MET-24: A Milder Extended Temperament

During late 2001-2002 it was my privilege to collaborate with George Secor on musical projects including his revisiting of his 1978 17-tone Well-Temperament (17-WT). Our articles growing out of this collaboration appear in Xenharmonikon 18 (2006). Here I would like to chronicle my own process of developing and optimizing a tuning genre called parapyth that represents a synthesis between 17-WT and another system to which George Secor introduced me, called the High Tolerance Temperament in its 29-note form which he developed for the Generalized Keyboard Scalatron in 1975 (29-HTT).

Originally I was asked to write about George Secor’s approach to optimization. Beyond saying that it involves creative use of irregular systems, a theme he ably develops in his XH 18 articles, I am not in the best position to speak for him, and so will explain my own approach to melodic optimization and a 24-note “near-superparticular” tuning based on ratios of primes 2-3-7-11-13. This is in contrast to as well as in emulation of 29-HTT, designed to realize near-just ogdoads of 4:5:6:7:9:11:13:15. It turned out that a 21-note subset of HTT-29 provided me with the tools I needed for my own musical agenda and project: much of my “tuning design” for what I term the parapyth genre of tuning was reverse-engineered from HTT-29.

1. Technology and stylistic tradition as shaping factors

Comparing an optimized parapyth tuning, the Milder Extended Temperament for 24 notes (MET-24), with 29-HTT may suggest how technical means and stylistic orientation can influence the evolution of tuning systems.

Secor’s 29-HTT was intended to support very complex isoharmonic chords with near-just accuracy on a generalized keyboard where the expectation and limitations of the 12-note Halberstadt keyboard would be irrelevant. Indeed the tuning was meant to illustrate the potentials of the new (in 1975) Generalized Keyboard Scalatron.

In contrast, MET-24 and it’s parapyth predecessors starting with Peppermint in 2002 were designed for an array of two 12-note Halberstadt keyboards “regularized” to have the same step and interval patterns, based on 12-note chains (Eb-G#, likely a popular 14th-century tuning) at an arbitrary spacing apart. In Peppermint, two 12-note chains of Wilson/Pepper fifths at 704.096 cents are spaced 58.680 cents apart to achieve pure 7/6 thirds. In MET-24 as ideally realized in 2048-ed2, 12-note chains of fifths at 703.711 cents (approximated on a 1024-ed2 synthesizer, borrowing a concept from Neidhardt, by alternating fifths at 703.125 and 704.297 cents) are spaced at 57.422 cents. By comparison, 29-HTT has 21 of its notes in two chains of regular fifths at 703.579 cents, spaced at 58.090 cents.

Stylistic orientations also played a vital role in this divergence of intonational designs, as the XH 18 articles may illustrate. Thus Secor started developing progressions for his 17-WT based on 18th-century tonal models, which might seem a more radically daring project: to emulate triadic harmony in a tuning system without triads based on ratios of 5! My familiarity with 14th-century style led me to start there, a less radical approach: I suspect that my mentor was doing what I was, taking the familiar as a point of departure. I must say that George Secor’s approach resulted in what he termed jamais vu or the “never seen” before, in contrast to the deja vu impression of “seen again.” Thus his sylistic location had radical potentials, whereas I felt that I was retuning familiar progressions or passages of a composer such as Guillaume de Machaut (1300-1377), and producing things certainly heard before, if only by medievalists like myself.

Another factor shaping the evolution of my tuning systems 2001-2011 was my desire for compatibility with medieval European norms. The authentic period practice on fixed-pitched instruments was evidently Pythagorean tuning where fifths, fourths, and tones and minor sevenths are at pure ratios of 3:2: 4:3: 9:8, and 16:9. Slightly tempering each fifth wide would better approximate this ideal, albeit with subtle compromises, then having some fifths pure and others wide by almost five cents, comparable in quality to a regular meantone or the meantone-like portion of an 18th-century well-temperament.

In addition to medieval European music, and especially the intonational subtleties described by Marchetto (Marchettus) of Padua in his Lucidarium of 1318, I was interested in the genera of Near Eastern music, and to a lesser degree in Ancient Greek music. John Chalmers admirably covers this field in Divisions of the Tetrachord, and the kind of melodic optimization used in parapyth tunings may fit a Chalmersian focus on melodic genera, or ajnas as they are known in the Arabic plural from Greek genus to the Arabic jins (and plural ajnas): thus the axiom “Follow the ajnas!” in Near Eastern music.

Themes that George discussed with me about the development of tuning systems were “cross-pollination” and “clash” of ideas. Here I might prefer to speak of divergence rather than “clash.” This is the fertile potential of musical and intonational ideals to take on new shapes as technologies and stylistic contexts change.

2. Secor’s 29-HTT as prototype for parapyth tunings: three facets

Although I have emphasized that George Secor may have developed 29-HTT as a generalized “near-just” temperament for the Generalized Keyboard Scalatron and similar instruments, as opposed to the highly specialized “near-just” focus of parapyth tunings on medieval European and Near Eastern music as implemented at least in the original conception on two standard 12-note keyboards, indeed 29-HTT provides a prototype for parapyth to the point where my modifications might be considered matters of fine-tuning in various senses.

