Yahoo Tuning Groups Ultimate Backup tuning Suz-i Dilara flavors (79MOS and Zest-24)

Hello, everyone.

Looking through the archives, I've been fascinated with some of the
discussions involving Ozan and others concerning the tunings of a
maqam called Suz-i Dilara.

Ozan, the discussions about producing this maqam by a chain of
differently sized fifths led me to realize that Zest-24 takes
precisely this approach, and to consider how this maqam might tie in
with others. Here I will take Bb* as Rast, although that may be an
arbitrary choice.

Anyway, in the 79-MOS, I gather from some of the previous articles
that this would be a good Suz-i Dilara:

! 79MOS_Suz-i_Dilara.scl
!
Tuning for Suz-i Dilara in Ozan Yarman's 79MOS
7
!
211.23644
422.92728
4/3
3/2
913.19144
1124.01947
2/1

In terms of the chain of fifths, this would be approximately:

F--(702)--C--(702)--G--(709)--D--(702)--A--(710)--E--(702)--B
498 0 702 211 913 422 1124

Let's assume that the major third step of this scale up from the 1/1
should be somewhere around 14/11 (418 cents) or 23/18 (424 cents), and
thus not too far from the region of maximum complexity between 5:4 and
9:7 at roughly 422-423 cents.

In Zest-24, the following two tunings of Suz-i Dilara are closest to
the above version in the 79MOS, and are found on Gb* or Ab* -- if Bb*
is Rast, then these steps would be approximately Bayyati (778 cents)
and Ajem (983 cents).

! zest24_Suz-i_Dilara_Gbup.scl
!
Zest-24 Suz-i Dilara near suggested values in Ozan Yarman's 79MOS
7
!
203.90624
421.87500
505.07812
696.09374
912.89062
1129.68749
2/1

Here the scheme of fifths is approximately:

Cb--(695)--Gb--(696)--Db--(708)--Ab--(709)--Eb--(709)--Bb--(708)--F
505 0 696 204 913 422 1130

Whereas fifths in the 79MOS may be precisely or virtually just, or
else about 1/3 of a Holdrian comma (1/53 octave), or a step of
159-EDO, either narrow or wide, in Zest-24 there are two basic types
of fifths, either narrow or wide by around 1/4 or 3/10 of a
Pythagorean comma (at 695/696 or 708/709 cents). Thus the impurity of
these decidedly tempered fifths is comparable in the two systems: but
the 79MOS, of course, additionally offers the attractions of many just
fifths and fourths.

Here is a slightly different although comparable version of Suz-i
Dilara on Ab*, or a 982-cent seventh (almost identical to 22-EDO,
about 30/17) above Bb* Rast:

! zest24_Suz-i_Dilara_Abup.scl
!
Zest-24 Suz-i Dilara similar to tuning in Ozan Yarman's 79MOS
7
!
217.96876
421.87500
492.18750
708.98438
925.78125
1117.96875
2/1

Here is the approximate chain of fifths:

Db--(708)--Ab--(709)--Eb--(709)--Bb--(708)--F--(696)--C--(696)--G
492 0 709 218 926 422 1117

While both of these Zest-24 forms have major thirds around 422 cents,
or within a cent or so of the 79MOS interval of 28 steps, another
option would be to try a septimal flavor with a 434-cent major third
built from four wide fifths, and very close to 9/7. Such a form occurs
on Db*, about 274 cents above Bb* Rast, which might be termed Nerm
Kurdi:

! zest24_Suz-i_Dilara_Dbup.scl
!
Zest-24 Suz-i Dilara, approximately septimal
7
!
216.79688
433.59375
503.90626
707.81250
925.78126
1129.68750
2/1

Here is the approximate chain of fifths:

Gb--(696)--Db--(708)--Ab--(709)--Eb--(709)--Bb--(708)--F--(696)--C
504 0 708 217 926 434 1130

All three Zest-24 versions of Suz-i Dilara, interestingly, are formed
from four wide and two narrow or meantone fifths. There are three
types of major seconds. A meantone at about 191/192 cents results from
two narrow fifths (e.g. F-C-G); a virtually just 9:8 tone at 204 cents
from a narrow plus a wide fifth (e.g. Gb-Db-Ab); and what I might term
a "septimal eventone fifth," equal to about half of a 9:7 major third,
or 217/218 cents, from two wide fifths (e.g. Eb-Bb-F).

Note that all these versions of Suz-i Dilara in Zest-24 are formed
from within a single 12-note circle of fifths; the same tunings would
also be available on the lower 12-note circle at Gb, Ab, and Db.

By comparison, the degree or perde of Bb* Rast itself has something
like the understanding of Maqam Rast as either a mode with small major
third and seventh around 5:4 and 15:8, or a mode with a middling to
large neutral third plus a neutral sixth (in a Zalzalian form) or
neutral seventh (in a common modern form).

1. Rast on Bb* with ~5:4 and ~15:8

Bb* C* D* Eb* F* G* A* Bb*
0 204 395 491 708 900 1091 1200

2. Rast on Bb* with neutral third and seventh
(Two disjunct Rast tetrachords)

Rast Rast
|----------------| |---------------|
Bb* C* D Eb* F* G* A Bb*
0 204 345 491 708 900 1041 1200

3. Zalzalian Rast on Bb* with neutral third and sixth
(Two conjunct Rast tetrachords)

Rast Rast
|----------------|---------------|
Bb* C* D Eb* F* G Ab* Bb*
0 204 345 491 708 850 983 1200

Note that all three flavors of Rast are available on Perde Rast: in
contrast, to move from the small-major-third flavor of Rast to Suz-i
Dilara would involve shifting the final or tonic up by a minor third,
sixth, or seventh if we wish to do this smoothly within a single
circle of fifths. The larger size and greater versatility of the 79MOS
obviously have their advantages.

Interestingly, the step Gb*, or Perde Bayyati if Bb* is Rast, has not
only the Suz-i Dilara shown above with a 422-cent major third, but
also an interesting form of Maqam Rast:

Rast Rast
|----------------| |---------------|
Gb* Ab* Bb Cb* Db* Eb* F Gb*
0 204 371 504 696 913 1079 1200

Rast Rast
|----------------|---------------|
Gb* Ab* Bb Cb* Db* Eb Fb* Gb*
0 204 371 504 696 862 1009 1200

These forms have a large neutral third around 26/21, and might be
styled "submajor," a term which also fits the large neutral sixth at
around 862 or 863 cents in the "Zalzalian" form, say 28/17.

Anyway, the submajor flavor of Rast provides a way to have both this
Maqam and Suz-i Dilara on the same final step -- although here two
circles of fifths are obviously involved.

With many thanks,

Margo
mschulter@...

🔗Margo Schulter <mschulter@...>

Hello, everyone.

Please let me attempt a response to the question about ratios of
factors such as 11 and 13 in "extended JI" from two perspectives. My
first perspective will be simply to urge that these intervals occur in
a variety of world musics, for example Near East maqam and dastgah
music of the Arab world, Turkey, Kurdistan, and Iran, where they have
been recognized in the theoretical literature for a millennium or
more. They are useful, practical, and beautiful -- whether presented
in JI, in a "near-JI" system like George Secor's HTT-29, or in more
decidedly tempered systems.

A different perspective is to point out also that Secor and others
have used these ratios more specifically in what I suspect is meant by
an "extended JI" style: one with very thickly saturated sonorities
like 4:5:6:7:9:11:13:15 (which HTT-29 very closely approximates),
and/or taking advantage of the special properties of isoharmonic
chords or sonorities like 9:11:13:15:17. While this level of density
is a bit different than my normal style, so is Wagner, for example --
and this kind of "extended" sonority offers a kind of 21st-century
complexity which needs neither simply to recaptulate nor laboriously
to avoid previous Western Europe practice, for those who come from
this tradition.

What I would like to urge strongly is that "ratios of 11, 13, etc."
can occur in many kinds of texture, ranging from a single melodic line
of traditional maqam or Persian dastgah music, to counterpoint for two
or three voices, to the thicker and denser textures which "extended
JI" might suggest, at least in my interpretation above. Many of these
applications are available with tunings of quite modest sizes, for
example a pentatonic or heptatonic mode.

Consider this beautiful tuning of Ibn Sina (980-1037), known in the
Latin west as Avicenna (Scala archive, avicenna_diat.scl):

1/1 14/13 7/6 4/3 3/2 21/13 7/4 2/1
0 128 267 498 702 830 969 1200

This set calls for only seven notes, and should be readily
recognizable without the listener being puzzled by too many steps for
a clear melodic pattern. An interesting point which I find has been
noted by Cris Forster on this newsgroup, and fits my own nonnative
intuitions, is that Ibn Sina's prominent use of small neutral or
middle intervals such as the 14:13 second and the 21:13 sixth could
quite possibly represent then, as now, a regional preference among
Persian and Kurdish musicians somewhat distinct from the taste in much
of the Arab world, for example, for rather larger sizes for these
intervals above the resting note or _Qarar_ (in European terms, the
"final" or "tonic").

Of course, as Ozan Yarman would have me emphasize, and I will do so
with a friendly amendment or two, neither European modality nor the
Near Eastern maqam and dastgah systems confine themselves to the
"textbook" heptatonic forms often cited for the various modes. Shifts
in the perceived modal center in a given passage or section of music,
customary inflections, and mutations or modulations from one mode to
another are characteristic of maqam and dastgah music as well as
14th-17th century European modal styles and 18th-19th century
tonality. However, it is eminently possible to use ratios of 11 and 13
in such a way that a given set of around seven notes is a salient
aural focus at a given time, sometimes with subtle inflections or
touches of "color" which decorate, but do not unduly obscure, the
prevailing pattern.

In a fascinating dialogue which took place in this forum, George Secor
recounted some of his experiments with a 17-tone well-temperament (his
17-WT of 1978), and Ozan Yarman remarked that someone exploring this
material (or its JI equivalent, as George suggests in a superb essay I
will offer a link to below) would likely discover or rediscover some
of the patterns of maqam music. Those posts moved me again to review
George's article published in _Xenharmonikon_ 18, where I noted indeed
that some of his scales involving ratios of 11 and 13 either match or
appear as logical variations from popular maqamat (plural of _maqam_)
in the medieval or modern tradition. Here is a link to this article,
and also to a version of a companion article appearing in the same
issue which I wrote with his invaluable collaboration and mentorship:

<http://xenharmony.wikispaces.com/space/showimage/17puzzle.pdf>
<http://www.bestII.com/~mschulter/Secor_17-WT_draft.zip>

It may also be useful briefly to note that while medieval Near Eastern
theory often specifies just ratios such as 14:13 or 13:12 or 12:11,
performers may often touch on or near these ratios without a specific
theoretical intent to do so. Also, performers may be in search of a
general type of color or nuance rather than any precise integer
ratio. For example, as someone often oriented toward rational ratios,
I may say to myself, "a tuning of around 1/1-13/12-13/11-4/3 is a good
one for a tetrachord of Arab Bayyati or Persian Daramad Shur" -- that
is, around 0-139-289-498 cents -- but this is merely one convenient
shade of intonation. In tuning systems I use, we might get these just
values (Zephyr 24, JOT-17); near-just ones of 0-138-288-496 cents
(Peppermint 24); or ones "in the general vicinity," say something like
0-141-287-503 cents (Zest-24).

The tetrachord described by Safi al-Din al-Urmawi in the 13th century
of 64:59:54:48 or 1/1-64/59-32/27-4/3 or 0-141-294-498 cents (Scala
archive, safi_diat2.scl) seems equally characteristic of Bayyati or
Shur -- so that a certain region of the spectrum, rather than specific
qualities of "11-ness" or "13-ness," may be the main point, although
Safi al-Din and others take a great interest in tunings built from
superparticular ratios, thus following the tradition of Ptolemy and
adapting it to Islamic civilization.

