Yahoo Tuning Groups Ultimate Backup tuning Turquoise-17: Another shade of maqam tuning

Hello, all.

In response to the recent discussion about Bleu17, I'd like to offer
Turquoise17, a strictly proper 17-note temperament based on the
division of 4/3 into 12:11-13:12-14:13-22:21 (33:36:39:42:44), and
with a variety of Zalzalian or neutral second steps for maqam or
dastgah music.

! turquoise17.scl
!
Turquoise 17, 1024-ED2: ~33:36:39:42:44 at steps 0, 7, 10
17
!
70.31249
151.17187
208.59375
289.45313
358.59375
416.01563
496.87500
566.01562
646.87500
704.29688
785.15625
854.29687
911.71875
992.57812
1061.71875
1119.14063
2/1

<http://www.bestII.com/~mschulter/turquoise17.scl>

There are 15 usual fifths, six at 703.125 cents and nine at 704.297
cents; these, together with the 2/1 octave and a small thirdtone at
57.422 cents, are the generators. The average size of these 15
"appreciable" fifths (as Scala calls them) is 703.828 cents,

While the intonation of Arab, Turkish, and Kurdish maqam music and
Persian music in the related dastgah tradition is almost infinitely
fluid and flexible, Turquoise17 does offer a variety of step sizes and
shadings for modalities such as Arab Rast and Bayyati or Persian Shur
and Bayat-e Esfahan. Just as the beauty and drama of meantone
chromaticism in European keyboard music around 1600 draws extra power
from the contrast between diatonic and chromatic semitone sizes, so
maqam and dastgah music gain beauty from a subtle contrast of larger
and smaller neutral steps.

As the name "Turquoise" may suggest, there's also a Rast in a bright
Turkish flavor using a septimal tetrachord (1/1-8/7-26/21-4/3 or
21:24:26:28) described by Ibn Sina, the great Persian philosopher and
musician, about a millennium ago. You'll find this at the ~14/11 step:

0 231 369 496 703 935 1073 1200
1/1 8/7 26/21 4/3 3/2 12/7 13/7 2/1

In short, while Bleu tempers out the commas distinguishing the steps
in 11:12:13:14, Turquoise seeks to observe these commas while
providing lots of fifths and fourths reasonably close to just. The two
approaches may be interesting to compare.

Most appreciatively,

Margo Schulter
mschulter@...

🔗hstraub64 <straub@...>

I have started a section on the xenwiki about unequal maqam temperaments, and added links to some of the recent discussions here.
I hope this is alright. Corrections and additions welcome, as usual.

http://xenharmonic.wikispaces.com/Arabic%2C+Turkish%2C+Persian

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> Hello, all.
>
> In response to the recent discussion about Bleu17, I'd like to offer
> Turquoise17, a strictly proper 17-note temperament based on the
> division of 4/3 into 12:11-13:12-14:13-22:21 (33:36:39:42:44), and
> with a variety of Zalzalian or neutral second steps for maqam or
> dastgah music.
>
> ! turquoise17.scl
> !
> Turquoise 17, 1024-ED2: ~33:36:39:42:44 at steps 0, 7, 10
> 17
> !
> 70.31249
> 151.17187
> 208.59375
> 289.45313
> 358.59375
> 416.01563
> 496.87500
> 566.01562
> 646.87500
> 704.29688
> 785.15625
> 854.29687
> 911.71875
> 992.57812
> 1061.71875
> 1119.14063
> 2/1
>
> <http://www.bestII.com/~mschulter/turquoise17.scl>
>
> There are 15 usual fifths, six at 703.125 cents and nine at 704.297
> cents; these, together with the 2/1 octave and a small thirdtone at
> 57.422 cents, are the generators. The average size of these 15
> "appreciable" fifths (as Scala calls them) is 703.828 cents,
>
> While the intonation of Arab, Turkish, and Kurdish maqam music and
> Persian music in the related dastgah tradition is almost infinitely
> fluid and flexible, Turquoise17 does offer a variety of step sizes and
> shadings for modalities such as Arab Rast and Bayyati or Persian Shur
> and Bayat-e Esfahan. Just as the beauty and drama of meantone
> chromaticism in European keyboard music around 1600 draws extra power
> from the contrast between diatonic and chromatic semitone sizes, so
> maqam and dastgah music gain beauty from a subtle contrast of larger
> and smaller neutral steps.
>
> As the name "Turquoise" may suggest, there's also a Rast in a bright
> Turkish flavor using a septimal tetrachord (1/1-8/7-26/21-4/3 or
> 21:24:26:28) described by Ibn Sina, the great Persian philosopher and
> musician, about a millennium ago. You'll find this at the ~14/11 step:
>
> 0 231 369 496 703 935 1073 1200
> 1/1 8/7 26/21 4/3 3/2 12/7 13/7 2/1
>
> In short, while Bleu tempers out the commas distinguishing the steps
> in 11:12:13:14, Turquoise seeks to observe these commas while
> providing lots of fifths and fourths reasonably close to just. The two
> approaches may be interesting to compare.
>
> Most appreciatively,
>
> Margo Schulter
> mschulter@...
>