In exploring the prototypical nature of 29-HTT for parapyth, I would like to focus on three features, the second of which would be relevant to a vast range of xenharmonic tuning systems, but the first and third are more particular to parapyth.

2.1. The Secorian comma (28672/28431) and Safi al-Din al-Urmawi’s 17-note linear tuning

A feature prototypical to 29-HTT and integral to the definition of its 703.579 cent fifth, also of paramount importance to parapyth, is the tuning of an augmented second or +9 fifths (e.g. F-G#) at a just 63/52 (9/8 x 14/13), larger than +9 pure 3:2 fifths at 19683/16384 by the Secorian comma or 28672/28431. Thus each fifth at 703.579 cents is tempered wide by 1/9 of this comma.

The Secorian comma may be demonstrated by thinking of a tuning with a just 13/11 minor third and a 14/11 major third, respectively 352/351 smaller than the Pythagorean 32/27 and 896/891 wider than the Pythagorean 81/64.

The chromatic semitone or apotome between 13/11 and 14/11 would be 14/13 (128.298 cents), 28672/28431 or (896/891 x 352/351) larger than the Pythagorean chromatic semitone or apotome between 32/27 and 81/64 or 2187/2048 (113.685 cents).

The result is that each chromatic semitone in 29-HTT approximates a 14:13 small middle second, while each diminished third or -10 fifths (e.g. C#-Eb) approximates a large middle second at 11:10 (165.004 cents).

Interestingly, Secor’s generator for a just 63/52 from +9 fifths, a fifth at 703.579 cents, is very close to 703.597 cents for a just 13/11 minor third from -3 fifths. The reason is that in Secor’s tuning, the Secorian comma at 28672/28431 is equal to 352/351 x 896/891, which may be further broken down to 352/351 x 352/351 x 364/363 (4.925 + 4.925 + 4.763 cents), with 352/351 minutely larger than 364/363 by a small factor called the harmonisma in honor of the harmoniai of Kathleen Schlesinger, 10648/10647 (0.163 cents). 104/63 (e.g. G#-F, 867.30 cents) is smaller by a hamonisma than three just 13/11 thirds (e.g. G#-B-D-F) at 13/11 (2197:1331 at 867.792 cents). Thus each minor third at 289.264 cents is narrow of 13:11 (289.210 cents) by 1/3 harmonisma (0.054 cents).

A result, although possibly not a designed feature of 29-HTT, is that one can play Near Eastern modalities calling for steps of 14:13 and 11:10 within a 12-note chain of fifths. Safi al-Din al-Urmawi (c. 1216-1294) proposed a 17-note linear tuning for the ‘oud based on Pythagorean intonation, but this had the problem from the viewpoint of everyday practice of making the small middle steps too small at the Pythagorean apotome of 2187/2048 or a 32805/32768 schisma larger than 16/15, and large middle steps too large at a Pythagorean diminished third or 65536/59049 or a schisma smaller than 10/9, where 14/13 and 11/10 would be more apt steps. Secor’s 29-HTT solves this problem, a solution which may be peripheral to the main purposes of 29-HTT but is central to parapyth.

More generally, tempering the fifth by an exact fraction of the Secorian comma produces an elegant scheme where errors from JI are symmetrical. Secor’s 1/9-comma, as in 29-HTT and more recent members of the High Tolerance Temperament (HTT) family ranging in size from 17 to 41 notes to my best knowledge, is one example, using a fifth size of 703.579 cents. Another “sweet spot” mentioned by Gene Ward Smith not too far from MET-24 (703.711 cents) is 1/8 comma at 703.782 cents, and another possibility very close to a tuning by Smith in 271-ed2 is 1/7-comma at 704.043 cents. The characteristic just interval for these tunings are 63/52 for 1/9-comma; 21/13 for 1/8-comma; and 14/13 for 1/7-comma. With 1/9-comma as in Secor’s 29-HTT, and 1/8 comma, spacing two chains of fifths to set 7/4 pure seems optimal, while 1/7-comma might call for a delicate compromise between setting 7/4 and 7/6 just. One mathematically simplified form of 1/7-comma temperament is Simplemint with a fifth at an even 704 cents, and two 12-note chains spaced at an even 58 cents apart. This sets 7/6 (266.871 cents) at an even 266 cents, and 7/4 (968.826 cents) at an even 970 cents.

A noteworthy feature of 29-HTT is the major sixth at a near-just 22/13, so that the spacing required for a just 7/4 is only greater by a third part of 10648/10647 than the 91/88 (58.036 cent) difference between 22/13 and 7/4. Secor’s spacing of 58.090 cents obtains a just 7/4.

2.2. Comma pairs and choices

Another feature that George Secor himself pointed out to me was two notes at a comma apart, permitting for example a choice between fourths of 4/3 and 21/16 above a note, and suspensions, for example, of 6:8:9 and 16:21:24. I found this feature most impressive.

Of course, the virtues of comma alternatives can apply to many xenharmonic tuning systems, and are a powerful refutation to overreliance on equal temperaments, where one must have enough notes, say around 50 in an octave, to have an equal tuning step as small as the comma distinction one wishes to observe.

In other of George Secor’s tuning systems, diversity is achieved through irregularity, as indeed it is in 29-HTT as a complete system. Another ingredient of diversity also present in a completely regular tuning such as parapyth is notes forming pairs at a comma — 24.023 cents in MET-24 — apart.

