, a realm further enriched by the extra steps and intervals of the full 24-note tuning. If we again take Eb as the note at the flat end of the 17-MOS chain, then D#, at 12 fifths or a diesis (less seven octaves) higher, will here serve as the 1/1. Following your account of the Abjad system of 17 perdeler based on the perde names of Nasir Dede, I take 1/1 to be Yegah and 4/3 to be Rast. 209 209 77 T T l |--------------------|--------------------|-------| l l d l l d l 77 77 55 77 77 55 77 0 77 154 209 286 363 418 496 D# ---- E ---- F --- E# ---- F# --- G --- Fx ---- G# Yegah Pes Pes Asiran Acem Arak Gevast Rast Beyati Hisar Asiran 209 209 77 209 T T l T |---------------------|--------------------|-------|--------------------| l l d l l d l l l d 77 77 55 77 77 55 77 77 77 55 495 572 649 705 782 859 914 991 1068 1145 1200 G# ---- A ---- Bb --- A# ---- B ---- C --- B# ---- C# ---- D ---- Eb -- D# Rast Suri Zirguleh Dugah Kurdi/ Segah Buselik Cargah Saba Hicaz/ Neva Nihavend Uzzal While regular diatonic intervals from chains of 1-6 generators (e.g. major and minor seconds, thirds, sixth, sevenths) correspond to those of the Pythagorean 17-MOS of Safi al-Din and Nasir Dede -- but with major intervals rather larger and minor ones rather smaller, as we have noted -- the schismatic or 5-limit flavors of the original are transformed here into neutral flavors, with large middle or neutral intervals abounding above the 1/1. These large neutral intervals include a middle second at 154 cents (D#-F); a third at 363 cents (D#-G); a sixth at 859 cents (D#-C); and a seventh at 1068 cents (D#-D). We may find it helpful to compare these neutral intervals with their 5-limit counterparts in the original Pythagorean 17-MOS of Safi al-Din and Nasir Dede, with note spellings given both as in the above scheme with perde yegah at D#, and in a widely followed modern custom placing this perde at G: -------------------------------------------------------------------- Interval Type Pythagorean cents 5-limit e-based Approx JI -------------------------------------------------------------------- D#-F or dim3 65536/59049 180 10/9 154 59/54 G-Bbb 182 153 -------------------------------------------------------------------- D#-G or dim4 8192/6561 384 5/4 363 21/17 G-Cb 386 366 -------------------------------------------------------------------- D#-C or dim7 32768/19683 882 5/3 859 23/14 G-Fb 884 859 -------------------------------------------------------------------- D#-D or dim8 4096/2187 1086 15/8 1068 63/34 G-Gb 1088 1068 -------------------------------------------------------------------- The small major or 5-limit flavors obtained in the Pythgorean 17-MOS from these vitally important diminished intervals involving chains of 7-10 generators are thus transformed in the e-based 17-MOS to large neutral intervals often on the order of a comma smaller. While the 154-cent neutral second is rather close to 12:11 at 151 cents, and so may be regarded as within a central neutral range, the large neutral third, sixth, and seventh at 363, 859, and 1068 cents could also be described as submajor flavors, rather "bright" and outgoing. Although notably different from the 5-limit flavors of Safi al-Din, these large neutral flavors also fit nicely into the world of maqam/dastgah music. A 154-cent middle second above the finalis gives Maqam Huseyni a pleasant quality, as does the 859-cent sixth, whose bright colors likewise fit the ethos of Acemli Rast, a form of Maqam Rast with two conjunct Rast tetrachords also known in the Arab world as Nirz Rast. In either this form of Rast or the form with disjunct Rast tetrachords, the submajor third at 363 cents should likewise be at home, while in the latter form the large neutral or submajor seventh at 1068 cents should pull nicely toward the octave of the finalis. Just as a good test for the Prosdocimus/Ugolino version of the 17-MOS was to seek out regular intensive and remissive cadences for each of the six "fixed" steps of the standard medieval European gamut, so one test for our tempered version of the Safi al-Din/Nasir Dede 17-MOS is to attempt a tuning of what I term "basic Rast": a 9-note set combining the steps of Maqam Rast in its forms with disjunct and conjunct tetrachords. Here I take perde rast at G# as the 1/1, and give possible modern perde names which may vary somewhat from those of Nasir Dede: Rast Rast |-------------------------| T |-----------------------| 0 209 363 495 705 914 1068 1200 G# A# C C# D# E# G G# rast dugah segah chargah neva huseyni evdj gerdaniye 209 154 132 209 209 154 132 T Jk Js T T Jk Js Rast |------------------------| T | 495 705 859 991 1200 C# D# F F# chargah neva dik hisar ajem gerdaniye 209 154 132 209 T Jk Js T Here the disjunct and conjunct versions of the upper Rast tetrachord are shown separately, with certain symbols borrowed from medieval and modern Near Eastern notations, and sometimes combined or modified, used to show step sizes. Thus T signifies a usual tone or tanini at 209 cents; Js a smaller mujannab or middle second, in this tuning system the apotome at 132 cents; and Jk a larger mujannab or middle second, here the diminished third at 154 cents. The desired form for a Rast tetrachord is thus T-Jk-Js or 209-154-132 cents; in a 53-comma notation, this form could be generalized as 9-7-6. In the more specific _and_ diverse -- and therefore more powerful --79-MOS notation, we would have a pattern of 14-10-9 steps or 211-151-136 