🔗Margo Schulter <mschulter@...>

🔗Jacques Dudon <fotosonix@...>

Hi again Margo,

This is brilliant, it sounds to me like a perfect well-temperament for Persian music indeed !

Since you compare it to "Bleu" (close or identical to a 60-edo frame), it made me wonder if there are not some other micro-temperaments it could relate to.
For one thing, Tsaharuk offers all distinctions needed between 12/11, 13/12 and 14/13 (those intervals, and their variations, belong to those wanted by Julien Jalaleddine Weiss). They simply differ by one equal step (= 12, 11, 10 steps in 94-edo).
Then when from "central Tsaharuk", which would be between the schismatic and the pure fifth circles, you get closer to the 17-edo ring (I refer to the Tsaharuk parametric horagram I mentionned before, that can be found in my TL folder), you extend gradually from more or less pure fifths to the ones you use in Turquoise, while you reach better approximations of both 14/13 and 12/11, and keep acceptable approximations of 13/12.
The mapping of [33:36:39:42:44] is [0, 18, 19, 3, -5] generators in all these variations.
There can be found some notable microtemperament versions :
94-edo (12 - 11 - 10 - 6 steps for the [33:36:39:42:44] division) will offer perhaps the best average approximations for 12/11, 13/12 and 14/13, and the best 14/13
111-edo (14 - 13 - 12 - 7 steps) and 128-edo (16 - 15 - 14 - 8 steps) will have about equally the best 12/11 approximations
145-edo (18 - 17 - 16 - 9 steps) will have the closest fifth (703.448275 c.) to your Turquoise temperament.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Jacques

Margo wrote :

> Hello, all.
>
> In response to the recent discussion about Bleu17, I'd like to offer
> Turquoise17, a strictly proper 17-note temperament based on the
> division of 4/3 into 12:11-13:12-14:13-22:21 (33:36:39:42:44), and
> with a variety of Zalzalian or neutral second steps for maqam or
> dastgah music.
>
> ! turquoise17.scl
> !
> Turquoise 17, 1024-ED2: ~33:36:39:42:44 at steps 0, 7, 10
> 17
> !
> 70.31249
> 151.17187
> 208.59375
> 289.45313
> 358.59375
> 416.01563
> 496.87500
> 566.01562
> 646.87500
> 704.29688
> 785.15625
> 854.29687
> 911.71875
> 992.57812
> 1061.71875
> 1119.14063
> 2/1
>
> <http://www.bestII.com/~mschulter/turquoise17.scl>
>
> There are 15 usual fifths, six at 703.125 cents and nine at 704.297
> cents; these, together with the 2/1 octave and a small thirdtone at
> 57.422 cents, are the generators. The average size of these 15
> "appreciable" fifths (as Scala calls them) is 703.828 cents,
>
> While the intonation of Arab, Turkish, and Kurdish maqam music and
> Persian music in the related dastgah tradition is almost infinitely
> fluid and flexible, Turquoise17 does offer a variety of step sizes and
> shadings for modalities such as Arab Rast and Bayyati or Persian Shur
> and Bayat-e Esfahan. Just as the beauty and drama of meantone
> chromaticism in European keyboard music around 1600 draws extra power
> from the contrast between diatonic and chromatic semitone sizes, so
> maqam and dastgah music gain beauty from a subtle contrast of larger
> and smaller neutral steps.
>
> As the name "Turquoise" may suggest, there's also a Rast in a bright
> Turkish flavor using a septimal tetrachord (1/1-8/7-26/21-4/3 or
> 21:24:26:28) described by Ibn Sina, the great Persian philosopher and
> musician, about a millennium ago. You'll find this at the ~14/11 step:
>
> 0 231 369 496 703 935 1073 1200
> 1/1 8/7 26/21 4/3 3/2 12/7 13/7 2/1
>
> In short, while Bleu tempers out the commas distinguishing the steps
> in 11:12:13:14, Turquoise seeks to observe these commas while
> providing lots of fifths and fourths reasonably close to just. The two
> approaches may be interesting to compare.
>
> Most appreciatively,
>
> Margo Schulter
> mschulter@...