2.3. A rich variety of middle or Zalzalian steps

Masur Zalzal (?-791), an ‘oudist of Baghdad, is famed for adding the wusta Zalzal or middle third finger fret to the instrument. I recall that George Secor mentioned Zalzal to me in connection with a discussion about the intonation of cadences in French-influenced music from the Court of Cyprus in the 14th century, the name Zalzal being familiar to him from Harry Partch.

As reasonably defined by the Persian writer Hormuz Farhat, the range of a middle or Zalzalian second is around 125-170 cents, which would include the superparticular ratios 14/13 (128.398 cents); 13/12 (138.573 cents); 12/11 (150.637 cents); and 11/10 (165.004 cents). All these ratios are represented in 29-HTT, and by definition in parapyth tunings.

In 29-HTT, each superparticular Zalzalian second has a near-just representation and mapping, that is derived from a given number of fifths along a given chain, possibly plus or minus the spacing between chains of fifths. In 29-HTT, the regular fifth is 703.57869 cents, and the spacing 58.08982 cents.

We get 14/13 and 11/10 from within a single chain of fifths:

14/13 (+7,0) at a tempered 125.051 cents, just value 128.298 cents, or 3.247 cents narrow. This is the maximum inaccuracy in a regular parapyth subset of 29-HTT, equal to about twice the temperament of the fifth. Here the Cartesian or ordered pair notation (+7,0) tells us that 14/13 is mapped up seven fifths (or by a chromatic semitone, e.g. C-C#), the zero indicates that we do not move from the original or lower of the two 12-note chains. In other words, the interval is found within the original chain of fifths.

11/10 (-10,0) at a tempered 164.213 cents, just value 165.004 cents, is more accurate, narrow by only 0.791 cents. Secor’s moderation in tempering the regular 29-HTT fifth facilitates a near-just 11/10 large Zalzalian second as well as more accurate tunings for 3/2, 4/3, 9/8 and 16/9, the basic Pythagorean intervals.

The central Zalzalian seconds at 12/11 and 13/12 involve the spacing generator of 58.090 cents and thus notes from both chains of fifths:

12/11 (+2,-1) at a tempered 149.068 cents is equal to the tone or major second at 207.157 cents less the spacing at 58.090 cents, as compared to the just value of 150.637 cents, and is narrow by 1.569 cents.

13/12 (-5,+1) at a tempered 140.196 cents, from an 82.107 cent diatonic semitone (-5 fifths or 5 fifths down) plus the 58.090 cent spacing, is 1.624 cents wide of the just value of 138.573 cents, comparable in accuracy to the fifth. As it happens, 13/9 and 18/13 are virtually just.

Thus all four superparticular seconds are distinctly represented in 29-HTT, a “design spec” of parapyth tunings generally as Scott Dakota puts it.

3. George Secor’s 29-HTT as the parapyth prototype and minimax solution (2/1, 703.579, 58.090)

Not only is 29-HTT a prototype for parapyth systems, meaning the earliest model (1975) of which I know for all subsequent parapyth tunings, but George Secor’s tuning is a superb implementation of parapyth, which I term the minimax solution for two reasons.

First, it is the solution permitting the least or minimal maximum error from JI among the supported intervals, those of interest for parapyth based on primes 2-3-7-11-13. This is without any ifs, ands, or buts, as for 9/8, also within Secor’s tolerance of twice the temperament (1.624 cents) of the fifth, or 3.247 cents. Some higher-odd ratios also fall within this tolerance, e.g. 21/16 and 63/32, which La Monte Young and others have used in a context based on primes 2-3-7.

The second sense in which Secor’s 29-HTT is a minimal solution is that it departs as little as possible from the ideal of medieval Pythagorean intonation, rendering the basic Pythagorean ratios of 3:2, 4:3, 9:8, and 16:9 as purely as possible while realizing the other supported intervals with comparable accuracy.

Jacques Dudon spoke to me in 2010-2011 about the importance of the Pythagorean ideal of pure fifths and fourths in Near Eastern maqam music, and this ideal is also central to medieval European music.

Figure 1 shows a 24-note parapyth tuning based on an expanded 21-note regular subset of 29-HTT.

keyboard layout of Parapyth Tuning
Fig.1 Parapyth tuning based on an expanded 21-note regular subset of 29-HTT.

This derivative parapyth set from 29-HTT, or “24-HTT” as I call it, illustrates the accuracy of George Secor’s tuning, and its mapping of ratios common to all parapyth tunings.

An important point is that while some approaches to temperament theory focus only on commas tempered out or fused, 24-HTT in particular and parapyth in general characteristically observe certain commas.

For example, four commas fused are 352/351; 364/363; their sum, 896/891; and their difference, the intriguing harmonisma at 10648/10647. The four ratios that the spacing of 58.090 cents may represent, 33/32 (e.g. 4/3 vs. 11/8); 91/88, (e.g.22/13 vs. 7/4); 121/117 (e.g. 13/11 vs. 11/9); and 28/27 (e.g. 9/8 vs. 7/6), have differences between each other illustrating these four fused ratios, with 91/88 and 121/117 differing by a harmonisma or 10648/10647; and 33/32 and 28/27 differing by 896/891.