Anyway, to sum up, I would emphasize two points. The first is that any
musical style or tradition which favors neutral intervals is quite
likely either deliberately or otherwise to use or approximate ratios
such as 13/12, 17/14, 11/9, 21/13, 13/8, 18/11, 11/6, 13/7, etc. --
whether the tuning is based on rational ratios, a fixed irrational
temperament, or the intuition of flexible-pitch performers. This is
not something new or radical or of questionable practicality: in
theory and practice, it's been a basic feature of maqam music for a
millennium and more.

As for the use of "dense" (e.g. complex isoharmonic) sonorities in
what I suspect could be meant by an "extended JI" setting, Secor's
article nicely covers the potentials both in JI or near-JI tunings,
and in not-quite-so-accurate tempered systems. Indeed, I would
recommend his essay as an ideal introduction to the subject, whether
from the standpoint of musical motivations or real-world practice.

Most appreciatively,

Margo Schulter
mschulter@...

🔗Margo Schulter <mschulter@...>

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
about
>in a "near-JI" system like George Secor's HTT-29

> an "extended JI" style: one with very thickly saturated sonorities
> like 4:5:6:7:9:11:13:15 (which HTT-29 very closely approximates),
> and/or taking advantage of the special properties of isoharmonic
> chords or sonorities like 9:11:13:15:17. While this level of density
> is a bit different than my normal style, so is Wagner, for example
...
> and this kind of "extended" sonority offers a kind of 21st-century
> complexity which needs neither simply to recaptulate nor laboriously
> to avoid previous Western Europe practice, for those who come from
> this tradition.
...
> What I would like to urge strongly is that "ratios of 11, 13, etc."
> can occur in many kinds of texture, ranging from a single melodic
> theory often specifies just ratios such as 14:13 or 13:12 or 12:11,
...
> qualities of "11-ness" or "13-ness," may be the main point, although
> Safi al-Din and others take a great interest in tunings built from
> superparticular ratios, thus following the tradition of Ptolemy and
> adapting it to Islamic civilization.
...
> likely either deliberately or otherwise to use or approximate ratios
> such as 13/12, 17/14, 11/9, 21/13, 13/8, 18/11, 11/6, 13/7, etc. --
> whether the tuning is based on rational ratios, a fixed irrational
> temperament, or the intuition of flexible-pitch performers.

Dears Margo, Geoge & all other friemds of 29,

especially these 'semi'-consonant 11&13-limit intervals,
are called in an Greek-German idiom as: "ecmelic"
http://www.moz.ac.at/~herf/sympos/symp_g.html
"Music with Microtones, Ecmelic Music;"
/tuning-math/message/3708

29 has an unique property, due to the series:
http://www.research.att.com/~njas/sequences/A060528
" A list of equal temperaments (equal divisions of the octave) whose
nearest scale steps are closer and closer approximations to the ratios
of two tones of musical harmony: the perfect 4th, 4/3, and its
complement the perfect 5th, 3/2. +

1, 2, 3, 5, 7, 12, 29, 41, 53,...."

hence an 5th in 29-EDO consists in 17steps:
29-EDO yields the next better approximation for 1.5
than the potty and meanwhile hacknayed 12-EDO:
700Cents = 2^(7/12) = ~1.49830708...

versus the improved:

2^(17/29) = ~1.50129438...

1200Cents*17/29 = 703+13/29 Cents = ~ 703.448276...Cents

for 5ths on the average in any 29-tone temperament.

Hence:
here comes an epimoric 29-tone tuning that interpolates
-on the one hand- hepatonic C-major JI and also
-on the other hand- the corresponding harmonic overtone series
1:3:5:7:9:11:13:15
both exactly unified at once.

Start from Werckmeister's choice '176'
http://www.rzuser.uni-heidelberg.de/~tdent/septenarius.html
http://en.wikipedia.org/wiki/Werckmeister_temperament#Werckmeister_IV_.28VI.29:_the_Septenarius_tunings
"Werckmeister writes it as 176." = 11 * 2^3
Scheibler derived from W. the modern standard pitch a4=440Hz :

in 5th's cycle
-1; F_-1= 11 22 44 88 F_3=176Hz := 440Hz/(5:2) ! 5th below unison
00; C_1 = 33 unison
01; G_2 = 99
02; D_3 = 297 (< 298 149)
03; A_4 = 447 (< 448...7)
04; E_0 = 21
05; B_2 = (63 126 <) 126.5 (< 127)
06; F#_4 = 381
07; C#_3 = 143 286 572 1144 (<1143) with 143:=11*13th partial on F_-1
08; G#_4 = 429 := 33 * 13th partial over its fundamental 00; C_1=33
09; D#_3 = 161 322 644 1288 (<1287)
10; A#_2 = 121 242 484 (>483) with 121=11^2, 11th partial on F_-1
11; F+_4 = (91 182 364 >) 363 := 33 * 11th partial over C_1=33
12; C+_4 = 273
13; G+_3 = 205 410 820 (<819)
14; D+_2 = 77 ... 616 (>615) that's the 7th harmonics over F_1=11
15; A+_3 = 231 = (C_1 33)*7
16; E+_5 = 693 = F-_5 enharmonic change happens here...
17; B+_1 = 65 130 260 ... 2080 (> 2079) = C-_2 ...or even there
18; G-_3 = 195
19; D-_4 = 293 586 (< 585)
20; A-_1 = 55 110 220 A-_4=440Hz 880 (> 879) tuning-fork
21; E-_3 = 165
22; B-_4 = 495
23; Gb_6 = 1485 begin division of the schisma into 6 epimoric factors
24; Db_7 = 2227 4454 (<4455 := 3*Gb_6)
25; Ab_5 = 835 1670 3340 6680 (< 6681 := 3*Db_7)
26; Eb_4 = 313 626 1252 2504 (< 2505 := 3*Ab_5)
27; Bb_5 = 939
28; F_-1 = 11...2816(< 2817:=3*939) end sub-schismatically epimorics
29=0;C_1 = 33

That fits exactly to hepatonic C-major JI,
but meets also as well the harmonic-overtone series:

F :G :A- :A# :C :C# :D+ :E- :F' == 11*(8:9:10:11:12:13:14:15:16)

over the fundamental F and also an 5th below the partials over C:

C :D :E- :E+ :F+ :G :G# :A+ :B- :C' == 33*(8:9:10:11:12:13:14:15:16)

When stepping in parcels of 12 times 5ths modulo 29,
we obtain an corresponding scale in ascending order.
with the frequencies in absolute-pitch:

00;-> 00: C. 264 = 8*33 middle-C5
12;-> 01: C+ 273
24;-> 02: Db 278
07;-> 03: C# 286 = 13*22 ! due to 3*12; = 36; == 07; in modulo 29
19;-> 04: D- 293
02;-> 05: D. 297 = 9*33 ! 2nd of 9/8
14;-> 06: D+ 308
26;-> 07: Eb 313
09;-> 08: D# 322
21;-> 09: E- 330 = 5*66 ! 3rd of 5/4
04;-> 10: E. 336
16;-> 11: E+ 346.5 = F- contains enharmonic exchange: E+=F-
28;-> 12: F. 352 = 32*11 = W's "176"*2 fundamental over F-chord! 4th
11;-> 13: F+ 363
23;-> 14: Gb 371.25
06;-> 15: F# 381
18;-> 16: G- 390
01;-> 17: G. 396 = 3*132 ! 5th of 3/2
13;-> 18: G+ 410
25;-> 19: Ab 417.5
08;-> 20: G# 429 = 13*33
20;-> 21: A- 440Hz = 5*88Hz ! 6th of 5/3 ! tuning-fork
03;-> 22: A. 447
15;-> 23: A+ 462
27;-> 24: Bb 469.5
10;-> 25: A# 484 = 11^2*4
22;-> 26: B- 495 = 15*33 ! 7th of 15/8
05;-> 27: B. 506
17;-> 28: B+ 520 = C- contains enharmonic exchange: B+=C- leads to...
29=0;=00: C' 528 ! 2/1 ...the concluding octave

And finally for those, that are not able to convert such
numerical frequency-values into memorizable absolute-pitches,
the same in relative ratios:

!septenarian29.scl
!by A. Sparschuh
C-major-JI and 2 harmonic overtone-series 1:3:5:7:9:11:15 over F & C
29
!
91/88 !___! 01: C+_4__273 / C_4__264
2227/2112 ! 02: Db_7_2227 / C_7_2112
13/12 !___! 03: C#_3__143 / C_3__132
293/264 !_! 04: D-_4__293 / C_4__264
9/8 !_____! 05: D__4__297 / C_4__264
7/6 !_____! 06: D+_2___77 / C_2___66
313/264 !_! 07: Eb_4__313 / C_4__264
161/132 !_! 08: D#_3__132 / C_3__132
5/4 !_____! 09: E-_3__165 / C_3__132
14/11 !___! 10: E__1___42 / C_1___33
231/176 !_! 11: E+F-5_693 / C_5__524
4/3 !_____! 12: F__1___44 / C_1___33
121/88 !__! 13: F+_4__363 / C_4__264
45/32 !___! 14: Gb_6_1485 / C_5_1056
63/44 !___! 15: F#_4__381 / C_4__264
65/44 !___! 16: G-_3__195 / C_3__132
3/2 !_____! 17: G__2___99 / C_2___66
205/132 !_! 18: G+_3__205 / C_3__132
835/528 !_! 19: Ab_5__835 / C_5__528
13/8 !____! 20: G#_4__329 / C_4__264
5/3 !_____! 21: A-_1___55 / C_1___33
149/88 !__! 22: A__4__447 / C_4__264
7/4 !_____! 23: A+_3__231 / C_3__132
313/176 !_! 24: Bb_5__939 / C_5__528
11/6 !____! 25: A#_2__121 / C_2___66
15/8 !____! 26: B-_4__495 / C_4__262
23/12 !___! 27: B__3__253 / C_3__132
65/33 !___! 28: (B+_2=C-_3)/C_2___33
2/1
!

Compare the above to George's
29-HTT 13&11-limit approximation
/tuning-math/message/7574
"
! secor29htt.scl
!
George Secor's 29-tone 13-limit high-tolerance temperament (5/4 & 7/4
exact)
29
.....
Have fun analyzing these,
and let me know if you think they sound
like JI!
"
Fully agreed, that your's "sound (almost? a)like JI".
But why using barely an imprecise JI-approximation,
when it's possible to use an exact JI-interpolation
that meets exactly the desired 13-limit ratios,
that matches and fits to my own practice on the:

http://en.wikipedia.org/wiki/Baroque_trumpet
"In order to play the out-of-tune 11th and 13th harmonics (notated f2,
and a2), for example, the player opens the thumb vent hole and plays
the f2 and a2 as the 8th and 10th harmonics of the new series. Some
baroque trumpets have extra "cheater" holes that allow the player to
make half-step transpositions and blow a relatively easy high C."

Attend that the notation of the overtone scale:
http://upload.wikimedia.org/wikipedia/de/b/b2/Barocktrompetenskala.png
appears only on the German wiki page:
http://de.wikipedia.org/wiki/Barocktrompete
http://de.wikipedia.org/wiki/Bild:Barocktrompetenskala.png

Have a lof of fun blownig yourself the partials 13&11 on the horn.
alike J.S.Bach's virtuoso:
http://en.wikipedia.org/wiki/Gottfried_Reiche
http://de.wikipedia.org/wiki/Gottfried_Reiche

Yours Sincerely
A.S.