cents. Combining the 9-note set from these two versions of Maqam Rast into a single series, we find other types of intervals as well: T Jk Js T Jk E B B Js 209 154 132 209 154 55 77 77 132 0 209 363 495 705 859 914 991 1068 1200 G# A# C C# D# F E# F# G G# rast dugah segah chargah neva dik huseyni ajem evdj gerdaniye hisar Here B for Turkish bakiye designates the usual limma or diatonic semitone of 77 cents, while E or "eksik bakiye" stands for the smaller or "diminished" semitone at 55 cents, actually in this tuning system the 12-diesis equal to 12 tempered fifths less 7 octaves, for example from dik hisar to huseyni or F-E#. In the full 17-MOS, with each adjacent step either a 77-cent limma (B) or a 55-cent diesis (E), we find another interesting interval of maqam/dastgah music which is recognized by Qutb al-Din al-Shirazi around 1300: a step of 7:6, or 267 cents, used as the middle step of a Hijaz tetrachord with a tuning of 1/1-12/11-14/11-4/3 or 0-151-418-498 cents, with steps of 12/11x7/6x22/21 or 151-267-81 cents. In the e-basd 17-MOS, this appears above the 1/1, yegah at D#, in a version quite close to JI: Jk A12 B 154 264 77 0 154 418 495 D# F Fx G# yegah dik mahur rast hisar Here the septimal or 7:6 flavor of minor third is shown by the Turkish sign "A12," meaning an "augmented" step, as in Hijaz, larger than a tone, and equal to about 12 commas. This 7:6 flavor of step also occurs in one popular form of the Persian tetrachord and dastgah now known as Chahargah, where Hormoz Farhat describes it as a "plus second." This small minor third or "plus second" is formed in this temperament from two apotome steps (Js) each at 132 cents, here F-F#-Fx. Since each apotome or small neutral second is formed from 7 generators up, this interval thus involves a chain of 14 such generators. Within the 17-MOS, another derivation is also possible: a regular 209-cent tone (T) plus a 55-diesis (E), here F-G-Fx. Interestingly, corresponding forms of the "eksik bakiye" or reduced limma (E) and the near-7:6 minor third or plus second (A12) occur in a 24-note or larger Pythagorean tuning, or its close approximation in 53-EDO, with the 7:6 flavor also occurring in a Pythagorean 17-MOS, where it results from 15 generators down, or a 32:27 minor third less a 23-cent comma. It thus results in the 17-MOS of Safi al-Din and Nasir Dede when an interval is formed from a chain of three adjacent limmas, each at 256:243 or 90.224 cents. Taking the 1/1 or perde yegah as G in a common modern spelling, we find these intervals between perdes gevart-zirgule or G-Cbb (limma steps G-Ab-Bbb-Cbb), and perdes buselek-hicaz/uzzal (E-Abb) in the Nasir Dede naming system (limma steps E-F-Gb-Abb). The eksik bakiye or reduced limma (E) does not quite occur in the Pythagorean 17-MOS, because it requires a chain of 17 generators, and is indeed well defined as the "17-diesis," the difference between 17 fifths at a pure 3:2 and 10 pure octaves, or about 66.76 cents (a usual 90-cent limma or 5 fifths down less a 23-cent comma or 12 fifths down). A conventional spelling for this small semitone would be, for example, E#-Gb. The interval with a not too dissimilar size in the e-based temperament, and present in a 17-MOS, is the 12-diesis at 55 cents, corresponding structurally with the Pythagorean comma (i.e. 12-comma). In this tempered system, the 17-MOS does not include as a direct step the structural counterpart of the Pythagorean 17-comma at 67 cents: a 17-diesis with a size curiously very close to that of the Pythagorean 12-comma, and yet closer to the Holdrian comma: 22 cents (e.g. Gx-Bb). The 23-cent Pythagorean comma, of course, does prominently appear as a direct step in the 17-MOS of Safi al-Din and Nasir Dede, where it vitally defines the difference, for example, between small or 5-limit flavor major third at 384 cents and a regular Pythagorean major third at 408 cents. With perde yegah at G, this distinctions occurs for example above perde rast at C between segah at Fb or 384 cents and buselik at E or 408 cents, In relation to yegah, these two steps are likewise at 882 and 906 cents, forming a 5-limit and a Pythagorean flavor of major sixth. To see how the comparably sized 17-comma of our e-based temperament also plays a vital role in better realizing maqam/dastgah music, we now move to a full 24-note tuning. -------------------------------------- 3. From the 17-MOS to a 24-note system -------------------------------------- From one perspective, a 24-note system may be considerably less tidy than a 17-MOS: it is not itself an MOS. Both in theory and as mapped to two conventional 12-note keyboards, however, it can be considered as a union of two 12-MOS systems placed at the distance of the comma or diesis defined by the difference between 12 fifths up and 7 pure octaves: here, with the 704.607-cent temperament, the 12-diesis of about 55.283 cents. In practice, the larger 24-note set is very helpful both in providing more intervals in flavors such as septimal (e.g. 7:6, 9:7, 7:4) requiring longer chains of generators, and in permitting a greater choice of subtly different maqam flavors from a given perde or step, so as to support more intricate and sophisticated modulations. In certain situations, it is also possible to use the 17-comma of about 22 cents in order to introduce small and expressive nuances to the tuning of a given perde as a performance proceeds, thus emulating to a limited degree the kind of