But the observed commas include those appearing only as melodic differences: e.g. 169/168 in C-C#-D^ or 14:13:12, and 144/143 in D^- E-F or 11:12:13; as well as those occurring as direct steps in the tuning, e.g. E^-F representing the 64/63 difference between C-E^ at a tempered 21/16 and C-F at a tempered 4/3.

Thus to borrow one of George Secor’s illustrations he gave me around 2001, it is possible to play a suspension of C-E^-G or 16:21:24, or a milder C-F-G at 6:8:9. This is a feature of parapyth generally, although 24-HTT beautifully realizes it with near-just accuracy. Here “near-just” means not only accuracy in tuning ratios, but sensitivity in preserving melodic as well as vertical nuances.

The various fifth generators and spacings of other parapyth schemes may be compared to the shadings of meantone. Just as Zarlino’s 2/7-comma meantone of 1558 is not only the first mathematically defined meantone but a superb example of the genre, so Secor’s 29-HTT includes within it a not only prototypical but superb parapyth.

4. From 29-HTT to MET-24: optimizing superparticular middle or Zalzalian seconds

MET-24 is a tuning that does not differ dramatically from George Secor’s 21-note parapyth subset within 29-HTT, or a 24-HTT system derived from that (2/1, 703.579, 58.890). Designed to optimize superparticular middle or Zalzalian second steps, MET-24 (2/1, 703.711, 57.422) is device-specific in that it was conceived for a synthesizer based in 1024-ed2 or 2048-ed2.

On a hypothetical 2048-ed2 synthesizer, each fifth would be tuned at 1201 steps or 703.711 cents. On a 1024-ed2 synthesizer, fifths at 600 steps or 703.125 cents alternate with fifths at 601 steps or 704.297 cents, with the first and last fifths in an 11-fifth or 12-note chain, Eb-Bb and C#-G#, being the smaller and more accurate size. The 2048-ed2 version has the advantage of being more accurate for ratios such as 13/11 and 7/4, while the 1024-ed2 version has six near-just fifths in each 12-note chain.

There is some humor in the name “Milder Extended Temperament” (MET) that I gave this tuning when I devised it in 2011. Originally in 2002, I had started exploring parapyth with Keenan Pepper’s noble fifth linear tuning of 2000, using two 12-note chains of fifths spaced for some pure ratios of 7/6 (2/1, 704.096, 58.680). This linear tuning was proposed by Pepper as a counterpart to Kornerup’s golden meantone, as it happens using a fifth identical to a horogram of Erv Wilson. In 2010, I narrowed the fifth to 703.893 cents with a spacing of 57.422 cents in a system called “Ozone” or O3 for short, a play on the name of the Turkish composer and scholar Dr. Ozan Yarman, whose “depletion” on the tuning lists because of his doctoral studies focused me on the question of perfecting a 24-note maqam tuning.

MET-24 was also inspired by Jacques Dudon and his exhortations that maqam tunings should seek to approximate just fifths as closely as possible. In Dudon’s language, an “extended” tuning has fifths wider than a just 3/2. There was some humor in the way I had gone from 704.096 cents to 703.893 cents (for a just 22/21 diatonic semitone) to 703.711 cents, when George Secor’s yet “milder” fifth at 703.579 cents in 29-HTT had marked my introduction to the parapyth concept.

If 29-HTT marks the parapyth minimax solution for all ratios of primes 2-3-7-11-13, MET-24 is close to the minimax solution for middle or Zalzalian seconds:

14/13               125.977           -2.312 cents

13/12               138.867           +0.295 cents

12/11               150.000.          -0.637 cents

11/10               162.891.          -2.114 cents.

As it happens, at around 703.723 cents 14/13 would as narrow as 11/10. Thus the MET-24 fifth at 703.711 cents may be seen as an approximation of this fifth in 1024/2048-ed2, while it may be also be seen as approximating the elegant 1/8 Secorian comma (28672/28431) tuning spaced like 29-HTT for a pure 7/4 (2/1, 703.782, 57.461).

Note that in all these tuning specifications, the pure 2/1 octave is specified as a generator along with the fifth and the spacing, since some schools of modern temperament regard the octave as an interval open to temperament.

In MET-24, the spacing of 57.422 cents (49 tuning units of 1024-ed2, or 98 of 2048-ed2) sets 7/6 (264.844 cents, -2.027 cents from just) almost identically impure to 13/8 (842.578 cents, +2.050 cents).

4.1. An overview of MET-24

Figure 2 shows the keyboard design for MET-24 on two Halberstadt keyboards. Unlike 29-HTT, MET-24 does not have the subtleties of the interplay of intervals impure by 1/3 of a 10648/10647 (0.163 cents) harmonisma or 0.054 cents. Thus in 29-HTT, 104/63 (-9 fifths) or 63/52 (+9 fifths) is just, from the fifth at 703.579 cents, while a fifth at 1/9 of a harmonisma larger or 703.597 cents would produce a pure 13/11 from -3 fifths.