🔗George D. Secor <gdsecor@...>

🔗Charles Lucy <lucy@...>

🔗Margo Schulter <mschulter@...>

🔗Kraig Grady <kraiggrady@...>

🔗Carl Lumma <carl@...>

🔗Kraig Grady <kraiggrady@...>

🔗Charles Lucy <lucy@...>

🔗Andreas Sparschuh <a_sparschuh@...>

🔗Kraig Grady <kraiggrady@...>

🔗Andreas Sparschuh <a_sparschuh@...>

🔗Margo Schulter <mschulter@...>

🔗robert thomas martin <robertthomasmartin@...>

🔗Ozan Yarman <ozanyarman@...>

Hi again Margo,

While Suz-i Dilara indeed employs a Pythagorean major scale, it involves many more scales that renders it unsuitable as a basis for comparisons. Instead, we ought to use Mahur, which is mostly a mixture of Pythagorean major on perde rast with Rast (which is naturally on perde rast). Thus, all instances of Suz-i Dilara should be replaced with Mahur. While AEU tends to Mahurize Chargah by converting the original Chargah to a Pythagorean major on perde kaba (sub) chargah, I find it synthetic and dismiss it entirely while staying true to the original. More comments below:

On Jul 2, 2008, at 9:44 AM, Margo Schulter wrote:

> Hello, everyone.
>
> Looking through the archives, I've been fascinated with some of the
> discussions involving Ozan and others concerning the tunings of a
> maqam called Suz-i Dilara.
>
> Ozan, the discussions about producing this maqam by a chain of
> differently sized fifths led me to realize that Zest-24 takes
> precisely this approach, and to consider how this maqam might tie in
> with others. Here I will take Bb* as Rast, although that may be an
> arbitrary choice.
>
> Anyway, in the 79-MOS, I gather from some of the previous articles
> that this would be a good Suz-i Dilara:
>
> ! 79MOS_Suz-i_Dilara.scl
> !
> Tuning for Suz-i Dilara in Ozan Yarman's 79MOS
> 7
> !
> 211.23644
> 422.92728
> 4/3
> 3/2
> 913.19144
> 1124.01947
> 2/1
>
>
> In terms of the chain of fifths, this would be approximately:
>
>
> F--(702)--C--(702)--G--(709)--D--(702)--A--(710)--E--(702)--B
> 498 0 702 211 913 422 1124
>
>
> Let's assume that the major third step of this scale up from the 1/1
> should be somewhere around 14/11 (418 cents) or 23/18 (424 cents), and
> thus not too far from the region of maximum complexity between 5:4 and
> 9:7 at roughly 422-423 cents.
>
> In Zest-24, the following two tunings of Suz-i Dilara are closest to
> the above version in the 79MOS, and are found on Gb* or Ab* -- if Bb*
> is Rast, then these steps would be approximately Bayyati (778 cents)
> and Ajem (983 cents).
>
> ! zest24_Suz-i_Dilara_Gbup.scl
> !
> Zest-24 Suz-i Dilara near suggested values in Ozan Yarman's 79MOS
> 7
> !
> 203.90624
> 421.87500
> 505.07812
> 696.09374
> 912.89062
> 1129.68749
> 2/1
>

A very fine Mahur, but it still needs to modulate to Rast toward the cadance of the maqam.

>
> Here the scheme of fifths is approximately:
>
>
> Cb--(695)--Gb--(696)--Db--(708)--Ab--(709)--Eb--(709)--Bb--(708)--F
> 505 0 696 204 913 422 1130
>
>
> Whereas fifths in the 79MOS may be precisely or virtually just, or
> else about 1/3 of a Holdrian comma (1/53 octave), or a step of
> 159-EDO, either narrow or wide, in Zest-24 there are two basic types
> of fifths, either narrow or wide by around 1/4 or 3/10 of a
> Pythagorean comma (at 695/696 or 708/709 cents). Thus the impurity of
> these decidedly tempered fifths is comparable in the two systems: but
> the 79MOS, of course, additionally offers the attractions of many just
> fifths and fourths.
>

Your fifths are very much in range.

> Here is a slightly different although comparable version of Suz-i
> Dilara on Ab*, or a 982-cent seventh (almost identical to 22-EDO,
> about 30/17) above Bb* Rast:
>
> ! zest24_Suz-i_Dilara_Abup.scl
> !
> Zest-24 Suz-i Dilara similar to tuning in Ozan Yarman's 79MOS
> 7
> !
> 217.96876
> 421.87500
> 492.18750
> 708.98438
> 925.78125
> 1117.96875
> 2/1
>
>
> Here is the approximate chain of fifths:
>
>
> Db--(708)--Ab--(709)--Eb--(709)--Bb--(708)--F--(696)--C--(696)--G
> 492 0 709 218 926 422 1117
>

This also works for Mahur. But don't forget the modulation to Rast, where the 3rd and 7th degrees are diminished by about a quarter-tone.

>
> While both of these Zest-24 forms have major thirds around 422 cents,
> or within a cent or so of the 79MOS interval of 28 steps, another
> option would be to try a septimal flavor with a 434-cent major third
> built from four wide fifths, and very close to 9/7. Such a form occurs
> on Db*, about 274 cents above Bb* Rast, which might be termed Nerm
> Kurdi:
>
> ! zest24_Suz-i_Dilara_Dbup.scl
> !
> Zest-24 Suz-i Dilara, approximately septimal
> 7
> !
> 216.79688
> 433.59375
> 503.90626
> 707.81250
> 925.78126
> 1129.68750
> 2/1
>

The 434 cent perde would be nishabur here, not nerm kurdi. Still, it has a buselik characteristic because of the chain of fifths. This scale also works for Mahur.

>
> Here is the approximate chain of fifths:
>
>
> Gb--(696)--Db--(708)--Ab--(709)--Eb--(709)--Bb--(708)--F--(696)--C
> 504 0 708 217 926 434 1130
>
>
> All three Zest-24 versions of Suz-i Dilara, interestingly, are formed
> from four wide and two narrow or meantone fifths. There are three
> types of major seconds. A meantone at about 191/192 cents results from
> two narrow fifths (e.g. F-C-G); a virtually just 9:8 tone at 204 cents
> from a narrow plus a wide fifth (e.g. Gb-Db-Ab); and what I might term
> a "septimal eventone fifth," equal to about half of a 9:7 major third,
> or 217/218 cents, from two wide fifths (e.g. Eb-Bb-F).
>
> Note that all these versions of Suz-i Dilara in Zest-24 are formed
> from within a single 12-note circle of fifths; the same tunings would
> also be available on the lower 12-note circle at Gb, Ab, and Db.
>

Admirably done. Congratulations!

> By comparison, the degree or perde of Bb* Rast itself has something
> like the understanding of Maqam Rast as either a mode with small major
> third and seventh around 5:4 and 15:8, or a mode with a middling to
> large neutral third plus a neutral sixth (in a Zalzalian form) or
> neutral seventh (in a common modern form).
>
> 1. Rast on Bb* with ~5:4 and ~15:8
>
> Bb* C* D* Eb* F* G* A* Bb*
> 0 204 395 491 708 900 1091 1200
>
>
> 2. Rast on Bb* with neutral third and seventh
> (Two disjunct Rast tetrachords)
>
> Rast Rast
> |----------------| |---------------|
> Bb* C* D Eb* F* G* A Bb*
> 0 204 345 491 708 900 1041 1200
>
>

In Turkish rendition of Rast, the 3rd and 7th degrees regularly fall between the values you have given above. segah is somewhere about 365 cents (21/17) and evdj about 1070 cents (13/7).

> 3. Zalzalian Rast on Bb* with neutral third and sixth
> (Two conjunct Rast tetrachords)
>
> Rast Rast
> |----------------|---------------|
> Bb* C* D Eb* F* G Ab* Bb*
> 0 204 345 491 708 850 983 1200
>
>
> Note that all three flavors of Rast are available on Perde Rast: in
> contrast, to move from the small-major-third flavor of Rast to Suz-i
> Dilara would involve shifting the final or tonic up by a minor third,
> sixth, or seventh if we wish to do this smoothly within a single
> circle of fifths. The larger size and greater versatility of the 79MOS
> obviously have their advantages.

Ah, you are not allowed to shift the tonic in either Suz-i Dilara or Mahur. In Yarman24, I sacrifice the circle of fifths in obtaining the Rast scale, but gain points for the correct representation of every maqam in at least on ahenk (diapason/pitch level) without transpositions.

>
>
> Interestingly, the step Gb*, or Perde Bayyati if Bb* is Rast, has not
> only the Suz-i Dilara shown above with a 422-cent major third, but
> also an interesting form of Maqam Rast:
>
> Rast Rast
> |----------------| |---------------|
> Gb* Ab* Bb Cb* Db* Eb* F Gb*
> 0 204 371 504 696 913 1079 1200
>
>
> Rast Rast
> |----------------|---------------|
> Gb* Ab* Bb Cb* Db* Eb Fb* Gb*
> 0 204 371 504 696 862 1009 1200
>
>
> These forms have a large neutral third around 26/21, and might be
> styled "submajor," a term which also fits the large neutral sixth at
> around 862 or 863 cents in the "Zalzalian" form, say 28/17.
>

These are just the Turkish inflexions of perde segah and evdj I was talking about.

> Anyway, the submajor flavor of Rast provides a way to have both this
> Maqam and Suz-i Dilara on the same final step -- although here two
> circles of fifths are obviously involved.
>

Excellent!

> With many thanks,
>
> Margo
> mschulter@...
>
>

Cordially,
Oz.

🔗Ozan Yarman <ozanyarman@...>

More comments below:

On Jul 24, 2008, at 1:30 AM, Margo Schulter wrote:

> Hello, everyone.
>
> Please let me attempt a response to the question about ratios of
> factors such as 11 and 13 in "extended JI" from two perspectives. My
> first perspective will be simply to urge that these intervals occur in
> a variety of world musics, for example Near East maqam and dastgah
> music of the Arab world, Turkey, Kurdistan, and Iran, where they have
> been recognized in the theoretical literature for a millennium or
> more. They are useful, practical, and beautiful -- whether presented
> in JI, in a "near-JI" system like George Secor's HTT-29, or in more
> decidedly tempered systems.
>
> A different perspective is to point out also that Secor and others
> have used these ratios more specifically in what I suspect is meant by
> an "extended JI" style: one with very thickly saturated sonorities
> like 4:5:6:7:9:11:13:15 (which HTT-29 very closely approximates),
> and/or taking advantage of the special properties of isoharmonic
> chords or sonorities like 9:11:13:15:17. While this level of density
> is a bit different than my normal style, so is Wagner, for example --
> and this kind of "extended" sonority offers a kind of 21st-century
> complexity which needs neither simply to recaptulate nor laboriously
> to avoid previous Western Europe practice, for those who come from
> this tradition.
>
> What I would like to urge strongly is that "ratios of 11, 13, etc."
> can occur in many kinds of texture, ranging from a single melodic line
> of traditional maqam or Persian dastgah music, to counterpoint for two
> or three voices, to the thicker and denser textures which "extended
> JI" might suggest, at least in my interpretation above. Many of these
> applications are available with tunings of quite modest sizes, for
> example a pentatonic or heptatonic mode.
>
> Consider this beautiful tuning of Ibn Sina (980-1037), known in the
> Latin west as Avicenna (Scala archive, avicenna_diat.scl):
>
> 1/1 14/13 7/6 4/3 3/2 21/13 7/4 2/1
> 0 128 267 498 702 830 969 1200
>

This scale seems to be constructed from the disjunct union of two permuted kavi (tense diatonic?) genera I mentioned earlier. Remember that the original kavi genus was 12:13:14:16.

