Re: A temperament for Maqam (fwd)

> I am pleased to share this very specific 24 notes/octave unequal
> tuning, based on a linear version of Tsaharuk temperament,
> especially designed for Maqam music.

Dear Jacques,

How curious that your 24-note Tsaharuk temperament is very similar to
a tuning I described last December in another forum -- but the
differences show the special logic of Tsaharum in meeting Julien
Jalaleddine Weiss's criteria for Maqam music, some of which do not
apply to my tuning despite the often identical or similar interval
sizes!

</justintonation/topicId_unknown.html#1005>
<http://www.bestII.com/~mschulter/bamm24b_C.scl>

> It contains five classical Rast heptaphones transpositions, plus 3
> Turkish Rast versions, one in between, and one folk style Rast,
> seven Bayyati heptaphones transpositions, two perfect Syntonon
> Diatonon (quartertone related), six Mustaqim heptaphones (from Ibn
> Sina and Safi al-Din tetrachords mentionned by Margo Schulter), and
> probably many other modes.

Here I should ask the ratios for the "folk style Rast." The other
categories are generally clear to me, and nicely sum up some of the
modes in this tuning. Note that my tuning mentioned above, Bamm24b,
does not have any just 5-limit ratios or Syntonon Diatonon tetrachords
or modes.

> It provides two sizes of each of the neutral intervals : seconds,
> thirds, etc., except that within this limited number of notes
> selected from the complete set of 77 notes, they can't be
> experienced with the same tonic.

This raises an important point you discussed in your earlier long post
on Maqam tunings. The vital thing is to have at least two sizes of
neutral intervals, even if only one is available from a given step.

Even in O3, a 24-note system which does have some steps with two sizes
of neutral intervals available at a comma apart, we can't get, for
example, a moderate Rast (357.4 or 358.6 cents) and a moderate
Mustaqim (345.7 or 346.9 cents) above the same note. That would
require a lower and higher segah only 12 cents or so apart. What we
can sometimes get, for example, is a high segah (369.1 cents) for a
Turkish Rast or moderately low segah (345.7 cents) for a typical Arab
Bayyati, a difference of a comma at 23.4 cents.

However, of course, a complete Tsaharuk does provide choices like a
virtually just 72/59 or 59/48 above the same note, which I found in a
94-note version with five generators equal to a pure 3/2.

> The neutral intervals of Rast will be the tonic of Bayyati or
> Mustaqim, and reversely.

This is also a pattern we see in Mohajira or Wilson's Rast-Bayyati
matrix, with a Rast heptatone appearing in a version extended to 10 or
more notes. And likewise an Arab Makam Sikah has the smaller neutral
intervals in contrast to Rast; while Persian Dastgah-e Segah has the
larger neutral intervals, in contrast to Mustaqim (or a modern gushe
such as Shekaste like the old Mustaqim).

> This has been concocted simply with a sequence of 11 schismatic
> fifths, transposed by 28/27.

Curiously, Bamm24b results from extending a variation on a 17-note
tuning for the `oud explained by Cris Forster in his book _Musical
Mathematics_, but may be concocted as two chains of 11 pure fifths at
a 531/512 apart (63.1 cents), or 14337:14336 more than 28/27!

Thus the only primes are 2, 3, and 59 -- while your Tsaharuk also has
5 and 7! The biggest difference may be that Tsaharuk 24 has pure
5-limit thirds, in line with Julien's standard, while in Bamm24b all
usual fifths are pure.

> So 28/27 appears 12 times as the larger step, alternatively with
> five 36/35 and seven septimal commas 64/63, all evenly
> distributed. Their relative size in 171-edo is 9, 7, and 4 steps.

In Bamm24b we have 12 larger steps at 531/512 (63.1 cents), five at
243/236 (50.6 cents), and seven at 131072/129033, smaller than 64/63
by 14337:14336.

> Nine quasi-pure 7/4 appear between the two fifth sequences, which
> can be surprising for a Maqam tuning.

On this point Tsaharuk-24 and Bamm24b are similar, and I agree that a
lot of modern Maqam theory doesn't emphasize septimal intervals. However, it said that in Maqam Ushshaq Masri (a variation on Nahawand
associated with Egypt, somewhat like Persian Nava, with an upper
Bayyati, e.g. re mi fa sol la si-d ut re, with "si-d" a half-flat, it
is sad that the third is a comma lower than usual, which could mean
around 7/6, with the seventh step maybe near 7/4. For a low Nahawand
or Ushshaq Masri, Tsaharuk-24 might be very useful!

Also, I've heard that Maqam Buselik in Turkey may favor a low third
around 7/6, and the 5-limit intervals of Tsaharuk-24 should fit a
Turkish style also.

> But as I said in my first post, an optimal generator would arrive at
> 7:1 in 24 reiterations, while it gives also almost pure fifths,
> which is the first criteria Julien Jalaleddine Weiss and other
> musicians want for Maqam tunings.

Indeed it will give a pure 7/4 and an almost pure 3/2 (very slightly
narrow) and 5/4 (likewise, with -40 generators at 385.290 cents).

> In fact, a pure 7:1 attained in 24 generators creates a slightly
> schismatic tuning of the fifths, that produces also quasi-pure 5/4
> thirds, and this is the idea of this tuning, in this central zone
> near 171-edo.

For this generator, Scala shows 140.368 cents, yielding a fifth around
701.839 cents. What I note is that 24 generators give a pure 7/4 and
-40 a near-pure 5/4. With a fifth generator around 3/2, getting 7/4
pure (-14 fifths) would require slightly extending the fifth, which
would make 5/4 (-8 fifths) less pure.