In this tuning, the spacing at 58.090 cents is equal to 91/88 plus 1/3 harmonisma, or 121/117 (58.198 cents) less 2/3 harmonisma. Thus while 13/11 is 1/3 harmonisma wide, 11/9, equal in its just size to 13/11 x 121/117, is 1/3 harmonisma narrow. The same relationship applies between 63/44 (+6 fifths) and 77/52 (+6, +1) or 6 fifths up plus spacing; 77/52 is equal to a 3/2 fifth less a 78/77 comma or 7/4 less 13/11.

MET-24, in contrast, is a tuning of delicately calculated compromises. The general approach is to set the fifth size so that 11/10 (+10 fifths) and 14/13 (+7 fifths) are about equally impure; and to set the spacing so that 7/6 (+2, +1) or 2 fifths up plus spacing and 13/8 (-4, +1) or 4 fifths down plus spacing are about equally impure. It happens that with pure 2/1 octaves a fifth at 1201 tuning units of 2048 and a spacing of 98 units (2/1,703.711, 57.422) approximates these results.

Fig. 2 Keyboard design for MET-24

Note that the simplest virtually just ratio is 77/52, in contrast to 29-HTT which has 7/4, 63/52, and 13/9 just, as well as 13/11 and 11/9 as well as 63/44 and 77/52 impure by only 1/3 harmonisma or 0.054 cents.

Both systems in 24-note parapyth form have a few ratios of 5, while 29-HTT includes some just locations for 5/4 and 7/4 as part of the six 4:5:6:7:9:11:13:15 ogdoads which the irregular design features.

5. Divisions of the Tetrachord or Multiple Division Without Equal Temperament

The alternative titles of this section are in honor of John Chalmers’ monument of scholarship Divisions of the Tetrachord (Frog Peak, 1993) and Chapter 7 of M. Joel Mandelbaum’s landmark of xenharmonic thought, Multiple Division of the Octave and the Tonal Resources of 19-Tone Temperament (Indiana University, PHD study, 1961), “Multiple Division Without Equal Temperament” (pp. 226-245).

The use of tetrachords with subtle differences in melodic step sizes is another xenharmonic path to a focus on “maximal evenness” or on scale sets or gamuts with only two sizes or intervals which form Moments of Symmetry (MOS) sets. The focus of parapyth systems is on medieval European and Near Eastern styles based on ratios other than those derived from prime 5, the mainstay of European music from the later 15th to the 19th centuries (or from the middle and later Renaissance through the Romantic Era), and so constitutes at once an historical offshoot and a “revolution” replacing tonality with modality and familiar meantone sonorities with the less familiar Gothic polyphony of the 13th-14th centuries, and the subtly varied Zalzalian steps and intervals as used in over a millennium of Near Eastern maqam and dastgah music.

The essence of varied Zalzalian steps as implemented in parapyth systems may be illustrated by a harmonic series I recall first posting to the Alternative Tuning List in July 2002, and named Turquoise (Figure 3) in 2010 or 2011, with discussion with Ozan Yarman and Jacques Dudon as a catalyst to a focus on this series.

Turquoise tuning
Fig. 3 Turquoise

Here I have used the letter J for the Just value of a melodic ratio or step in cents; H for its value in 29-HTT; and M for its value in MET-24. The lowercase “h” in 33h, 36h, 39h, 42h, and 44h is precautionary, to warn that these numbers measure a harmonic series of ratios, rather than string or monochord lengths.

It is notable that 29-HTT is more accurate with ratios for thirds such as 13/11 and also 7/6 as well as the pure 7/4 or 8/7; while MET-24 is more accurate for the superparticular Zalzalian steps 12:11, 13:12, 14:13, and also the 22:21 diatonic semitone.

The naming of this 33:36:39:42:44 series as Turquoise was a play on a tuning system called Bleu with generators of 140 cents, one variation on Jacques Dudon’s Tsaharuk system based on the division of a just or slightly wide fifth into five equal parts. Tsharuk was based on many generators forming a tuning of around 80 notes or more, as I recall; the idea of the Turquoise series is with a relatively small parapyth tuning to realize accurately a subtle variety of superparticular steps.

The Turquoise series also includes within itself a collection of tetrachords both historical and new; in 2002 I was aware through the writings of Owen Wright of Qutb al-Din’s (1236-1311) tuning of Hijaz as 12:11-7:6-22:21 or 1/1-12/11-14/11-4/3, i.e. 33:36:42:44 in terms of a subset of the Turquoise series.

George Secor laid the foundation for my exploration in this direction by calling my attention to the 12:13:14 division of a 7:6 minor third, which led me to the 11:12:13 division of the 13:11 minor third, both included in the Turquoise series (as 36:39:42 and 33:36:39). The following section looks at some tetrachordal subsets of Turquoise.

5.1. Tetrachords contained within the Turquoise series or its extension

The Turquoise series, 33:36:39:42:44, includes within itself three tetrachords of interest, two old and one new.

The first is William Lyman Young’s “Exquisite 3/4-tone Hellenic Lyre”, 33:36:39:44 or 12:11-13:12-44:39, i.e. 1/1-12/11-13/11-4/3 (see Chalmers, Divisions of the Tetrachord, 1993, p. 189, tetrachord #443; and scale archive for Manuel Op de Coul’s application Scala, young-wt.scl). This tetrachord features the 11:12:13 division, apparently a modern development.