> This set calls for only seven notes, and should be readily
> recognizable without the listener being puzzled by too many steps for
> a clear melodic pattern. An interesting point which I find has been
> noted by Cris Forster on this newsgroup, and fits my own nonnative
> intuitions, is that Ibn Sina's prominent use of small neutral or
> middle intervals such as the 14:13 second and the 21:13 sixth could
> quite possibly represent then, as now, a regional preference among
> Persian and Kurdish musicians somewhat distinct from the taste in much
> of the Arab world, for example, for rather larger sizes for these
> intervals above the resting note or _Qarar_ (in European terms, the
> "final" or "tonic").
>
> Of course, as Ozan Yarman would have me emphasize, and I will do so
> with a friendly amendment or two, neither European modality nor the
> Near Eastern maqam and dastgah systems confine themselves to the
> "textbook" heptatonic forms often cited for the various modes. Shifts
> in the perceived modal center in a given passage or section of music,
> customary inflections, and mutations or modulations from one mode to
> another are characteristic of maqam and dastgah music as well as
> 14th-17th century European modal styles and 18th-19th century
> tonality. However, it is eminently possible to use ratios of 11 and 13
> in such a way that a given set of around seven notes is a salient
> aural focus at a given time, sometimes with subtle inflections or
> touches of "color" which decorate, but do not unduly obscure, the
> prevailing pattern.

Righteously!

>
>
> In a fascinating dialogue which took place in this forum, George Secor
> recounted some of his experiments with a 17-tone well-temperament (his
> 17-WT of 1978), and Ozan Yarman remarked that someone exploring this
> material (or its JI equivalent, as George suggests in a superb essay I
> will offer a link to below) would likely discover or rediscover some
> of the patterns of maqam music. Those posts moved me again to review
> George's article published in _Xenharmonikon_ 18, where I noted indeed
> that some of his scales involving ratios of 11 and 13 either match or
> appear as logical variations from popular maqamat (plural of _maqam_)
> in the medieval or modern tradition. Here is a link to this article,
> and also to a version of a companion article appearing in the same
> issue which I wrote with his invaluable collaboration and mentorship:
>
> <http://xenharmony.wikispaces.com/space/showimage/17puzzle.pdf>
> <http://www.bestII.com/~mschulter/Secor_17-WT_draft.zip>
>

I had the oppurtunity to skim through the Xenharmonikon article of George Secor. I am saddened by the fact that he considers better melodic intervals those where the semitones are narrowed to 70 cents or so. This is a prejudice. In Turkish maqam music, which is totally a melodic genre by the way, the better melodic intervals are those where the semitone is an apatome or greater.

> It may also be useful briefly to note that while medieval Near Eastern
> theory often specifies just ratios such as 14:13 or 13:12 or 12:11,
> performers may often touch on or near these ratios without a specific
> theoretical intent to do so. Also, performers may be in search of a
> general type of color or nuance rather than any precise integer
> ratio. For example, as someone often oriented toward rational ratios,
> I may say to myself, "a tuning of around 1/1-13/12-13/11-4/3 is a good
> one for a tetrachord of Arab Bayyati or Persian Daramad Shur" -- that
> is, around 0-139-289-498 cents -- but this is merely one convenient
> shade of intonation. In tuning systems I use, we might get these just
> values (Zephyr 24, JOT-17); near-just ones of 0-138-288-496 cents
> (Peppermint 24); or ones "in the general vicinity," say something like
> 0-141-287-503 cents (Zest-24).
>

These tetrachords are all very fine for Ushshaq.

> The tetrachord described by Safi al-Din al-Urmawi in the 13th century
> of 64:59:54:48 or 1/1-64/59-32/27-4/3 or 0-141-294-498 cents (Scala
> archive, safi_diat2.scl) seems equally characteristic of Bayyati or
> Shur -- so that a certain region of the spectrum, rather than specific
> qualities of "11-ness" or "13-ness," may be the main point, although
> Safi al-Din and others take a great interest in tunings built from
> superparticular ratios, thus following the tradition of Ptolemy and
> adapting it to Islamic civilization.
>

Admirably too, I might add! As a side note, let me remark that the tetrachord you mention is listed as Class III in the third table on Kavi (tense or even diatonic) genera based on the subtraction of 9:8 from 4:3 and the division of 32:27 to two equal parts. This could be interpreted as an Ushshaq if read the way you have specified or a Rast if read the other way around (48:54:59:64).

> Anyway, to sum up, I would emphasize two points. The first is that any
> musical style or tradition which favors neutral intervals is quite
> likely either deliberately or otherwise to use or approximate ratios
> such as 13/12, 17/14, 11/9, 21/13, 13/8, 18/11, 11/6, 13/7, etc. --
> whether the tuning is based on rational ratios, a fixed irrational
> temperament, or the intuition of flexible-pitch performers. This is
> not something new or radical or of questionable practicality: in
> theory and practice, it's been a basic feature of maqam music for a
> millennium and more.
>

Very well put Margo.

> As for the use of "dense" (e.g. complex isoharmonic) sonorities in
> what I suspect could be meant by an "extended JI" setting, Secor's
> article nicely covers the potentials both in JI or near-JI tunings,
> and in not-quite-so-accurate tempered systems. Indeed, I would
> recommend his essay as an ideal introduction to the subject, whether
> from the standpoint of musical motivations or real-world practice.
>
> Most appreciatively,
>
> Margo Schulter
> mschulter@...
>

Cordially,
Oz.

🔗Ozan Yarman <ozanyarman@...>

🔗Caleb Morgan <calebmrgn@...>

🔗Danny Wier <dawiertx@...>

🔗Ozan Yarman <ozanyarman@...>

🔗Carl Lumma <carl@...>

🔗Ozan Yarman <ozanyarman@...>

🔗Torsten Anders <torsten.anders@...>

🔗Ozan Yarman <ozanyarman@...>

🔗Herman Miller <hmiller@...>

🔗Carl Lumma <carl@...>

🔗Ozan Yarman <ozanyarman@...>

🔗Carl Lumma <carl@...>

🔗Ozan Yarman <ozanyarman@...>

🔗Torsten Anders <torsten.anders@...>

🔗Carl Lumma <carl@...>

🔗Torsten Anders <torsten.anders@...>

🔗Carl Lumma <carl@...>

🔗Margo Schulter <mschulter@...>

> Dear Margo,

> My apologies for the very late reply. I have been enjoying a well-
> earned summer's rest. I hope the importance of the topic has not
> faded while I was away. My comments are below:

Dear Ozan,

Please let me say that no apologies are needed, and that I am
delighted that you have been enjoying your summer after writing such a
germinal dissertation! Of course, the topic is as exciting for me as
ever, and exploring the curious tuning that I give below and some Arab
maqam theory, as well as your papers on your website, has heightened
that interest yet more.

And so we return to our discussion of Dariush Anooshfar's
iran_diat.scl in the Scala archives, offered as an interpretation of
an unspecified tuning of Safi al-Din al-Urmavi in the later 13th
century.

Please let me apologize in advance for trimming a bit: I've tried to
edit mainly my own remarks, or passages where we are agreeing on
things already said in previous discussions.

> Ah, I am sad to say that the Scala 125-EDO subset does great injustice
> to the original Ushshaq scale of Urmavi, which I believe goes:

> 9:8 x 9:8 x 256:243 x 9:8 x 9:8 x 256:243 x 9:8

> Notice how the limma has been reduced in size by half in the Scala
> file so as to destroy the qualities of the half-tone step, making it
> a quarter-tone step. Bear in mind that this Ushshaq has nothing to
> do with the Ushshaq we perform today. I am at a lost to explain how
> a diatonic major scale got transformed to a neutral minor during the
> centuries that followed Urmavi.

This looks like a very interesting conversation! This summer I found
on the Web a paper by Dr. Fazli Arslan of the Turkish Ministry of
Education and others, _Safi al-Din al-Urmawi and the Theory of Music_
(March 2007), which leads to the conclusion that Anooshfar might have
had a different tuning of Urmavi in mind -- but first this Ushshaq.

Certainly we agree that if Anooshfar had intended this tuning, then
his 125-EDO version would seem very curious! I wonder if, instead, he
had this septimal flavor of Urmavi, again with two conjunct
tetrachords plus an upper 9:8 tone, in mind, based on what Urmavi
called "the first conjunct" tetrachord of 8/7x9/8x28/27, to which I
will add an indication of approximate Pythagorean-53 or 53-EDO commas:

10 9 3 10 9 3 9
commas: 0 10 19 22 32 41 44 53
1/1 8/7 9/7 4/3 32/21 12/7 16/9 2/1
cents: 0 231 435 498 729 933 996 1200
8:7 9:8 28:27 8:7 9:8 28:27 9:8
231 204 63 231 204 63 204
|-----------------|----------------------|.....|
lower jins upper jins 9:8

This 0-231-435-498 or 231-204-63 tuning seems to me considerably
closer to Anooshfar's tetrachord of about 0-221-442-490 or 221-221-48
than Urmavi's Ushshaq at 0-204-408-498 (or 9-9-4 commas). If we start
from this septimal flavor of Urmavi, then three basic changes might
lead to an interpretation like Anooshfar's.

The first change is to use some fifths tempered about 1/3-comma wide,
or fourths 1/3-comma narrow, so that we have Urmavi's 4/3 placed at
around 491 cents and his 16/9 at around 982 cents -- as in 22-EDO.
The second is a very dramatic narrowing of the original 28:27 step (as
in the diatonic of Archytas, of which Urmavi's tuning is a
permutation) from 63 cents to 50 cents, a quirk to which both
Anooshfar and I seem curiously attracted <grin>:

Bb* Db Eb Eb* Gb Ab Ab* Bb*
cents: 0 224 441 491 728 932 982 1200
224 217 50 237 204 50 218
|-----------------|----------------------|.....|
lower jins upper jins 9:8

To get an approximation of Anooshfar's 0-221-442-490-710-931-979-1200,
we need additionally to substitute a wide tempered 3/2 for Urmavi's
32/21, interestingly almost just in the last Zest-24 example at 728
cents:

Bb* Db Eb Eb* F* Ab Ab* Bb*
cents: 0 224 441 491 708 932 982 1200
224 217 50 217 224 50 218
|-----------------|----------------------|.....|
lower jins upper jins 9:8

Humorously, I might suggest that my attraction to this Anooshfar
tuning comes from the routine license in Zest-24 of often tempering
28:27 at 50 cents, actually a virtually just 35:34! A milder version
of this license would occur in 22-EDO:

C D E F G A Bb C
cents: 0 218 436 491 709 927 982 1200
218 218 55 218 218 55 218
|-------------|--------------|.....|
lower jins upper jins tone

The liberty involved here could be put another way: Urmavi's 16/9 is
being tempered with narrow fourths down to something like 982 cents,
which represents both 16/9 and 7/4. Thus his interval 12/7-16/9
becomes something like 12/7-30/17, or about 35:34 -- almost exactly
what it is in Zest-24! If the idea is to get a somewhat less
inaccurate version of Urmavi's steps, then in Zest-24 we might try:

|---------------|---------------|......|
C D* E* F G A* Bb C
0 242 434 504 696 937 996 1200
242 192 70 192 241 59 204

Here either 70 cents or 59 cents is a more or less reasonable
approximation of 28:27 -- although not as close as with your 17
perdeler! We have a virtually just 16/9 without any pretensions of
representing 7/4. The 241/242-cent steps are rather wider than 8:7;
but one might argue that getting 28:27 more accurately is most
important if the idea is faithfully to represent Urmavi's tuning.