practice advocated by theorists such as al-Sabbagh and scientifically documented by investigators such as Can Akkoc who have measured the pitches and intervals produced by performers on flexible-pitch instruments such as the ney. The qualification "to a limited degree" merits emphasis: at most, from a given position in the tuning, we will have two steps available which could fit the category of a "neutral third" (at 341 or 363 cents), for example. In contrast, the 79-MOS typically provides three subtly different flavors of neutral thirds from a given step, thus better modelling the "cluster" of pitches or interval sizes produced in practice by flexible-pitch performers. Comparing the 17-MOS and 24-note versions of the perdeler or steps of maqam music in the octave from perde rast to perde gerdaniye may help to demonstrate these points, and to illustrate two common musical situations where the difference of a 22-cent comma can be used to emulate the "cluster" effect documented by Akkoc and adopted as one important basis for mapping out steps and intervals in the 79-MOS. To arrive at a 17-MOS version of the rast-gerdaniye octave, we need only take the Safi al-Din/Nasir Dede mapping we have just explored, moving the lower tetrachord from yegah to rast up an octave so that it becomes an upper tetrachord from neva to gerdaniye. Values in cents are accordingly revised so as to be measured from perde rast, with some perde names revised both to reflect current practices and to accommodate for certain aspects of the 704.607-cent tuning: 209 209 77 209 T T l T |---------------------|--------------------|-------|--------------------| l l d l l d l l l d 77 77 55 77 77 55 77 77 77 55 0 77 154 209 286 363 418 496 572 649 705 G# ---- A ---- Bb --- A# ---- B ---- C --- B# ---- C# ---- D ---- Eb --- D# Rast Suri Zirguleh Dugah Kurdi Segah Buselik Cargah Hijaz Saba Neva 209 209 77 T T l |---------------------|--------------------|-------| l l d l l d l 77 77 55 77 77 55 77 705 782 859 914 991 1068 1123 1200 D# ----- E ---- F --- E# ---- F# --- G --- Fx ---- G# Neva Beyati Dik Huseyni Acem Evdj Mahur Rast Hisar Adding the seven extra notes of the 24-note system produces this mapping, using for the moment conventional note spellings like those of our mappings so far, with "c" showing a 22-cent comma (the 17-comma): 209 209 77 T T l |-----------------------------------|---------------------------|---------| l l d l l d l |--------------|------------|-------|---------|-----------|-----|---------| d c d c d l d c d l 55 22 55 22 55 77 55 22 55 77 0 55 77 132 154 209 286 341 363 418 496 G# ---- F#x -- A ---- Gx -- Bb ---- A# ------ B ---- Ax - C --- B# ------ C# Rast Nerm Suri Nerm Zengule Dugah Kurdi Dik Segah Buselik Chargah Shuri Zengule Kurdi 209 T |-----------------------------------| l l d |--------------|-------------|------| d c d c d 55 22 55 22 55 496 551 572 628 649 705 C# ----- Bx -- D ---- Cx -- Eb ---- D# Chargah Nerm Hijaz Saba Dik Neva Hijaz Saba 209 209 77 T T l |-----------------------------------|----------------------------|--------| l l d l l d l |--------------|-------------|------|---------|-----------|------|--------| l d c d l d c d l 77 55 22 55 77 55 22 55 77 705 782 837 859 914 991 1046 1068 1123 1200 D# ----------- E ----- Dx -- F ---- E# ----- F# ---- Ex -- G --- Fx ----- G# Meva Beyati Hisar Dik Huseyni Acem Nerm Evdj Mahur Gerdaniye Hisar Evdj In mapping out this maqam/dastgah system and proposing names for the perdeler, placing perde rast at G# may be a convenient choice because this step has a chain of 11 fifths available either up or down -- as likewise D#, here perde yegah or neva, another customary reference point for Near Eastern tuning systems and gamuts. As cardinal points of reference, indeed, yegah and rast may be somewhat analogous to the steps G and C in the European gamut, with the former as the foundation of the medieval hexachord system as the lowest regular note gamma-ut (thus the term "gamut"), and the latter as the lowest tone of the natural hexachord (C-D-E-F-G-A, the six "fixed" steps in the regular or _musica recta_ gamut) and the frequently chosen place in medieval and later times for a clef in staff notation (along with F, and in later periods also G). The diagram above divides the 24-note octave from rast to gerdaniye into two disjunct tetrachords plus a middle tone. We find three types of adjacent steps: the 77-cent limma and 55-cent diesis familiar from the 17-MOS, and the new 17-comma step at 22 cents. The diagram also shows the 17-MOS division of the gamut into limmas and dieses; and a possible diatonic division, as in the earlier diagram of the 17-MOS, into five regular tones and two limmas. Within the first whole tone from rast to dugah (G#-A#), and likewise in the tone from chargah to neva (C#-D#) that occurs between the two tetrachords in the diagram, we have a situation where all adjacent intervals are either 55-cent dieses or 22-cent commas, with each limma of the 17-MOS thus divided into these two smaller steps. Elsewhere, however, five limmas from the 17-MOS remain undivided. If this regular 704.607-cent tuning were carried to 29 notes, then we would reach another MOS, with all adjacent steps throughout the system either dieses or commas. The conventional notation above is intended to show the chains of fifths; but another system especially practical when using two 12-note keyboards, each with its 12-MOS, is to use a familiar