In view of the frequent use of 12:11 and 13:12 in medieval Near Eastern theory, one might ask why 11:12:13 did not become a common division. My guess is that the 13/11 third, and its 11:12:13 or 13:12:11 division, did not take a prominent role in this theory because of the prevalent Pythagorean division of a 4:3 fourth into 9:8 tone and 32:27 minor third rather than a 44:39 tone larger by 352:351 (4.925 cents) and a 13:11 third smaller by 352:351 than the standard 32:27.

In Young’s modern tetrachord, which nicely fits the tetrachord above the final of a contemporary Arab maqam Huseyni, we have Figure 4.

Fig. 4 Young’s tetrachord compared with 29-HTT and MET-24

The second tetrachord (Figure 5) is a permutation of Ibn Sina’s “very noble genus” most simply expressed as a monochord division of 16:14:13:12 or 8:7-14:13-13:12, i.e. 1/1-8/7-16/13-4/3. Here the 12:13:14 division of a 7/6 minor third stands out.

Fig. 5 Ibn Sina’s “very noble genus” tetrachord compared with 29-HTT and MET-24

Note the minor thirds of 13/11 and 7/6 are closer to just in 29-HTT (or the 24-note parapyth set derived from it), while the melodic steps 12:11, 13:12, and 14:13 are closer to just in MET-24.

The third tetrachord (Figure 6) is a chromatic from Qutb al-Din al-Shirazi (c. 1236-1311), and the oldest known form of chromatic Hijaz. Here a 7:6 step is undivided, this step being larger than half a 4:3 fourth (approximately 248 cents). As Chalmers notes (1993, p. 15) an author named Gevaert in 1875 proposed the term “neochromatic” for this Hijaz type of tetrachord where the large interval is the middle of the three melodic steps, in contrast to the chromatic of Classic Greek theory where the large step tends to be the last, as in Ptolemy’s Intense Chromatic, of which Qutb al-Din is aware his Hijaz tetrachord is a permutation (21:22:24:28 or 22:21-22:11-7:6, i.e. 1/2-22/21-8/7-4/3 in its original form):

Fig. 6 Chromatic tetrachord from Qutb al-Din al-Shirazi compared with 29-HTT and MET-24

Both tempered systems, a parapyth from 29-HTT and MET-24, feature major seconds closer to Young’s 44:39 than to 9:8, but still closer to a just 9:8 than 12n-ed2. In George Secor’s view, as he expressed it to me in 2001-2002, this accuracy for 9:8 should be regarded as one of the marks of a “near-just” tuning.

Thus if circulation in 12 notes or enharmonic equivalence is not required or assumed, parapyth should be an attractive tuning for 20th-century quartal/quintal harmony. At the same time, stimulating variations are available like 9:12:16:21, e.g. G3-C4-F4-A^4, a variant “fourth chord” with an outer 7/3 minor seventh and an upper 21/16 narrow fourth.

My own interest was a tuning that would approximate medieval European Pythagorean intonation, and at the same time support medieval and later Near Eastern modalities with their varied Zalzalian steps and intervals. But such a tuning system supports also certain aspects on contemporary 20th-century European music on neomodal lines not dependent on prime 5, as with perhaps certain aspects of Bartok. Given indiscriminate complaints that 12-ed2 is “out of tune,” it is well to remember that near-just fifths and fourths are its mainstay, so that parapyth tuning has been humorously defined as “tempering fifths about as much as in 12n-ed2, but in the more interesting direction.” Of course, this “more interesting” direction is the wrong one if one seeks the circulation of 12-ed2 coupled with its imprecise but acceptable implementation of meantone. Thus parapyth is guaranteed not to replace meantone or 12-ed2 for that matter, and may represent a contrarian voice to trends that seek out one tuning system to solve all musical problems and address all stylistic needs.

Note that none of Ibn Sina’s four tetrachords is a permutation of another — with the identical three melodic step sizes rearranged in order. Indeed there are eight sizes of middle second steps, pairs of which “resemble” each other in size. We can group these pairs into four categories shown in Figures 7 and 8.

Fig.7 Ibn Sina’s four tetrachords
Fig. 8 Four Zalzalian thirds

By comparison, in the 24-note parapyth tuning of either MET-24 or an expanded subset of 29-HTT which I here term 24-HTT, shadings 1 and 4 are precise permutations, as likewise shadings 2 and 3 (Figure 9).

Fig. 9 Shadings in MET-24 and HTT-24

Thus eight JI shadings are reduced to four tempered shadings, a form of “color blindness.” The precise equation in parapyth tuning is that a tone at a tempered 9:8 tone plus a large Zalzalian second is equivalent to a tempered 4:3 fourth less a small Zalzalian second; and similarly that a tempered 9:8 tone plus a large central Zalzalian second is equivalent to a tempered 4:3 fourth less a smaller central Zalzalian second.

In Ibn Sina’s JI, these are “resemblances” subtly differentiated by the 2080:2079 and 352:351 commas; thus 9:8 x 11:10 ~= 4:3 / 14:13, with the 2080:2079 marking the difference between 99/80 and 26/21; likewise 9:8 x 12:11~=4:3 / 13:12, with the 352:351 marking the difference between 27/22 and 16/13. In parapyth, these commas are fused, so that there is an exact equation.