> I should state further, that it is most unfortunate that Turks have
> not written a single academic work on their conception of Maqam
> music for the auspices of the international music community since
> Rauf Yekta. My thesis should serve as a brick that closes the
> centennial gap.

Yes, and clearly I am encountering a software limitation in
Ghostscript that causes a VMerror (not enough virtual memory) in
processing the file. Obviously it is vital that I find some solution
-- possibly a later version of this software that might handle more
recent PDF files (I tried this some months ago and it solved a problem
with a set of PDF files for John Chalmers' book on tetrachords.)

This is _my_ problem, not yours, and it's obvious that I need to
figure out some way of reading and reviewing your dissertation.

[...]

[On avoiding the region around 170 cents, or "equitable heptatonic"
close to 7-EDO, in maqam music]

> Indeed so. I remember having stated the necessity to avoid the
> equitable heptotonic in Yarman24.

Yes, over the summer I found your 24-note system where you mentioned
this. Personally, I tend to regard a 170-cent step as a "Persian
liberty," more fitting to a dastgah style than a usual maqam style.
Since it often occurs in a Zest-24 tetrachord like 0-191-333-504 cents
(e.g. C*-D*-E-F*) with steps of 191-141-171 cents, where the small
neutral third around 40:33 would fit a Persian style, this view might
have some connection to Near Eastern practice. An interesting question
is whether a tetrachord like F*-G*-A-Bb* or 0-192-333-492 cents, with
steps of 192-141-159 cents, might fit into an Arab or Turkish maqam
style. You've mentioned that these small neutral thirds do occur.

> One place 26/21 comes up is in Ibn Sina's 28:26:24:21 tetrachord.

> I checked Ibn Sina's passages. The actual tetrachord is given as 8:7
> x 14:13 x 13:12. The tetrachord you have mentioned permutes the last
> two intervals. In the original form, we have 16:13 instead of
> 26:21. But I suspect the tetrachord should be read the other way
> around, which yields 12:13:14:16. Ibn Sina calls this a Kavi genus
> which is of quality and nobility, I am not sure what this
> corresponds to in music theory of the Ancient world. Ibn Sina says
> that this is the genus preferred by Ptolemy.

This tetrachord comes up in Urmavi also, and the string ratios given
confirm the same order you give for Ibn Sina's version; Urmavi's
tuning uses conjunct tetrachords, with a 32/21 above the final, as in
the 8/7x9/8x28/27 tuning discussed above:

lengths: 64 56 52 48 42 39 36 32
|----------------|------------------|.....|
1/1 8/7 16/13 4/3 32/21 64/39 16/9 2/1
0 231 359 498 729 857 996 1200

Indeed if read the tetrachord "the other way," with the string lengths
taken instead as frequencies (to put it in one modern way), we get
12:13:14:16, a tetrachord much favored by George Secor. John Chalmers
has the 28:26:24:21 or 1/1-14/13-7/6-4/3 I mentioned above in his
avicenna_diat.scl in the Scala archive; maybe this assumes the common
although not invariable Greek custom of putting the smallest interval
first.

>> Ah, but supraminor thirds are not restricted to Persian music. For
>> you see, they are important flavours in such maqams as Hijaz, Nikriz
>> and Neveser. I admit, though, that Persians are more fond of
>> supraminor seconds compared to middler seconds.

Might I ask for a few examples of these maqam flavors? I know some
general forms of Hijaz and Nakriz as practiced in some Arab regions,
but have learned that the names and tunings can vary in different
regions. (Suz-i Dilara is a very curious example which I maybe should
save for another post.) Below we discuss this further, and I guess at
possible examples in Hijaz and Ushshaq on Dugah. Nakriz I think of as
a pentachord with a lower tone plus a Hijaz tetrachord, and if I place
this on Neva in Zest-24, then the example might serve our purposes,
with the middle step of the pentachord at 333 cents or 40/33:

Neva Huseyni Dik Ajem Sehnaz Muhayyer
F* G* A B* C*
0 192 333 575 696
192 141 242 121

I am not sure what Neveser is, and this example would be especially
interesting.

>
>> 14:17:21 makes a wonderful neutral chord.
>
> Agreed!

> This is a fascinating design: 12 notes from a circulating temperament
> of Rameau, and the other 12 added to provide some delicious Maqam
> flavors!
>
> <[34]/tuning/topicId_76333.html#76333>
>

> Don't forget the 17-tone closed cycle achieved via superpythagorean
> fifths for 7-limit major and minor chords!

Yes! It's really beautiful to look at in Scala; the flavor of Hijaz on
the 1/1 at 0-142-409-498 looks especially tempting. By the way, starting
from your 27/16 or Perde Huseyni, we have a quite reasonable version
of Urmavi's 8/7x9/8x64/63, much more accurate than either Anooshfar's
or mine In Zest-24 <grin>:

|----------------------|-----------------------|........|
0 214 437 498 703 925 990 1200
214 223 61 205 222 65 210

Of course, Urmavi's septimal steps are tempered, and we have a 3/2
rather than the original 32/21; but such are the inherent compromises
of a temperament, and in fact the vital 28:27 steps are within about
two cents of the just size, rather than 13 cents or more narrow. To
quote Anooshfar's tuning for comparison from the Scala archive:

! iran_diat.scl
!
Iranian Diatonic from Dariush Anooshfar, Safi-a-ddin Armavi's scale from 125 ET
7
!
220.800 cents
441.600 cents
489.600 cents
710.400 cents
931.200 cents
979.200 cents
2/1

> I urge you not to consider this subset of 125-EDO as a basis for any
> scale construction. The correct Urmavi scale is the one I provided
> above.

Yes, likely either that Ushshaq or else 8/7x9/8x28/27. As I might put
it, while according to Wright a step like 36/35 does sometimes appear
in sources for maqam tunings around 1250-1300, something like
iran_diat.scl might reflect either some Iranian popular tradition of
intonation or else a xenharmonic variation. It's too bad that Dariush
Anooshfar isn't involved in this conversation, because then he could
explain the connection with Urmavi.

>> Also, the structure of the Zest-24 tuning itself imposes some
>> arbitrary limitations. For example, your 24-note system includes
>> neutral thirds at both 17/14 and 16/13; or, in your 79-MOS, sizes
>> of both 332 and 362 cents above Rast. In Zest-24, however, I
>> realized that it is impossible to have two sizes of neutral
>> thirds above the same perde or step. The situation might be
>> analogous to that of some equal temperaments where only a single
>> size of middle or neutral third is available -- although here,
>> the size may change as one moves around the system.

> 17/14 is not only requisite of Ushshaq as a very low perde segah,
> but also as the second degree of Hijaz as a fairly steep perde
> kurdi.

Please let me say that if only I could process your dissertation with
my PDF software, then I would read that first and then ask questions!
However, for now, I realize that examples of Ushshaq and Hijaz would
help me in understanding how a 17/14 step is requisite for either.
With Hijaz on Dugah, would this be a correct understanding?

from Rast: 9/8 17/14 1024/729 3/2
from Dugah: 1/1 68/63 8192/6561 4/3
Dugah Dik Kurdi Hijaz Neva
0 132 384 498
132 252 112

This flavor of Hijaz would be a bit like Tala`i's Persian Chahargah
tetrachord (to use the dastgah name) at around 0-140-380-498 or
140-240-120.

With Maqam Ushshaq, to ask for an example might be wisest. If this
also starts on Dugah as final, then we would have something like:

from Rast: 9/8 17/14 4/3 3/2
from Dugah: 1/1 68/63 13/11 4/3
Dugah Nerm Segah Chahargah Neva
0 132 289 498
132 157 209

Is this close to a correct example of 17/14 in Ushshaq?

[My possible names of steps for a Maqam tuning in Zest-24]

> --------------------------------------------------------------------
> approx tentative approx approx
> degree cents 159-EDO perde name 288-EDO RI
> --------------------------------------------------------------------
> 0: 0.00000 0 Rast 0 1/1
> 1: 32.81250 4 (Sarp Rast) 8 64/63
> 2: 83.20312 11 Shuri 20 22/21
> 3: 153.51562 20 (Zengule cluster) 37 12/11
> 4: 203.90624 27 Dugah 48 9/8
> 5: 223.82812 30 (Sarp Dugah) 54 8/7
> 6: 274.21874 37 Nerm (soft) Kurdi 66 75/64
> 7: 344.53124 46 Segah of Ushshaq 83 11/9
> 8: 394.92187 52 Segah 95 5/4
> 9: 440.62500 58 Dik Nishabur 106 9/7
> 10: 491.01562 65 Chargah 118 4/3
> 11: 536.71875 71 Nim Hijaz 129 15/11
> 12: 587.10937 78 Hijaz 141 7/5
> 13: 657.42187 87 (Saba cluster) 158 19/13
> 14: 707.81249 94 Neva 170 3/2
> 15: 727.73437 96 (Sarp Neva) 175 32/21
> 16: 778.12500 103 Bayyati 187 11/7
> 17: 849.60937 113 (Hisar/Huzzam cluster) 204 18/11
> 18: 899.99999 119 Huseyni 216 27/16
> 19: 931.64062 123 (Sarp Huseyni) 224 12/7
> 20: 982.03124 130 Koutchouk (little) Ajem 236 30/17
> 21: 1040.62500 138 (Evdj cluster) 250 51/28
> 22: 1091.01562 145 Evdj 262 15/8
> 23: 1149.60937 152 Mahurek 276 64/33
> 24: 1200.00000 159 Gerdaniye 288 2/1
>
>

[Your reply]

> Since there are no more steps between rast and shuri, you should
> name the second degree as dik rast. The same goes for sarp dugah,
> sarp neva, and sarp huseyni. They should all be dik in
> parantheses. Also, you should name 154 cents as Zengule without the
> paranthesis. 274 cents makes a very low kurdi, but kurdi
> nonetheless. Similarly, 345 cents is could be a dik kurdi if you
> prefer. 441 cents is buselik and nishabur at the same time since
> there are no other options. I'd rather you called it a buselik. Saba
> cluster should be plainly saba. You could drop one y from Bayyati to
> make it more Turkish. 845 cents is plainly hisar without the
> paranthesis. Little ajem is needless since we took kurdi
> plainly. Let's call it simply ajem. Evdj cluster is a misnomer here,
> it should be a dik ajem. Mahurek ought to be mahur.

This does make a lot of sense to me: if we are looking at this
temperament as a system in itself, then if there is only one possible
tuning for a given perde, why not just say "ajem" or "mahur," etc.?
Your point about the spelling of Bayati may be especially fitting
because I've noticed that at least according to Marcus, the modern
Arab system doesn't have a perde Bayati (or Bayyati)!

As for "Evdj cluster" at 1041 cents, I realize that my concept of a
"neutral seventh" might better fit the Arab perde Awj, although your
"Evdj kumesi" at around 11:6 or 1049 cents might fit a typical Arab
tuning better than 1041 cents. The difference is that whereas Awj is
typically some kind of middle seventh, Turkish Evdj is evidently at
around 15/8 or 4096/2187 above Rast -- or 48 commas, rather than
somewhere around 46 or 47 commas for Arab perde Awj. This might be a
difference related to Segah at around 17 commas (5/4 or 8192/6561)
above Rast in Maqam Rast as understood in Turkey, but 16 commas or so
in a typical Egyptian interpretation (e.g. something like Zalzal's
27/22, which you have mentioned as one practice).

> But you see, you cannot play a decent Hijaz with this setup, because
> you are either consigned to a very low kurdi or a very high dik
> kurdi.