spelling of Eb-G# for the notes on each keyboard, with a diesis sign (*) showing a note on the upper keyboard raised by the 55-cent diesis. Thus D# could also be written as Eb*, and F#x as G#* -- in the last example, a simpler spelling to read and interpret, especially at the keyboard. Of course, a more sophisticated system such as Sagittal can be used with great success; it is interesting to consider whether or how a generalized keyboard might affect the matter of a convenient notation. In any event, I must mention that some crude mechanical problems I have encountered when using two standard MIDI controller keyboards, which must necessarily be at a large vertical distance apart, make the above logical scheme of the perdeler often more relevant in theory than in practice. Specifically, I have found that playing maqam/dastgah music satisfactorily requires the ability to move between notes on the two keyboards easily and fluently with a single hand. The great obstacle to this is a situation where one must move from an accidental on the upper keyboard to a natural on the lower keyboard. Thus the practical solution is to choose a perde for the finalis which will best accommodate the seyir of a given maqam, including likely modulations; or likewise will best fit a given dastgah and its gusheh-ha. Either a generalized keyboard, or a solution involving a more ergonomic arrangement of two standard 12-note keyboards, would permit a more consistent approach to the perdeler when actually playing. ------------------------------------------------------ 3.1. Septimal flavors: Ibn Sina's soft diatonic tuning ------------------------------------------------------ The matter of F#x or G#* leads us to an important musical aspect of the full 24-note system: septimal flavors of a kind featured, for example, in a beautiful soft diatonic tuning, as John Chalmers has described it, with two possible interpretations. The first, which Chalmers follows in a Scala scale archive file (avicenna_diat.scl), has a tetrachord of 1/1-14/13-7/6-4/3 (0-128-267-498 cents) with the smaller middle or neutral second step at 14:13 preceding the larger at 13:12, followed by a large or 8:7 tone (128-139-231 cents). These disjunct tetrachords would have string ratios of 28:26:24:21: 1/1 14/13 7/6 4/3 3/2 21/13 7/4 2/1 0 128 267 498 702 830 969 1200 14:13 13:12 8:7 9:8 14:13 13:12 8:7 128 139 231 204 128 139 231 Another reading which you have favored, Ozan, places the larger neutral second first in these tetrachords, thus producing a frequency ratio of 12:13:14:16, an arrangement also favored by George Secor (whose use of it led me to look into its possible history and discover Ibn Sina's tuning): 1/1 13/12 7/6 4/3 3/2 13/8 7/4 2/1 0 139 267 498 702 841 969 1200 13:12 14:13 8:7 9:8 13:12 14:13 8:7 139 128 231 204 139 128 231 Whatever views people might take as to which of these interpretations is correct, the 704.607-cent tuning neatly offers a diplomatic solution by using two tempered steps at an equal 132 cents each, as in this version starting from C on the lower keyboard, a very likely position if Ibn Sina's beatiful tuning is treated in modern terms as a variation on the Daramad of Shur Dastgah: C C# Cx/D* F G G# Gx/A* C 0 132 264 495 705 837 969 1200 132 132 231 209 132 132 231 As the conventional spellings show, the 132-cent neutral second is the apotome or chromatic semitone, so that the first two intervals of each tetrachord are formed by successive apotome steps: C-C#-Cx or G-G#-Gx. On two standard 12-note keyboards, this sequence is conveniently notated as C-C#-D* or G-G#-A*, and involves a motion up from F to F#, for example, and then to G* on the upper keyboard. While the conventional notation shows that the near-7:6 third is equal to two apotome steps, the keyboard notation shows that it is equal to a regular tone plus a diesis or small semitone at 55 cents, e.g. C-D-D* or G-A-A*. All transpositions of this beautiful tuning in the 24-note system must place the 1/1 on the lower keyboard, since a chain of 15 fifths up is required to obtain all of the steps and intervals including the 7:4 minor seventh (virtually just at 969 cents). The most sharpward of these transpositions assigns F#x or G#*, a note whose spelling we have been considering, this vital role of a 7/4 step. B B#/C* Cx/C#* E F# Fx/G* F#x/G#* B 0 132 264 495 705 837 969 1200 132 132 231 209 132 132 231 As noted above, this is not the likeliest transposition in practice, at least with my present complications of keyboard ergonomics, since the vertical space between the two 12-note keyboards makes it difficult to play an interval like C#*-E or G#*-B smoothly and fluently with a single hand. However, the position above with C as the finalis, taken as a form or variation on the modern Daramad Shur, is quite practical and has some interesting ramifications. 