For a relatively small set of 24 notes, the availability of four shadings of near-just Zalzalian intervals may be noteworthy, but a subtle compromise is involved. Note as the note spellings for the tetrachord examples illustrate, that large and small Zalzalian intervals in parapyth result from notes within the same chain of fifths, while central Zalzalian intervals, large or small, result from mixing notes in different chains. These mappings are the same in MET-24 and 24-HTT.

11:10 or 208:189 (-10,0). 99:80 or 26:21 (-8,0)

12:11 or 128:117 (+2,-1). 27:22 or. 16:13 (+4,-1)

13:12 or 88:81. (-5,+1). 39:32 or 11:9 (-3, +1)

14:13 or 320:297. (+7,0). 63:52 or 40:33 (+9,0)

5.3. Miscellaneous monochord approximations: the division of the tone and the diesis in MET-24

This section looks at some fine points regarding string length or monochord divisions as they serve as ex post facto explanations for some pattern in the MET-24 tuning system. That is, these monochord divisions were not part of the tuning design, but may lend an interesting perspective.

One aspect of the division of the tone might win a competition for the least likely interpretation of Marchettus of Padua’s (or Marchetto, or Marcheto) division of the tone described in his Lucidarium (1318) having usual Pythagorean proportions of 9:8 in a scheme where what we would term the chromatic semitone (e.g. C-C#) has “three parts,” while the limma or diatonic semitone has “two parts.” Note that we are dividing string lengths or dealing, so to speak, with wavelength rather than frequency ratios, so that smaller numbers represent higher notes (Figure 10).

Fig. 10 Monochord division of tone into chromatic and diatonic semitones, MET-24 approximation

Thus MET-24 divides a tone at approximately 44:39 into a diatonic semitone or limma at a near-just superparticular ratio of 22:21 (44:42) and a chromatic semitone at a near-just 14:13 (42:39). A delicate compromise is obtaining a size of fifth (703.711 cents) and thus tone (207.422 cents) not far from either 44:39 or a just 9:8 (203.910 cents) for smooth quartal/quintal sonorities such as 6:8:9 (e.g. G-C-D), 8:9:12 (e.g. D-E-A), 4:6:9 (e.g. G-D-A), and 9:12:16 (e.g. G-C-F) regarded as relatively concordant or blending in many 13th-century and some 14th-century European styles (as with 4:6:9 in the style of Guillaume de Machaut, c. 1300-1377), and as fully concordant in some 20th-century classical styles. In MET-24, 9:8 is wide by 3.512 cents, while 29-HTT is more accurate, keeping 9:8 as well as a range of other ratios within its tolerance of twice the temperament of the fifth (at 703.579 cents or 1.624 cents wide), i.e. 3.247 cents. Either tuning has a more accurate 9:8 than 12n-ed2 at 200.000 cents, or 3.910 cents narrow.

Jay Rahn,“Practical Aspects of Marchetto’s Tuning” explains Marchetto’s division as a variant on Phololaus’s division of the tone as recounted by Boethius into “four diaschismata and a comma” where a diaschisma is a theoretical half of a Pythagorean diatonic semitone or limma at 256:243 (90.225 cents), and thus around 45.112 cents in size. Dividing a ratio like 256:243 into precisely identically sized halves might be something of a “what if” problem, since such a strictly equal division cannot be brought about with the rational ratios to which the music theory of Boethius was largely limited, but Marchetto in Rahn’s view brings about such a division in practice, using subtly unequal ratios and manageably small numbers and monochord lengths (Figure 11).

Fig. 11 Monochord division of tone, possible interpretation of Marchetto of Padua

Thus Rahn’s interpretation of Marchetto’s monochord division is remarkably close to Philolaus’s division of the tone into “four diaschismata” (shown by “D” in the diagram, diaschismata being the Greek plural of diaschismata, a term in ancient theory referring to half a limma) at an identical 45.112 cents each and a Pythagorean comma at 531441:524288 (23.460 cents). As it happens, placing the comma in the middle of the division and diagram at 77-76 gives it a size slightly smaller than a Pythagorean comma, but almost identical to the Holderian or Holdrian comma named after the author John Holder and equal to 1/53 of an octave (22.641 cents), a standard measure in Turkish and Syrian theory. The two limmata (a Greek plural for limma) are 81:77 or 87.676 cents and 76:72 (19:18) at 93.603 cents, as compared with the 256:243 limma at 90.225 cents.

Thus a follower of Boethius might cite the division of Marchetto to demonstrate that such divisions by integer ratios will inevitably be unequal, while their close approximation to an equal division is also noteworthy.

The 12-note diesis of MET-24 appears only as a difference between interval sizes, not as a direct interval, because it represents an interval of +12 fifths (e.g. Eb-D#), and there are only 11 fifths in a 12-note chain of fifths (here Eb-G#). But it would occur in a MET-34 system with two 17-note chains (say Gb-A#). The size of a tone (+2, 0) near 44:39, and of a diminished third near 11:10 (-10, 0), suggest this monochord interpretation (Figure 12).

Fig. 12 Enharmonic monochord division of tone as approximated in MET-24

Perhaps the divisions illustrate a theme of MET-24: the association of the approximated just ratio of 44:39 for a tone with superparticular ratios, or its division into superparticular ratios. The same tempered interval can also represent 9:8, a result of the fusing of the 352:351 comma.