🔗hstraub64 <straub@...>

🔗Jacques Dudon <fotosonix@...>

🔗Margo Schulter <mschulter@...>

> Hi again Margo,

> This is brilliant, it sounds to me like a perfect well-temperament
> for Persian music indeed !

> Since you compare it to "Bleu" (close or identical to a 60-edo
> frame), it made me wonder if there are not some other
> micro-temperaments it could relate to.

Dear Jacques,

Thank you for a most generous reply as my Turquoise 17, as well as for
your Soria tunings which, as you have discussed, also offer some fine
shadings for Persian music.

I should apologize for responding first to this discussion when your
wonderful story of Julien might justly claim priority. However, I feel
that it might be wise to take a bit more time and attempt a response
which may more adequately show my appreciation of his breathtaking art
as a performer.

Your discussion of Tsaharuk as well as of simpler systems makes some
very interesting comparisons. One point I am realizing, however, is
that the chains to get intervals such as 14/11 can be quite different
than with related fifth generators! Thus 14/11 will be 3 generators
up, rather than 20 (i.e. 4 fifths up) -- curiously, a kind of
"17-comma" since it involves a difference of 17 generators!

As far as Turquoise 17, it is actually almost identical to 104-EDO --
or 104-ED2, if we wish to emphasize that the octave is 2:1. While
Turquoise is itself based on 1024-ED2, we can see it as a slightly
irregular variation on the strictly linear 104-ED2, a "shaggy"
variation one might say, rather like a Bouvier Bernois. Here is the
33:36:39:42:44 division in 104-EDO which you discuss below in some
possible shadings of Tsaharuk:

104-equal: 150.0 138.5 126.9 80.8
just: 150.6 138.6 128.3 80.5
12:11 13:12 14:13 22:21
33 : 36 : 39 : 42 : 44
1/1 12/11 13/11 14/11 4/3
just: 0 150.6 289.2 417.5 498.0
104-equal 0 150.0 288.5 415.4 496.2

While 104-EDO, like Turquoise, is quite accurate for these ratios, it
is notably less so for 3/2 (703.846 cents), 4/3, or 9/8 (207.692
cents) than the various Tsaharuk shadings you discuss below.

For the curious, here is a version of Turqouoise in 104-ED2:

! turquoise17-104ed2.scl
!
Turquoise 17 in 104-tET/ED2, ~33:36:39:42:44 at steps 0 7 10
17
!
69.23077
150.00000
207.69231
288.46154
357.69231
415.38462
496.15385
565.38462
646.15385
703.84616
784.61539
853.84616
911.53846
992.30769
1061.53846
1119.23077
2/1

> For one thing, Tsaharuk offers all distinctions needed between
> 12/11, 13/12 and 14/13 (those intervals, and their variations,
> belong to those wanted by Julien Jalaleddine Weiss). They simply
> differ by one equal step (= 12, 11, 10 steps in 94-edo).

Yes, we have in 94-equal:

12 153.2 cents 11 140.4 cents
10 127.7 cents

> Then when from "central Tsaharuk", which would be between the
> schismatic and the pure fifth circles, you get closer to the 17-edo
> ring (I refer to the Tsaharuk parametric horagram I mentionned
> before, that can be found in my TL folder), you extend gradually
> from more or less pure fifths to the ones you use in Turquoise,
> while you reach better approximations of both 14/13 and 12/11, and
> keep acceptable approximations of 13/12.

What is clear to me is that Tsaharuk is a very subtle and intricate
system, going far beyond Turquoise or any similar tuning. Looking at
your horagram, I see generators for the schismatic and pure fifth
circles, as well as others including 2^11/94 -- a "septimal
schismatic" circle, as I mention below. Clearly your system is a
kind of magnificent tribute to Julien which does honor to you both.

One point here is that in Turquoise or 104-ED2, 14/13 is approximated
by the regular apotome or 7 fifths up -- but 9/8 is over twice as
impure as 14/13! In Tsaharuk, as you observe, 14/13 is 18 generators
up, or a generator less the "17-comma."

Here I give your corrected version of the Tsaharuk generators for
neutral seconds:

> Sorry, I just inversed of the order of the neutral seconds here :

> the mapping of [33:36:39:42:44] is [0, -16, -15, 3, -5] tsaharuk
> generators, in all its variations.