Certainly we agree that for Maqam Hijaz starting on Dugah,
0-70-383-504 cents or C*-Db*-E*-F* (Dugah-Kurdi-Hijaz-Neva) would place
the second step of the tetrachord "very low," contrary to the usual
understanding that the lowest step should be smaller than the
highest (here 70-313-121). However, at least if the first step of
Hijaz is understood to be a middle second, I might guess that a "very
high dik kurdi" could be fairly idiomatic:

Dugah Dik Kurdi Hijaz Neva Huseyni Ajem Gerdaniye Muhayyer
C* D E* F* G* Ab* Bb* C*
0 141 383 504 696 778 996 1200
141 242 121 192 82 218 204

Here the Hijaz tetrachord is quite close to Tala`i's 140-240-120, and
the first step is larger than the highest. Flexibly we might state
this conception of Hijaz as something around 6-11-5 commas. Of
course, for the classic kind of Hijaz described by Shirazi at
12/11x7/6x22/21 (a permutation of Ptolemy's Intense Chromatic), or
151-267-81, we would have to transpose elsewhere, for example, using a
spelling with some enharmonic equivalents (in Zest-24, F#=Gb, etc., in
each 12-note circle):

Beyati Dik Huseyni Rast Shuri Kurdi Segah Nim Hijaz Tiz Beyati
F#* G# A#* B* C#* D E F#*
0 154 422 505 696 766 959 1200
154 268 83 191 70 192 241

This classic type of Hijaz is more like 6.5-12-3.5 commas.

With many thanks,

Margo
mschulter@...

🔗Margo Schulter <mschulter@...>

Dear Jim and all,

Please let me, as you have invited people to do, offer some
suggestions as to the Thummer and the "Matrix" concepts which you and
your colleagues have presented in your paper of that title. What I
would like to focus on especially are some currents of world music and
of recent explorations specifically directed to "alternative tunings"
which might further enrich the Matrix paradigm.

Rather than attempting at this point to approach the Matrix in terms
of general theory, as interesting as that might be, I'd consider the
most productive strategy at this point from my point of view simply to
survey some tunings and styles which might be relevant to that theory.

One important theme in this survey is outlooks on melody on harmony
based on sets of partials or primes like 2-3-7, or 2-3-7-11-13. One
very readable presentation of this theme, written in 2002 although
published considerably later, is an article by George Secor, who in
1978 designed a 17-tone well-temperament ideally fitting such an
outlook:

<http://xenharmony.wikispaces.com/space/showimage/17puzzle.pdf>

As we'll see in the second part of this presentation, a regular
19-note Thummer tuning at 705 cents nicely serves many of the same
purposes, if not quite so simply or neatly in certain respects.
However, the topic of irregular temperaments such as George's does
lead to my main techical recommendation: support scale files in the
format used by Manuel Op de Coul's free scale generation and analysis
program known as Scala <http://www.xs4all.nl/~huygensf/scala/>.

Of course the Thummer, unlike Scala, is evidently limited to 19 notes
per octave; but 12-note and 17-note irregular temperaments are one
very useful application, and the Scala file format is a community
standard. A special advantage is that people can simply download the
Scala scale archive with over 3000 scales from various world musics
both old and new, or posted here, send them with Scala to a supported
device, and play. This option would greatly enhance the utility of the
Thummer, supplementing rather than replacing its "native" strategies
for defining a tuning and timbre.

Experience has taught me that I should not simply give a scale with
its chains of generators and sizes of intervals, assuming that the
stylistic possibilities will be self-explanatory. The range of world
musical styles, and of diverse backgrounds contributed by people here
and elsewhere in the "alternative tuning" community, is simply too
great. Especially if an area is familiar to me, but might not be so
familiar to many users or potential users of the Thummer, I'll try to
give some examples of how intervals and chords can be used in
practice, making this presentation longer but possibly more
informative.

-------------------------------------------
1. Gamelan at 694.74 cents: 19-tET slendro
-------------------------------------------

On an instrument with 19 notes per octave, it's only natural that
someone would try 19-tET. Let's see what happens when we apply an
interval mapping most commonly associated with the territory around
Pythagorean tuning with pure fifths, where beta is at about 702 cents,

The basic idea is that we will define our "minor third" as an
augmented second formed from nine betas or fifths up less five
octaves, or [-5, +9]. Using these special minor thirds, we can form a
scale like this:

[0, 0] [-5, 9] [1,-1] [0, 1] [-5, 10] [1, 0]
C D# F G A# C
0 253 505 695 948 1200
253 253 189.5 253 253

Note that in 19-tET, this mapping divides the fourth at slightly
larger than 505 cents into two equal intervals -- each an augmented
second, or very small minor third, at 253 cents. Four of these small
minor thirds, plus a usual 19-tET major second or tone [-1, +2] at F-G
with a size of around 189.5 cents, make up an octave of 2:1 or 1200
cents, our period or alpha. The 253-cent third has a size close to a
just 22:19 (254 cents), and the 948-cent seventh to 19:11 (946
cents). The small third could possibly also be taken for a very large
major second or whole tone, and the small seventh also for a very
large major sixth! Intervals of this type around 250 and 950 cents are
very common in gamelan music.

This beautiful slendro tuning differs in two important ways from a
typical traditional slendro of Java or Bali. First, traditional
tunings are more or less subtly irregular, with the steps differing
from those of 5-tET (with five 240-cent steps) or any other regular
tuning, and the octaves often slightly wider or narrower than 2:1.
Also, the smallest step of a slendro is typically rather wider than
190 cents, although there is immense diversity and steps of around
this size may sometimes occur.

As with 5-tET, what we have is a charming variation on traditional
slendro -- which, in 19-tET, is available as a kind of alternative to
or change of pace from the usual Renaissance meantone color which was
Guillaume Costeley's main concern when he described the tuning in
1570. Let's move to a different region which likewise offers some
fascinating alternative mappings.

--------------------------------------------------
2. Encounter with Archytas and Wilson at 702 cents
--------------------------------------------------

The term "Pythagorean tuning" refers to a system of just intonation
(JI) based on the pure ratios of the 2:1 octave, 3:2 fifth, 4:3
fourth, and 9:8 tone or major second (1200, 702, 498, and 204 cents).
This style of tuning has been used in a variety of world musics in
places such as China and Japan, very possibly Babylonia, Greece, the
medieval and modern Near East, and medieval Europe.

Thus tunings at or near 702 cents do very well in styles where fifths
and fourths are the main consonances, sometimes contrasting very
effectively with more complex intervals which invite directed
resolutions to these stable consonances, as in 12th-14th century
European compositions. Additionally, extended Pythagorean tuning can
closely approximate some other simple JI ratios, as we'll see here.

One musically engaging possibility is presented by the ancient Greek
theorist Archytas, who describes a tuning which Ptolemy later adopted
as the "Tonic Diatonic." This tuning features some pure ratios based
on primes 2, 3, and 7: a minor third at 7:6, a major third at 9:7, and
a minor seventh at 7:4 (267, 435, and 969 cents).

To approximate these ratios of primes 2-3-7 using only chains of pure
or virtually pure fifths (3:2 is 701.955 cents), we'll need rather
long chains -- which, fortunately, our 19-note tuning is just large
enough to accommodate so that at one unique location, G#, we can find
all seven notes of our desired Archytas tuning. Our arrangement occurs
on the seventh step of the original Archytas/Ptolemy version (closely
approximated in this version if one proceeds A#-A#):

[0, 0] [-1, 2] [9, -15] [1, -1] [0, 1] [-1, 3] [9, -14] [1, 0]
G# A# Cb C# D# E# Gb G#
0 204 271 498 702 906 973 1200
204 67 227 204 284 67 227

Although it is debatable whether or to what degree the harmony of two
or more voices sounding at the same time may have been a factor in
ancient Greek music, today this and related tunings are often used in
styles of harmony rather different from major/minor tonality based on
the 4-5-6 triad. Let's consider this three-voice progression:

D# E#
Cb A#
G# A#

The first chord at 0-271-702 cents closely approximates partials
6-7-9, with a small 7:6 minor third below and a large 9:7 major third
above. The small minor third then contracts to a unison, while the
large major third expands to a fifth, bringing us to the stable fifth
A#-E#. We can expand this to a four-voice progression, and bring in
the additional attraction of a 7:4 minor seventh; like our 7:6 minor
third, this interval will be wide of just by about 3.80 cents, a
rather small amount. The opening chord at 0-271-702-973 cents thus
offers a near-just rendition of partials 12-14-18-21:

Gb E#
D# E#
Cb A#
G# A#

We get a somewhat different color by rearranging the voices of the
first chord so that a large or 9:7 major third above the lowest voice
expands to a fifth, while a large major sixth around 12:7 expands to
an octave:

G# A#
Gb E#
D# E#
Cb A#

The lower three voices of this chord can be used in an "Archytan"
variation of a cadence very popular in 13th-century European music,
where a major third contracts to a unison while a minor third expands
to a fifth, which each voice moving by some size of major second or
tone:

Gb E#
D# E#
Cb A#

The opening chord with a 9:7 major third _below_ the 7:6 minor third
is in a typical harmonic timbre quite active and strident -- causing
some to call 9:7 the "car horn third," although more favorably stirred
listeners may call it the "clarion third." This chord in our 702-cent
tuning is very close to a just 14:18:21. Here the tension is fitting,
because the chord leads into a resolution to a stable fifth, the
favorite consonance in medieval European music.

However, as the above examples show, the effect is smoother if we
either place the minor 7:6 third below the major 9:7 one -- 6-7-9 --
or else take 14:18:21 and add a 12:7 major sixth, making it
14:18:21:24. as in the example before the last.

In fact, as George Secor and others have pointed out, a very
harmonious five-note or pentatonic tuning is available using these
ratios which Lou Harrison, among others, has favored as a kind of
slendro.

[0, 0] [9, -15] [1, -1] [0, 1] [9, -14] [1, 0]
G# Cb C# D# Gb G#
0 271 498 702 973 1200
271 227 204 271 227
approx JI: 12 14 16 18 21 24
1/1 7/6 4/3 3/2 7/4 2/1

As Secor suggests, all five notes of this scale played at once make a
harmonious chord, approximating partials 12-14-16-18-21. We can, if we
like, focus on fifths and fourths as the main stable concords -- as
often happens in Javanese or Balinese slendro, where, however, these
intervals are typically tuned at some distance from a pure 3:2 or 4:3
in a way that nicely fits the inharmonic timbres! However, Secor calls
attention to some distinctly "modern" possibilities which the Archytan
or 2-3-7 prime approach opens to musicians seeking something new.

Erv Wilson, a music theorist and instrument designer much interested
in generalized keyboards who recently celebrated his 80th birthday,
has arrived at a beautiful scale related to this last, but with six
notes per octave rather than five -- and therefore called a "hexany."
Hexanies may be based on various JI ratios or factors: this is the
1-3-7-9 hexany, since it uses these four factors to generate the
tuning.

Hexanies and other "Combination Product Sets" or CPS systems for short
do this by multiplying subsets of the factors, here 1-3-7-9, to arrive
at products which represent the numbers of partials, and when reduced
to fit within a single octave, define the ratios and intervals of the
tuning. In a CPS system proper, although the number 1 is used in
generating the set, it does not actually itself appear in the tuning
as a "fundamental" note: all of the notes bring into play other
factors. Here the CPS identity for each note as defined by a product
of two of the four factors (1-3-7-9), as well as the Thummer generator
chain, is shown for each note:

[0, 0] [-1, 2] [9, -15] [8, -13] [0, 1] [9, -14] [1, 0]
G# A# Cb Db D# Gb G#
1.3 3.9 1.7 7.9 1.9 3.7 1.3
0 204 271 475 702 973 1200
204 67 204 227 271 227

The hexany features some sonorities or chords we have already
encountered like 6:7:9 (G#-Cb-D#) or 12:14:18:21 (G#-Cb-D#-Gb). We
additionally encounter a wonderful chord featuring an interval with a
very special and striking effect: the narrow fourth at 21:16, here
tempered like other ratios of 7 at about 3.8 cents wide, or 475 cents:

Gb
D#
Cb
G#

This sound closely approximates partials 16-21-24-28: the 3:2 fifth
and outer 7:4 minor seventh "anchor" the chord, while the 21:16 above
the lowest voice adds a certain feeling of "tension" or energy.