6. Maqam music: intonational nuances and adornments

One theme of George Secor’s approach to optimization is what might be termed after Mark Lindsey’s “modal color,” often associated with circulating irregular temperaments such as Secor’s 17-tone well-temperament or 17-WT, but also found in some regular schemes, where a given mode is available at different locations in different intonational shadings or colors. Another, which Secor associated with his 29-HTT tuning, is the availability of alternate versions of a note at a comma apart.

Both facets play a role in the realization of maqam music in MET-24. Here a “nuance” means the choice of two positions for a note at a comma apart: while an “adornment” means the availability of an intonation deemed especially apt for a given jins or genus.

6.1. Solving Dudon’s Rast/Bayyati quandary: F^ Rast or B Rast

In a Tuning List post of July 11, 2011, Jacques Dudon shares a charming anecdote of performing in his ecovillage with a folk group called Embryo. As he describes, and as scholars such as Amine Beyhom and Scott Marcus have documented in more detail, there is a widespread understanding among Arab musicians that the third step of Rast is higher than the second step of Bayyati. Dudon’s instrument was tuned properly to place this step as in Bayyati, but not in the higher position for Rast.

In MET-24, a possible solution keeping an apt third step for a moderate Arab shading of Maqam Rast might be as follows (Figure 13).

Fig. 13 Maqam Rast on F^ and Maqam Bayyati on G^

In terms of F^, here the step rast or the step of the gamut that serves as the usual final for Rast, we place the final of Maqam Bayyati or the step dugah/duka (the Persian or Turkish and Arabic forms of the same name, meaning the “second” step of Rast) on G^, but the third step segah/sika as a comma higher in Rast (357.4 cents above rast) than in Bayyati (334.4 cents), the size of this comma being 24.023 cents in MET-24.

This solution offers an adornment in Maqam Nahawand, one common modulation from Maqam Rast. The third step of Nahawand is smaller than a regular minor third at 288.9 cents in MET-24, at 276.0 cents, a stylish distinction in some Arab practices (Figure 14).

Fig. 14 Ascending Maqam Nahawand on F^

Another possibility for solving the Rast/Bayyati quandary is to place step rast, the final of Maqam Rast, on B on the lower keyboard and chain of fifths rather than F^ on the upper chain (Figure 15).

Fig. 15 Maqam Rast on B and Maqam Bayyati on C#

This option results in a higher Rast, with the third step at 370.3 cents or 26/21 in just intonation terms; 357.4 cents in the F^ option might represent the 27/22 of al-Farabi, the 16/13 of Ibn Sina, or the 59/48 of Safi al-Din al-Urmawi. The first option provides a Lebanese “folk Bayyati” of the kind described by Amine Beyhom, with the step above the final of Bayyati at 130 cents, here 126.0 cents, or in just terms 14/13 (128.298 cents). In the second option, we have a Bayyati rather like that described by Scott Marcus among traditional musicians of Egypt, with the step above the final at around 135-145 cents; here 138.9 cents is close to a just 13/12 (138.573 cents).

As it happens, the second option yields a usual modulation to Maqam Nahawand with a regular minor third B-D at 288.9 cents, more a near-just 13/11 (Figure 16).

Fig. 16 Maqam Nahawand on B

What are the main features of MET-24 as a tempered tuning system? The first might be its approximation of what in 2000 I described as “nuanced rational intonation (RI),” seeking fine gradations of simple and complex integer ratios. This nuancing or shading can be vertical, as with thirds in MET-24 at 264.8 cents (~7:6), 276.0 cents (~168:143), and 288.9 cents (~13:11), as well as a few locations at 312.9 cents (~6:5), present as an “unintentional feature” of this tuning system primarily oriented to prime factors of 2-3-7-11-13. A good example of nuancing is the contrast between an augmented fifth like C-G# at 829.7 cents (~21:13, and not too far from Phi at a value of 833.1 cents), and a minor sixth plus spacing like D-Bb^ at 842.6 cents (~13:8).

While this nuancing can allow fine shadings of vertical consonance/assonance/dissonance, in keeping with contemporary music theorist Ludmila Uhelha’s emphasis on “the control of dissonance,” it can also be melodic, as in maqam music, bringing into focus the importance of melodic optimization in MET-24. One of the design criteria is that the superparticular middle seconds all be within 3 cents of just: thus 14:13 (tempered 126.0 cents); 13:12 (tempered 138.867 cents); 12:11 (tempered 150.0 cents); and 11:10 (tempered 162.9 cents). The Turkish composer and theorist Ozan Yarman has eloquently written about this quest for variety and diversity of middle intervals in what is classically termed in Near Eastern writings in reference to middle second steps “the mujannab zone” MET-24 demonstrates what is possible in a relatively small tuning set focusing on the mujannab zone absent of such constraints. The result is a bimusical system for medieval European music and a subset of Near Eastern maqam and dastgah music.

Editor’s appendix

scala version of Secor's 29-HTT
Fig. 17 Cent values for George Secor’s 29 HTT as represented in Scala. Thanks to David Kennan for pointing out that the C 0.000 cents of this Scala file is treated as Bb (992.843 cents) in Schulter’s mode of 29-HTT used in this paper.