> There can be found some notable microtemperament versions : 94-edo
> (12 - 11 - 10 - 6 steps for the [33:36:39:42:44] division) will
> offer perhaps the best average approximations for 12/11, 13/12 and
> 14/13, and the best 14/13

This would be:

94-equal 153.2 140.4 127.7 76.6
just: 150.6 138.6 128.3 80.5
12:11 13:12 14:13 22:21
33 : 36 : 39 : 42 : 44
1/1 12/11 13/11 14/11 4/3
just: 0 150.6 289.2 417.5 498.0
94-equal 0 153.2 293.6 421.3 497.9

An interesting point of symmetry is that just as your horagram shows a
generator for a 5-limit schismatic temperament, so 94-ED2 provides a
septimal schismatic temperament for virtually pure fifths and near-pure
ratios of 7/6, 7/4, and 9/7! These two schismatic shadings, along with
the pure fifths, are what I take to be "central Tsaharuk."

> 111-edo (14 - 13 - 12 - 7 steps) and
> 128-edo (16 - 15 - 14 - 8 steps)
> will have about equally the best 12/11 approximations

These would be:

111-equal: 151.3 140.5 129.7 75.7
just: 150.6 138.6 128.3 80.5
12:11 13:12 14:13 22:21
33 : 36 : 39 : 42 : 44
1/1 12/11 13/11 14/11 4/3
just: 0 150.6 289.2 417.5 498.0
111-equal 0 151.3 291.9 421.6 497.3

128-equal: 150.0 140.6 131,2 75.0
just: 150.6 138.6 128.3 80.5
12:11 13:12 14:13 22:21
33 : 36 : 39 : 42 : 44
1/1 12/11 13/11 14/11 4/3
just: 0 150.6 289.2 417.5 498.0
128-equal 0 150.0 290.6 421.9 496.9

A curious point is that Turquoise has as one of its generators the
fifth of 128-equal at 703.125 cents -- 75 steps of 128 or 600 of 1024.
This provides fifths closer to pure than in 104-equal at certain
locations -- although less pure fifths of 704.297 cents at others!

> 145-edo (18 - 17 - 16 - 9 steps) will have the closest fifth
> (703.448275 c.) to your Turquoise temperament.

This would be:

145-equal: 149.0 140.7 132,4 74.5
just: 150.6 138.6 128.3 80.5
12:11 13:12 14:13 22:21
33 : 36 : 39 : 42 : 44
1/1 12/11 13/11 14/11 4/3
just: 0 150.6 289.2 417.5 498.0
145-equal 0 149.0 289.7 422.1 496.6

> the mapping of [33:36:39:42:44] is > [0, -16, -15, 3, -5] tsaharuk generators, > in all its variations.

This is an important point. When I first looked at the example in
145-ED2, I wondered if there was some "inconsistency," since with a
fourth of 496.55 cents or so, 14/11 should be around 414 cents rather
than 422 cents. However, looking at the Tsaharuk mapping made me
realize that 14/11 will be three generators up rather than 20!

> For example, with a 140.7 cents generator > (= a fifth of 703.5 c.), and pure octaves,
> [33:36:39:42:44] is approximated by : > 0, 148.8, 289.5, 422.1, 496.5 cents

A curious point is that 13/11 at -15 generators is the same as the
usual diatonic mapping of -3 fifths -- but 14/11 is different, as is
22/21 at -8 generators rather than -25 (-5 fifths).

> if compared to Turquoise's 0, 151.17187, 289.45313, 416.01563, 496.875 cents,
> Turquoise is closer to 13-limit JI : 0, 150.637, 289.2097, 417,508, 498,045
> but Tsaharuk will have other advantages, as a linear temperament.

Yes, with the generally purer fifths as one very important advantage.

> A fifth of 703.5 c. is close to what we get in a 145-edo microtemperament
> (703.448275 c.)
> and even closer is the algebraic solution of the sequence
> x^10 = 3x - 1 (140.69789471196 c.,
> generating a fifth of 703.4894735598 c.),
> expressing the equal-beating of 9/8 and 13/8.

Of course, the accuracy of 9/8 is a critical advantage of this shade
of Tsaharuk temperament, or Secor's 29-HTT at 703.579 cents, where 9/8
and 14/13 are equally impure and 63/52 thus pure, as opposed to
Turquoise or similar systems which are comparable in the impurity of
the fifths to 12n-equal, although in the opposite direction!

> Other useful Tsaharuk sequences with extended fifths :

> x^21 = 2x^10 + 1 near 128-edo (140.63340515328 c.)
> x^21 = x^5 + 4 near 111-edo (140.54922573072 c.)
> x^29 = 4x^6 + 4 near 94-edo (140.4134555676 c.)

Your discussion above gives an idea of the neutral steps in these
sequences -- although I realize that Tsaharuk deals with far more than
this. Julien's desire for pure fifths and 5-limit thirds are two very
relevant parameters satisfied by "central Tsaharuk," for example.

Most appreciatively,

Margo
mschulter@...