132 628 760 F#/Gp G* C C# Cx/D* F G G# Gx/A* C 0 132 264 495 705 837 969 1200 132 132 231 209 132 132 231 The first nuance involves the flexible nature of the sixth degree when Ibn Sina's tuning is used as a stimulating variation on the modern form of Daramad Shur. Since a minor rather than neutral sixth is typically cited in "textbook" accounts of Shur showing a seven-note octave, these two forms are very likely to alternate -- with the minor version of this step, G* at 760 cents or actually about 5 cents smaller than 14/9 (765 cents), colorfully reflecting the septimal flavor of the tuning. The small semitone or diesis G-G* at 55 cents offers a very expressive melodic interval at a vital location in this favorite Persian dastgah known for its spiritual intensity -- although it remains an open question whether steps quite this small are used in traditional Persian music, or how Iranian musicians might regard them in such a context. Another step vital to the usual practice of Shur is the lowered fifith degree at a step conventionally spelled F#, but where the functional Persian spelling Gp (the "p" symbol indicating the koron, lowering a note often by about 50-70 cents, or something approaching a third of a tone) is much more communicative. While "F#" might suggest a leading tone to G, like buselik-chargah or mahur-gerdaniye in maqam music, in a Persian setting the lowered fifth degree of Shur, here Gp, pulls strongly downward toward the lower tetrachord and the finalis. The descending figure G*-Gp-F is especially characteristic, and involves two small neutral seconds at 132 cents spanning the distance of about a 7:6 minor third between the steps F at a tempered 4/3 and G* at a tempered 14/9. This touch mirrors the intonational patterns of Ibn Sina's soft diatonic. ----------------------------------------------- 3.2. Subtle variations and "cluster" techniques ----------------------------------------------- While the potential for fine variations in intonation are rather modest in our 24-note regular temperament by comparison to the more ample 79-MOS, still there are some pleasant opportunities for such nuances which may often occur. One of them involves the tuning of a Hijaz (or in the Turkish spelling, Hicaz) tetrachord. As already noted, the classic tuning of this tetrachord by Qutb al-Din as 12/11x7/6x22/21 seems ideally to fit this tempered system, or vice versa, with perde names "of convenience" (not the same as in the earlier diagrams) to fit a very recent situation where I used this tetrachord as part of the seyir of Maqam Saba with the finalis placed on D*, which is thus regarded as perde dugah, and C* as rast: Jk A12 B 154 264 77 0 154 418 495 C* D E* F* gerdaniye shehnaz tik tik buselik chargah In the descending portion of the seyir, as you explain in your dissertation at page 121, another tetrachord of this general nature, called Chargah, also occurs, realized in the 704.607-cent tuning as follows: Js A13 B 132 286 77 0 132 418 495 F* F#* A* Bb* chargah saba huseyni acem In this version of Maqam Saba, as in the 79-MOS version, this Chargah jins or genus has a middle step equal to about 13 commas or a bit smaller, here 286 cents or a near-just 13:11 (289 cents) or 33:28 (284 cents), in contrast to the narrower step of about 12 commas or 7:6 in the previous Hijaz tetrachord at the beginning of the seyir's descending portion from tik chargah down to gerdaniye. The Turkish sign "A13" shows this larger middle or "augmented" step, by comparison to the "A12" in the previous example. If, in a different musical context, we desired a 13-comma genus at the first location C*, or a 12-comma version at F*, the 24-note system makes them available. Here I refrain from giving perde names, since these could vary with the situation and I do not immediately have any specific situation in mind: Js A13 B 132 286 77 0 123 418 495 C* C# E* F* Jk A12 B 154 264 77 0 154 418 495 F* G A* Bb* Although users of the 79-tone qanun or ney players might regard the term "cluster" as somewhat overblown for a tuning system with only, at most, two versions of a given basic perde separated by less than the 55-cent diesis, one situation where I find a quasi-clustering technique very apt and pleasant is in realizing the tetrachord below the finalis of Dastgah Bayat-e Esfahan, often regarded as an avaz or tributary of Dastgah Homayun, although Hormoz Farhat prefers to regard it as an independent dastgah or modal family. The tuning of this lower tetrachord of Bayat-e Esfahan leading up to the finalis has been reported to show considerable variations in practice between some of the leading musicians whose intonations have been measured, and has also been a topic of some discussion among Persian and other theorists. In what is known as the traditional or "old" Esfahan, there is a small neutral second, a whole tone of around 9:8 or possibly larger, and then a larger neutral second as the leading tone to the finalis ("leading tone" meaning here simply the step of approach, and not implying as in some European and related theory specifically the interval of a semitone, possibly obtained by an accidental inflection): Js T Jk 132 209 154 B C* D* E 0 132 341 495 In what is known as the "new Esfahan," the tetrachord is regarded as having a structure similar to that of Chahargah Dastgah and also prominently featured in Homayun Dastgah: a lower small neutral second, middle "plus second" often somewhere around 11-12 commas, and upper semitone. Taking recent views of some Persian theorists that the regular major third is often close to Pythagorean at 81:64 (408 cents), and a small neutral second might average around 135 cents, we would have as one type of Chahargah or a similarly tuned Esfahan something like 135-280-90 cents. However, Dariush Tala`i recommends a Chahargah _dang_ or tetrachord at around 140-240-120 cents, and some of Jean Darling's measurements suggest that an upper step of around 120 cents may indeed fit some recent practice. As a kind of pleasant intermediate option between the "old" and