> The average Tsaharuk generator in the tuning below, five of them
> giving a fifth, is 1.08444952 ~20/171 of octave, itself very close
> to the meta-temperament version of Tsaharuk, 1.084451679092 ; it
> often appears as 64/59 in this rational version of the temperament.

An interesting point is that Bamm24b takes a pure 64/59 and 59/54 as
one of its starting points (using generators of 3/2 and 531/512), but
your Tsaharuk has generators of varying sizes around 3/2^1/5: for
example 64/59 and 243/224, differing by 14337:14336, as well as two
others a bit smaller helping to produce the pure 5-limit thirds.

> You will notice there are neither 11 nor 13 primes in this tuning,
> but only 2, 3, 5 , 7 and 59.

It's curious that in Bamm24b, we have only 2, 3, and 59 -- but with
the same types of approximations you mention next.

> One of the reasons is that Tsaharuk tempers 352/351, and 59 finds
> its place precisely in between (59/48 for example is in the middle
> of 27/22 and 16/13).

Yes, 59 is very close to exactly between 11 and 13. A 59/48, for
example, is larger than 27/22 by 649:648, and smaller than 16/13 by
768:767, respectively 2.670 and 2.256 cents. Early this year I did a
lattice for Bamm24b showing some of these near-equivalences:

<http://www.bestII.com/~mschulter/hexapentadic17.txt>

> I am only giving a rational version here, the more useful for
> fretting applications ; but it expresses a temperament.

The pure thirds, fitting Julien's criteria, do make the tempering
clear.

With Bamm24b, we have basically two chains of 11 pure fifths at a
531/512 apart, almost identical to a 28/27, which could be tuned by
ear, so the only "tempering" is the 14337:14336 difference (0.121
cents). In 1024-ED2, the generators are 599 steps (703.953 cents) and
54 steps (63.281 cents) -- a virtually pure 3/2 I suspect Julien would
accept, and an approximation of 531/512 about 0.200 cents wide.

However, while we have lots of 72/59 and 59/54 thirds for rast-segah
in Bayyati or Mustaqim and Rast (but, as with Tsaharuk-24, not both positions above a single step), and nine virtually pure 7/4's, Julien
would notice that there are no pure ratios of 5. They are not a part
of the design, although some schismatic thirds will occur -- in
contrast to the pure 5/4 and 6/5 thirds of tsaharuk-24, or near-pure
thirds with complete Tsaharuk tunings using 7^1/24, for example.

I also notice that a Tsaharuk-94 with the 7^1/24 generator has some
other near-just intervals like 13/11 and 17/14 -- and lots of places
supporting both Rast with segah near 59/54 and Mustaqim or Bayyati
with segah near 72/59!

Best,

Margo

Dear Tuning List Colleagues,

Please let me express my sense of intellectual delight and artistic
camaraderie in sharing with Jacques Dudon in our independent
discoveries of 24-note JI tunings involving the small schisma of
14337:14336 which defines the difference, for example, between the
septimal ratio of 28/27 (the beloved thirdtone step of Archytas) and
the ratio of 531/512 based on prime 59.

His most intricate and subtle Tsaharuk 24 tuning, and my earlier
Bamm24b published on a smaller tuning group which Jacques had not
seen, are a wonderful illustration of how certain ideas at certain
times seem almost to be "floating in the air," where different people
in different parts of the world can independently recognize and
embrace them. That is exactly what has happened here.

Following his creative path as a lover of Maqam and other musical
traditions, and artfully uniting a musical sensitivity to these
traditions with a fine aural and mathematical craft in designing
differentially coherent (-c) JI tunings as well as most subtle
temperaments, Jacques and such colleagues as Julien Jalaleddine Weiss
along with some of the most renowned practitioners of this tradition,
are setting a new state of the art.

While Jacques developed his Tsaharuk 24 quite unaware of my Bamm24b, I
must add that my Bamm24b evolved in good part out of the discussions
that Jacques and I had had last year in connection with the
magnificent Ethno2 collection and the related topic of his -c series
for Maqam and Persian Dastgah music involving ratios of 59.

Last December, when Cris Forster's book _Musical Mathematics_ gave me
the idea of designing a 24-note JI tuning for the `oud (or keyboard)
expanding on classic `oud tunings of the Islamic or Arabistic
Renaissance of the 8th-15th centuries as measured by the European
calendar, my earlier dialogues with Jacques helped to focus my mind on
ratios such as 72/59 and 59/48 for neutral thirds, leading me to
design a system with two chains of 11 pure fifths at 531/513 (around
63.1 cents) apart. Immediately I recognized that this was extremely
close to the very familiar ratio of 28/27, and quickly used Manuel Op
de Coul's Scala to find that ratio for this small schisma was
14337:14336 (about 0.121 cents).

In contrast to my quest for a 24-note JI superset of a classic `oud
tuning, Jacques has independently designed his Tsaharuk 24 as a
subset, very useful in itself, of a fuller Tsaharuk system, an
amazingly comprehensive tuning or gamut for the qanun which may
advance the state of an art already flourishing with incredible
subtlety when Ibn Sina described some of its tunings and modes about a
millennium ago: the consummate art of Maqam, of which the Dastgah
system of Persian music is also a most fruitful branch.

The independent nature of his conception is shown by the fine
schismatic nuances which Jacques has introduced into Tsaharuk 24,
something quite outside the conception or implementation of the
simpler Bamm24b. Thus, through his deftly chosen subset, one may
sample some of the refinements of a complete Tsaharuk system.

As George Secor has well said, "cross-pollination" is one of the basic
creative processes in music and its intonation, and it is a great
honor to share in the demonstration of this truth with such a
distinguished colleague as Jacques Dudon.

Most appreciatively,

Margo Schulter
mschulter@...