the "new," the 24-note system offers what I might term a Buzurg-Hijaz tetrachord since it is very similar to the lower tetrachord of Qutb al-Din's Buzurg genus which spans a fifth: 1/1 14/13 16/13 4/3 56/39 3/2 0 128 359 498 626 702 14:13 8:7 13:12 14:13 117:112 128 231 231 128 76 From this just division of the fifth, a tempered version of the lower tetrachord results as follows: Js T10 Js 132 231 132 B C* Eb E 0 132 363 495 Since a tone or "T" is often presumed to be a regular diatonic step of around 9:8 or 9 commas, the explicit sign "T10" identifies the larger middle interval here as around 10 commas or 8:7; in Qutb al-Din's Buzurg, as in Ibn Sina's soft diatonic, the subtly different 14:13 and 13:12 steps are both realized by the same tempered step of 132 cents. The third step Eb at 363 cents above the lowest note of the tetrachord, and 132 cents or a small neutral second below the finalis E, is at once somewhat brighter and more strongly directed toward the finalis than in the "old Esfahan" version with this step a 22-cent comma lower, and more nuanced than a yet smaller semitone of some kind in the "new" fashion. These alternative tempered renditions of Esfahan at 132-209-154 cents or 132-231-132 cents can illustrate a point about interval notations: either version could be notated generally in the medieval manner as JTJ, a lower and an upper mujenneb or neutral second step with the middle interval of a tone of some kind. Using comma notation, these two forms would be 6-9-7 and 6-10-6. While in a given performance of Esfahan one might lean toward either the 6-9-7 or 6-10-6 form, I have found it very pleasant to alternate frequently between the two positions for the third step which distinguish these forms otherwise using the same notes: playing now D* at 341/154 cents from the lowest note of the tetrachord and the finalis respectively, and now Eb at 363/132 cents. Apart from a tendency, not surprisingly, to prefer the higher position for cadences to the finalis, I find that these steps may freely alternate, the variation being in good part for its own sake to "humanize" a keyboard rendition a bit by introducing a touch of the "clustering" which is a pervasive feature of flexible pitch instruments including the human voice. Of course, this technique may also be taken as an enthusiastic affirmation of the reality that there is more than one way to tune Bayat-e Esfahan. ------------- 4. Conclusion ------------- In exploring the 704.607-cent tuning both as a 17-MOS which may be compared with medieval 17-note Pythagorean MOS systems in the Near East and Europe, and as a full 24-note system, I have highlighted one salient feature in which it varies dramatically from both types of medieval antecedents: the pervasive role of middle or neutral intervals. This feature results from a technique evidently not on the conceptual map of the Near Eastern or European theory of the time: temperament, and more specifically a rather gentle regular temperament in which fifths are slightly widened and fourths narrowed in order to produce neutral flavors with chains of only 7-10 generators, rather than the 19-22 generators which would be required with a pure 3:2 fifth. Of course, the same technical means can be adopted to quite different artistic ends: and this holds true both with the Pythagorean 17-MOS, and with the modern technique of temperaments, regular or irregular, which may involve either widening or narrowing the fifth, or sometimes may combine both options in the same system, as with the 79-MOS. For Prosdocimus and Ugolino (Section 2.1), the Pythagorean 17-MOS provides a gloriously perfected gamut (Gb-A#) expanding the standard 14th-century universe of regular Pythagorean intervals and polyphonic resolutions to more remote melodic and harmonic outposts. The focus is above all on regular diatonic intervals of 1-6 generators; for these authors, neither the augmented and diminished intervals of 7-10 generators with their schismatic or 5-limit flavor, nor the nonexistent neutral intervals, seem of much consequence. For Safi al-Din, in contrast, both the regular Pythagorean and 5-limit flavors of his 17-MOS are of vital interest, as in his version of Rast; and so also are the middle or neutral intervals absent from this tuning, but receiving great attention elsewhere in his discussions of interval categories, tetrachords, and tuning methods for instruments such as the `ud. It remains an open question whether his emphasis on the 17-MOS might reflect a view that 5-limit flavors take a certain priority over neutral ones; or possibly simply a preference for the elegance of a regular tuning with an unbroken chain of fifths. In the absence of temperament, such a preference would necessarily result in a 17-note system favoring regular Pythagorean and schismatic flavors rather than neutral ones, which would require a larger tuning size. How does the 704.607-cent temperament in its 17-MOS and 24-note forms tie in with these different medieval worlds? From the perspective of a 14th-century European style like that celebrated by Prosdocimus and Ugolino, a tempered 17-MOS provides a kind of accentuated variation on the Pythagorean Gb-A# gamut, with regular major intervals yet wider, minor ones yet smaller, and the compact Pythagorean limma at 90 cents, so important to directed resolutions, narrowed to a yet more incisive 77 cents. With these distinctions of color, the regular steps and progressions sought by these theorists can map smoothly from the Pythagorean to the tempered system. While the neutral flavors of augmented or diminished intervals (7-10 generators) would not appear on a period keyboard tuned in pure fifths, they might well have arisen in an "exuberant" style of intonation by singers and other flexible-pitch performers described by Marchettus of Padua (1318) and favored by such a noted modern performer and scholar as Christopher Page, where cadential sharps may be raised by a comma or more above their usual Pythagorean positions. It should be added that these augmented and diminished intervals play mostly an incidental or ornamental role; but the possibility that a momentary sonority of C#-F or F-C#, for example, might sometimes have been sung in 14th-century Europe as a neutral third or sixth is intriguing. From perspective of Safi al-Din's Pythagorean 17-MOS gamut, however, the 704.607-cent temperament is not a reasonable equivalent or variation, because it lacks the 5-limit flavors that are needed for his tunings of basic maqamat such as Rast. Rather, this temperament would need to seek its historical footing in his discussions of middle or neutral intervals, which of course draw richly upon such earlier musicians and theorists of the Islamic tradition as Zalzal, al-Farabi, and Ibn Sina. Expanding either the Pythagorean or tempered 17-MOS to 24 notes gives us a reasonable number of locations for another family of intervals important to medieval Islamic theory: the septimal intervals arising from 14-17 Pythagorean generators or 12-15 tempered generators. Here it is interesting that Safi al-Din mentions 7:6 as a relatively concordant ratio; septimal flavors also play an important role in the tunings of Ibn Sina, and of Safi al-Din's younger contemporary Qutb al-Din al-Shirazi. Just as the Pythagorean 17-MOS structure is used for rather different musical ends by Prosdocimus and Ugolino in Europe, and by Safi al-Din in the Near East, so the technique of temperament can be used to produce different kind of maqam tunings. Whlle the 704.607-cent tuning at once emulates the regular structure of the Pythagorean 17-MOS while emphasizing the neutral intervals that in a medieval approach would be generated from two or more chains of fifths, the 79-MOS system makes it possible to obtain a 5-limit Rast reasonably close to that of Safi al-Din with a single unbroken chain of fifths. This latter technique, of course, is similar to that of European Renaissance meantone, but used as one aspect of a far more intricate tuning system and intonational style to realize the diverse musical world of the maqamat. In its full 24-note version, the 704.607-cent tuning seeks to survey an interesting and useful subset of two musical worlds, that of medieval Europe and that of maqam/dastgah music as it has developed through medieval and modern times. The 79-MOS draws its footing not only from the maqam/dastgah tradition but from European common practoce of the 18th-19th centuries, offering for example a 12-note chromatic cycle, and also combining just fifths with wide and narrow ones in order to permit maximum flexibility in the fine intonation of the maqamat and dastgah-ha. Elsewhere I have compared the 79-MOS to a palatial estate, and the 704.607-cent tuning to a much more modest garden. Each scale of organization may have its own charms. A larger system may place the traits and preferences of a smaller one in a fuller perspective, while a smaller one may encourage us more thoroughly and familiarly to explore a congenial region which may then be better integrated into a larger view. ----------------------------------------------------------------- Appendix: Interval categories and symbols for maqam/dastgah music ----------------------------------------------------------------- In some of the examples above, we have used some categories and symbols to show step sizes in maqam/dastgah music, based in part on medieval and modern Near Eastern theory often based in Pythagorean intonation or 53-EDO, an in part on some specific needs of the 704.607-cent tuning. A particular goal of this proposed synthesis is to retain compatibility as much as possible with the concepts and symbols of a modern Turkish approach, while covering the range of intervals and steps from 1 to 13 commas and, especially, adding categories and symbols for the vital middle or neutral second steps of approximately 6 or 7 commas. We deal here with a "middle-resolution" system based on 53 commas to the octave, taken for mathematical convenience to be the equal or Holdrian commas of 53-EDO. A "lower-resolution" notation, in comparison, might be based on 17 thirdtones or 24 quartertones to the octave; while a "higher-resolution" system could, of course, use cents, or the JI ratio precisely defining or closely approximated by a given interval, with the 79-MOS and its measurement of the yarman, equal to approximately 2/3 of a Holdrian comma, tending in this direction. An advantage of the 53-comma system is that it is reasonably familiar as developed in Syrian as well as Turkish theory, and at the same time general enough that it leaves room for describing the nuances of a given maqam or gusheh, etc., while leaving open precise realizations in cents. Thus a "679" tetrachord might mean 141-153-204 (Safi al-Din's division of 64:59:54:48), or 132-154-209 (e-based), or many other fine shadings; but the pattern of a smaller followed by a larger neutral second, and then a tone, is clear. In the following scheme, intervals of 1-2 commas would typically indicate interval differences or inflections rather than direct melodic steps. Intervals of around 3 commas, or sometimes a bit smaller, however, might serve as small or septimal semitones equivalent to a JI ratio such as 33:32 (53 cents) or 28:27 (63 cents), the latter much favored by Archytas. A step of around 4 commas, of course, is the usual limma. Steps in the range of 5-8 commas might all be considered in a broad sense "intermediate" between the limma and the regular tone at around 9:8, although in medieval and modern tunings such as 5-limit Rast, the 5-comma step may serve as a usual semitone, with some of the complications regarding modern Turkish accidentals stemming from this fact. The proposed notation proceeds from "small" (5 commas) to "small-middle" (6 commas) to "large middle" (7 commas) to "large" (the small whole-tone of 8 commas). This is not the only possible solution, but, given the existing history, may be the "least astonishing" one, to borrow a vivid phrase from computer science. Next follow the regular tone at 9 commas, the large or septimal tone at 10 commas, and "augmented" steps at 11-13 commas of a kind used most notably in a Hijaz tetrachord, or the corresponding Chahargah or Chargah genus associated with the Persian dastgah of that name. Following tradition, these categories are oriented around Pythagorean or 53-EDO intervals -- but with equivalents here shown, where applicable, for the 24-note regular tempered at 704.607 cents, since these two styles of tuning have been the main focuses of this article. ---------------------------------------------------------------------- Middle-resolution categories based on 53-comma system ---------------------------------------------------------------------- 53-EDO type commas cents symbol e-based type 53-commas cents ---------------------------------------------------------------------- 12-comma 1 22.64 F 17-comma 0.96 21.68 ...................................................................... demi-limma 2 45.28 D -------- ----- ----- ...................................................................... reduced limma 3 67.92 E 12-diesis 2.44 55.28 ---------------------------------------------------------------------- limma 4 90.57 B limma 3.40 76.97 ---------------------------------------------------------------------- apotome 5 113.21 S limma + comma 4.36 98.65 ...................................................................... small neut 2nd 6 135.85 Js apotome 5.84 132.25 ...................................................................... large neut 2nd 7 158.49 Jk dim 3rd 6.80 153.93 ...................................................................... dim 3rd 8 181.13 K reduced tone 8.28 187.53 ---------------------------------------------------------------------- tone 9 203.77 T tone 9.24 209.21 ...................................................................... large tone 10 226.42 T10 large tone 10.20 230.90 ---------------------------------------------------------------------- hemifourth 11 249.06 A11 ---------- ----- ------ ...................................................................... small min 3rd 12 271.70 A12 small min 3rd 11.68 264.50 ...................................................................... min 3rd 13 294.34 A13 min 3rd 12.64 286.18 ---------------------------------------------------------------------- A note is in order regarding the 2-comma interval, here carrying the symbol "D" -- for "demi-limma," or for the "diaschisma" of Philolaus and Boethius equal to half of a limma, not to be confused with the smaller interval of 2048:2025 (19.55 cents) also called a diaschisma in later theory. One could also call this in English a "half-limma." The term "hemifourth" may likewise fit an interval or around 11 commas, about midway between 8:7 and 7:6, and equal in 53-EDO precisely to half of the 22-comma fourth. Following a strategy that could appy to some other temperaments also, e-based categories are mapped based on functional equivalence with the Pythagorean or 53-EDO types rather than on a simple mathematical rounding to the nearest 53-comma. Thus "E" in either system is the best representation of 28:27, and "B" the regular limma. When maqamat or dastgah-ha calling for regular Pythagorean or septimal flavors are mapped from one system to the other, this approach may achieve "least astonishment" -- rather as the Sagittal system of staff notation seeks to do with considerable success. For the sake of completeness, I should add that the above table includes two equivalent categories present at rare and remote locations in a 24-note version of the regular e-based system: S at 98.65 cents, and K at 187.53 cents. While not especially accurate approximations of their schismatic Pythagorean or 53-EDO counterparts, these intervals can be used to produce a variation on Safi al-Din's Rast: Rast T Rast |--------------------|........|-------------------| Bb C C#* Eb F G G#* Bb 0 209 397 495 705 914 1101 1200 209 188 98 209 209 188 98 T K S T T K S This tuning, with some intervals comparable to meantone or 12-EDO, is certainly the exception rather than the rule for the 24-note system. However, exploring some of the remote corners of a system may give us a better understanding of the whole. With many thanks, Margo Schulter mschulter@calweb.com