While hexanies and similar tunings are often intended to encourage an
exploration of "consonance for its own sake," so to speak, rather than
conventionally directed harmony of one sort or another, the hexany
does support a characteristic 13th-century Euyropean cadence where a
minor third expands to a fifth and a minor sixth to an octave. Here
these intervals have proportions very close to a just 7:6 and 14:9
(267 and 765 cents in JI, and here 271 and 769 cents):

Gb G#
Db D#
A# G#

We can also add a seventh note to our hexany, making it a related
heptatonic or "seven-note" scale:

[0, 0] [-1, 2] [9, -15] [8, -13] [0, 1] [-1, 3] [9, -14] [1, 0]
G# A# Cb Db D# E# Gb G#
0 204 271 475 702 906 973 1200
204 67 227 204 284 67 227

This scale adds a usual Pythagorean major sixth E# above our resting
note or "final" of G# -- a note also present in our earlier Archytan
tuning. The difference is that here, as in the hexany, we have a
narrow 21:16 fourth at Db rather than a usual 4:3 Pythagorean fourth
at C#.

With this tuning, we have a characteristic resolution available for
the 16:21:24:28 chord featured in the hexany:

Gb E#
D# E#
Cb A#
G# A#

As in some of the earlier progressions above, the outer 7:4 minor
seventh contracts to a fifth. This cadence very nicely releases the
"tension" of the 16:21:24:28 chord; but from another viewpoint,
encouraged by the hexany, we can take it simply as an energetic kind
of consonance that doesn't necessarily need to "go" anywhere. Our
Thummer system pleasantly allows both viewpoints.

Let's return for a moment to our original Archytan scale to make a
final point about Pythagorean tuning and its musical possibilities.

[0, 0] [-1, 2] [9, -15] [1, -1] [0, 1] [-1, 3] [9, -14] [1, 0]
G# A# Cb C# D# E# Gb G#
0 204 271 498 702 906 973 1200
204 67 227 204 284 67 227

We've gone to considerable lengths in seeking out those 7-based
intervals: in fact, in order to get this simple scale, we've had to
draw on the full length or span of our 19-note gamut, from Cb to E#!
However, many of the progressions we've explored with Archytan or
2-3-7 prime intervals and chords are also effective with the usual
Pythagorean intervals such as thirds and sixths -- as they should be,
since some of them are taken directly from 13th-14th century European
compositions written in a period when Pythagorean was the usual tuning
system.

Consider, for example, this progression:

E# Gb
C# Cb
A# Cb

Our opening chord is a regular Pythagorean version of a lower minor
third at 294 cents or 32:27, and an upper major third C#-E# at 408
cents or 81:64. As in earlier examples, the minor third contracts to a
unison while the major third expands to a fifth. For many listeners,
this type of chord, known in medieval times as a "split fifth" where
an outer fifth is divided or "split" into two thirds, may sound more
harmonious with the minor third placed below -- as we found also with
6:7:9 and 14:18:21. However, the form with the quite active 81:64
major third placed below is also very common and effective in
13th-century writing:

G# A#
E# D#
C# D#

This outlook and the major third as a _relatively_ "concordant" but
rather tense interval pulling dynamically toward some kind of
resolving cadence is quite different from the view of major/minor
tonality, where it is the essence of rest and stability in a system
based on partials 4-5-6 -- and the different tunings fit naturally
with these different worldviews.

Also, while we've rightly focused a lot of attention on the 7:4 minor
seventh, the Pythagorean tuning has another very important type of
minor seventh at the ratio of 16:9, equal to two Pythagorean 4:3
fourths, or two fifths down plus two octaves [2, -2]. Around 1200,
this seventh is featured by Perotin and other composer in a most
impressive closing cadence where the lowest voice remains stationary:

G# A#
D# E#
A#

The cadence has a 4:3 fourth and 16:9 minor seventh (498 and 996
cents) above the lowest voice, with the upper voices progressing
smoothly in parallel fourths to the fifth and octave, arriving at the
ideal medieval harmony defined by partials 2-3-4. In period theory
the minor seventh is often called an "imperfect" interval -- often an
"imperfect discord," but sometimes an "imperfect concord" -- meaning
that it is rather tense in itself, but not acutely so, and has a
certain degree of blend also. Combining it with two pure fourths, as
in A#-D#-G#, brings out this "somewhat blending" aspect.

In 20th-century terms, this would be called a "fourth chord," often
used by composers such as Debussy and Bartok, and also favored in some
styles of jazz harmonization. In fact, sonorities built in fourths or
fifths are common not only in medieval and modern European practice,
but in various other world musics, with Japanese gagaku or traditional
orchestral court music taking this process to an impressive degree of
complexity.

Happily, a 19-note Pythagorean tuning abounds in pure fifths and
fourths, and also in other regular intervals such as thirds and sixths
that can be used in the familiar patterns of 13th-14th century
European and other world musical styles -- as well as some "special"
intervals such as the Archytan.

Of course, it's additionally possible to use a schismatic mapping in
order to obtain pure or near-pure ratios of 5 (e.g. thirds at 5:4 and
6:5). However, a regular Pythagorean mapping offers the option of
exploring a very rich territory that has nourished many centuries of
world musical experience; and the Archytan intervals are there for
those with the patience to seek them out. As we'll see, there are
other places on the continuum where these 2-3-7 intervals abound; but
the allure of combining them with pure fifths and fourths makes the
search at 702 cents worth the effort.

(To be continued)

Most appreciatively,

Margo Schulter
mschulter@...

🔗Herman Miller <hmiller@...>

🔗Billy Gard <billygard@...>

🔗Ozan Yarman <ozanyarman@...>

O Margo,

Fazli is my kinsman and colleague, who for the first time in our history translated Risala Ash-Sharafiyyah to Turkish. Only after reading Arslan's article (which he wanted me to check beforehand) and the said book have I seen how Urmevi employs very high limit intervals in his divisions of tetrachords and pentachords. You might want to examine Arslan's article, even though it is in Turkish:

http://www.fazliarslan.com/safiyyuddin_serefiyyesinde_musiki_matematigi.pdf

The tetrachord 8:7 x 9:8 x 28:27 is called first muttasil (contiguous). Using this tetrachord in an adjunct manner, Urmevi constructs and octave species he calls the fourth Munfasil al-ahad. It is clear that Urmevi is thinking in terms of JI here, and that 125 logarithmic divisions of the octave cannot yield a subset to satisfactorily represent all intervals, while Yarman24, as you discovered, can (best on the 7th, 16th, 17th, 20th, 21st degrees).

Though, I admit with enthusiasm that Zest-24 does a very good job with many maqam flavours!

Also, bear in mind that Urmevi was demonstrating mental exercises to attain tetrachord and pentachord divisions compatible with practice. It is clear that he implies that not all of the divisions he formulated are to be used. He even categorizes them vaguely from most consonant to the least.

I agree that small neutral thirds are a specialty of Persian music under the sphere of Eastern maqam musics, but I do not agree that 170 cents is a particular feature of the genre.

Dear Margo, I am experiencing mental fatigue and beseech you to postpone your request for more maqam flavours in Hijaz, Nikriz and Neveser. Let me briefly state that Turkish musicians tend to raise the second degree of Hijaz and lower the third. Supraminor seconds can be seen in Ushshaq and its variants (possibly Karjighar) as well. Everywhere there is a variant of Hijaz or Ushshaq, you are likely to find supraminor seconds.

I found this site, with a link to Neveser too:

http://www.makamlar.net/makamlar.html

The pentachord you have given resembles a transposed intro to Nikriz or Neveser.

Yes, the Hijaz and Ushshaq tetrachords you have given are spot on! However, I do not use 1024/729 specifically, but a 380 cent interval. Don't forget 16/13 as an Ushshaq flavour when ascending melodically.

Turkish evdj indeed echoes Turkish segah, as it is the fifth complement of the latter.

Ah, 141 cents for dik kurdi is way too steep I'm afraid. But for the lack of any better interval, 141 cents might be substituted in place of 68/63. Shirazi's version would require perde hijaz to rise along with perde segah. But when hijaz is as low as 5/4 from dugah, dik kurdi has to be lower than 140 cents.

Cordially,
Oz.

On Aug 12, 2008, at 4:11 AM, Margo Schulter wrote:

>> Dear Margo,
>
>> My apologies for the very late reply. I have been enjoying a well-
>> earned summer's rest. I hope the importance of the topic has not
>> faded while I was away. My comments are below:
>
> Dear Ozan,
>
> Please let me say that no apologies are needed, and that I am
> delighted that you have been enjoying your summer after writing such a
> germinal dissertation! Of course, the topic is as exciting for me as
> ever, and exploring the curious tuning that I give below and some Arab
> maqam theory, as well as your papers on your website, has heightened
> that interest yet more.
>
> And so we return to our discussion of Dariush Anooshfar's
> iran_diat.scl in the Scala archives, offered as an interpretation of
> an unspecified tuning of Safi al-Din al-Urmavi in the later 13th
> century.
>
> Please let me apologize in advance for trimming a bit: I've tried to
> edit mainly my own remarks, or passages where we are agreeing on
> things already said in previous discussions.
>
>> Ah, I am sad to say that the Scala 125-EDO subset does great >> injustice
>> to the original Ushshaq scale of Urmavi, which I believe goes:
>
>> 9:8 x 9:8 x 256:243 x 9:8 x 9:8 x 256:243 x 9:8
>
>> Notice how the limma has been reduced in size by half in the Scala
>> file so as to destroy the qualities of the half-tone step, making it
>> a quarter-tone step. Bear in mind that this Ushshaq has nothing to
>> do with the Ushshaq we perform today. I am at a lost to explain how
>> a diatonic major scale got transformed to a neutral minor during the
>> centuries that followed Urmavi.
>
> This looks like a very interesting conversation! This summer I found
> on the Web a paper by Dr. Fazli Arslan of the Turkish Ministry of
> Education and others, _Safi al-Din al-Urmawi and the Theory of Music_
> (March 2007), which leads to the conclusion that Anooshfar might have
> had a different tuning of Urmavi in mind -- but first this Ushshaq.
>
> Certainly we agree that if Anooshfar had intended this tuning, then
> his 125-EDO version would seem very curious! I wonder if, instead, he
> had this septimal flavor of Urmavi, again with two conjunct
> tetrachords plus an upper 9:8 tone, in mind, based on what Urmavi
> called "the first conjunct" tetrachord of 8/7x9/8x28/27, to which I
> will add an indication of approximate Pythagorean-53 or 53-EDO commas:
>
> 10 9 3 10 9 3 9
> commas: 0 10 19 22 32 41 44 53
> 1/1 8/7 9/7 4/3 32/21 12/7 16/9 2/1
> cents: 0 231 435 498 729 933 996 1200
> 8:7 9:8 28:27 8:7 9:8 28:27 9:8
> 231 204 63 231 204 63 204
> |-----------------|----------------------|.....|
> lower jins upper jins 9:8
>
> This 0-231-435-498 or 231-204-63 tuning seems to me considerably
> closer to Anooshfar's tetrachord of about 0-221-442-490 or 221-221-48
> than Urmavi's Ushshaq at 0-204-408-498 (or 9-9-4 commas). If we start
> from this septimal flavor of Urmavi, then three basic changes might
> lead to an interpretation like Anooshfar's.
>
> The first change is to use some fifths tempered about 1/3-comma wide,
> or fourths 1/3-comma narrow, so that we have Urmavi's 4/3 placed at
> around 491 cents and his 16/9 at around 982 cents -- as in 22-EDO.
> The second is a very dramatic narrowing of the original 28:27 step (as
> in the diatonic of Archytas, of which Urmavi's tuning is a
> permutation) from 63 cents to 50 cents, a quirk to which both
> Anooshfar and I seem curiously attracted <grin>:
>
> Bb* Db Eb Eb* Gb Ab Ab* Bb*
> cents: 0 224 441 491 728 932 982 1200
> 224 217 50 237 204 50 218
> |-----------------|----------------------|.....|
> lower jins upper jins 9:8
>
> To get an approximation of Anooshfar's 0-221-442-490-710-931-979-1200,
> we need additionally to substitute a wide tempered 3/2 for Urmavi's
> 32/21, interestingly almost just in the last Zest-24 example at 728
> cents:
>
> Bb* Db Eb Eb* F* Ab Ab* Bb*
> cents: 0 224 441 491 708 932 982 1200
> 224 217 50 217 224 50 218
> |-----------------|----------------------|.....|
> lower jins upper jins 9:8
>
> Humorously, I might suggest that my attraction to this Anooshfar
> tuning comes from the routine license in Zest-24 of often tempering
> 28:27 at 50 cents, actually a virtually just 35:34! A milder version
> of this license would occur in 22-EDO:
>
> C D E F G A Bb C
> cents: 0 218 436 491 709 927 982 1200
> 218 218 55 218 218 55 218
> |-------------|--------------|.....|
> lower jins upper jins tone
>
> The liberty involved here could be put another way: Urmavi's 16/9 is
> being tempered with narrow fourths down to something like 982 cents,
> which represents both 16/9 and 7/4. Thus his interval 12/7-16/9
> becomes something like 12/7-30/17, or about 35:34 -- almost exactly
> what it is in Zest-24! If the idea is to get a somewhat less
> inaccurate version of Urmavi's steps, then in Zest-24 we might try:
>
> |---------------|---------------|......|
> C D* E* F G A* Bb C
> 0 242 434 504 696 937 996 1200
> 242 192 70 192 241 59 204
>
> Here either 70 cents or 59 cents is a more or less reasonable
> approximation of 28:27 -- although not as close as with your 17
> perdeler! We have a virtually just 16/9 without any pretensions of
> representing 7/4. The 241/242-cent steps are rather wider than 8:7;
> but one might argue that getting 28:27 more accurately is most
> important if the idea is faithfully to represent Urmavi's tuning.
>
>> I should state further, that it is most unfortunate that Turks have
>> not written a single academic work on their conception of Maqam
>> music for the auspices of the international music community since
>> Rauf Yekta. My thesis should serve as a brick that closes the
>> centennial gap.
>
> Yes, and clearly I am encountering a software limitation in
> Ghostscript that causes a VMerror (not enough virtual memory) in
> processing the file. Obviously it is vital that I find some solution
> -- possibly a later version of this software that might handle more
> recent PDF files (I tried this some months ago and it solved a problem
> with a set of PDF files for John Chalmers' book on tetrachords.)
>
> This is _my_ problem, not yours, and it's obvious that I need to
> figure out some way of reading and reviewing your dissertation.
>
> [...]
>
> [On avoiding the region around 170 cents, or "equitable heptatonic"
> close to 7-EDO, in maqam music]
>
>> Indeed so. I remember having stated the necessity to avoid the
>> equitable heptotonic in Yarman24.
>
> Yes, over the summer I found your 24-note system where you mentioned
> this. Personally, I tend to regard a 170-cent step as a "Persian
> liberty," more fitting to a dastgah style than a usual maqam style.
> Since it often occurs in a Zest-24 tetrachord like 0-191-333-504 cents
> (e.g. C*-D*-E-F*) with steps of 191-141-171 cents, where the small
> neutral third around 40:33 would fit a Persian style, this view might
> have some connection to Near Eastern practice. An interesting question
> is whether a tetrachord like F*-G*-A-Bb* or 0-192-333-492 cents, with
> steps of 192-141-159 cents, might fit into an Arab or Turkish maqam
> style. You've mentioned that these small neutral thirds do occur.
>
>
>> One place 26/21 comes up is in Ibn Sina's 28:26:24:21 tetrachord.
>
>> I checked Ibn Sina's passages. The actual tetrachord is given as 8:7
>> x 14:13 x 13:12. The tetrachord you have mentioned permutes the last
>> two intervals. In the original form, we have 16:13 instead of
>> 26:21. But I suspect the tetrachord should be read the other way
>> around, which yields 12:13:14:16. Ibn Sina calls this a Kavi genus
>> which is of quality and nobility, I am not sure what this
>> corresponds to in music theory of the Ancient world. Ibn Sina says
>> that this is the genus preferred by Ptolemy.
>
> This tetrachord comes up in Urmavi also, and the string ratios given
> confirm the same order you give for Ibn Sina's version; Urmavi's
> tuning uses conjunct tetrachords, with a 32/21 above the final, as in
> the 8/7x9/8x28/27 tuning discussed above:
>
> lengths: 64 56 52 48 42 39 36 32
> |----------------|------------------|.....|
> 1/1 8/7 16/13 4/3 32/21 64/39 16/9 2/1
> 0 231 359 498 729 857 996 1200
>
> Indeed if read the tetrachord "the other way," with the string lengths
> taken instead as frequencies (to put it in one modern way), we get
> 12:13:14:16, a tetrachord much favored by George Secor. John Chalmers
> has the 28:26:24:21 or 1/1-14/13-7/6-4/3 I mentioned above in his
> avicenna_diat.scl in the Scala archive; maybe this assumes the common
> although not invariable Greek custom of putting the smallest interval
> first.
>
>>> Ah, but supraminor thirds are not restricted to Persian music. For
>>> you see, they are important flavours in such maqams as Hijaz, Nikriz
>>> and Neveser. I admit, though, that Persians are more fond of
>>> supraminor seconds compared to middler seconds.
>
> Might I ask for a few examples of these maqam flavors? I know some
> general forms of Hijaz and Nakriz as practiced in some Arab regions,
> but have learned that the names and tunings can vary in different
> regions. (Suz-i Dilara is a very curious example which I maybe should
> save for another post.) Below we discuss this further, and I guess at
> possible examples in Hijaz and Ushshaq on Dugah. Nakriz I think of as
> a pentachord with a lower tone plus a Hijaz tetrachord, and if I place
> this on Neva in Zest-24, then the example might serve our purposes,
> with the middle step of the pentachord at 333 cents or 40/33:
>
> Neva Huseyni Dik Ajem Sehnaz Muhayyer
> F* G* A B* C*
> 0 192 333 575 696
> 192 141 242 121
>
> I am not sure what Neveser is, and this example would be especially
> interesting.
>
>
>>
>>> 14:17:21 makes a wonderful neutral chord.
>>
>> Agreed!
>
>> This is a fascinating design: 12 notes from a circulating temperament
>> of Rameau, and the other 12 added to provide some delicious Maqam
>> flavors!
>>
>> <[34]/tuning/topicId_76333.html#76333>
>>
>
>> Don't forget the 17-tone closed cycle achieved via superpythagorean
>> fifths for 7-limit major and minor chords!
>
> Yes! It's really beautiful to look at in Scala; the flavor of Hijaz on
> the 1/1 at 0-142-409-498 looks especially tempting. By the way, > starting
> from your 27/16 or Perde Huseyni, we have a quite reasonable version
> of Urmavi's 8/7x9/8x64/63, much more accurate than either Anooshfar's
> or mine In Zest-24 <grin>:
>
> |----------------------|-----------------------|........|
> 0 214 437 498 703 925 990 1200
> 214 223 61 205 222 65 210
>
> Of course, Urmavi's septimal steps are tempered, and we have a 3/2
> rather than the original 32/21; but such are the inherent compromises
> of a temperament, and in fact the vital 28:27 steps are within about
> two cents of the just size, rather than 13 cents or more narrow. To
> quote Anooshfar's tuning for comparison from the Scala archive:
>
> ! iran_diat.scl
> !
> Iranian Diatonic from Dariush Anooshfar, Safi-a-ddin Armavi's scale > from 125 ET
> 7
> !
> 220.800 cents
> 441.600 cents
> 489.600 cents
> 710.400 cents
> 931.200 cents
> 979.200 cents
> 2/1
>
>> I urge you not to consider this subset of 125-EDO as a basis for any
>> scale construction. The correct Urmavi scale is the one I provided
>> above.
>
> Yes, likely either that Ushshaq or else 8/7x9/8x28/27. As I might put
> it, while according to Wright a step like 36/35 does sometimes appear
> in sources for maqam tunings around 1250-1300, something like
> iran_diat.scl might reflect either some Iranian popular tradition of
> intonation or else a xenharmonic variation. It's too bad that Dariush
> Anooshfar isn't involved in this conversation, because then he could
> explain the connection with Urmavi.
>
>>> Also, the structure of the Zest-24 tuning itself imposes some
>>> arbitrary limitations. For example, your 24-note system includes
>>> neutral thirds at both 17/14 and 16/13; or, in your 79-MOS, sizes
>>> of both 332 and 362 cents above Rast. In Zest-24, however, I
>>> realized that it is impossible to have two sizes of neutral
>>> thirds above the same perde or step. The situation might be
>>> analogous to that of some equal temperaments where only a single
>>> size of middle or neutral third is available -- although here,
>>> the size may change as one moves around the system.
>
>> 17/14 is not only requisite of Ushshaq as a very low perde segah,
>> but also as the second degree of Hijaz as a fairly steep perde
>> kurdi.
>
> Please let me say that if only I could process your dissertation with
> my PDF software, then I would read that first and then ask questions!
> However, for now, I realize that examples of Ushshaq and Hijaz would
> help me in understanding how a 17/14 step is requisite for either.
> With Hijaz on Dugah, would this be a correct understanding?
>
> from Rast: 9/8 17/14 1024/729 3/2
> from Dugah: 1/1 68/63 8192/6561 4/3
> Dugah Dik Kurdi Hijaz Neva
> 0 132 384 498
> 132 252 112
>
> This flavor of Hijaz would be a bit like Tala`i's Persian Chahargah
> tetrachord (to use the dastgah name) at around 0-140-380-498 or
> 140-240-120.
>
> With Maqam Ushshaq, to ask for an example might be wisest. If this
> also starts on Dugah as final, then we would have something like:
>
> from Rast: 9/8 17/14 4/3 3/2
> from Dugah: 1/1 68/63 13/11 4/3
> Dugah Nerm Segah Chahargah Neva
> 0 132 289 498
> 132 157 209
>
> Is this close to a correct example of 17/14 in Ushshaq?
>
> [My possible names of steps for a Maqam tuning in Zest-24]
>
>> --------------------------------------------------------------------
>> approx tentative approx approx
>> degree cents 159-EDO perde name 288-EDO RI
>> --------------------------------------------------------------------
>> 0: 0.00000 0 Rast 0 1/1
>> 1: 32.81250 4 (Sarp Rast) 8 64/63
>> 2: 83.20312 11 Shuri 20 22/21
>> 3: 153.51562 20 (Zengule cluster) 37 12/11
>> 4: 203.90624 27 Dugah 48 9/8
>> 5: 223.82812 30 (Sarp Dugah) 54 8/7
>> 6: 274.21874 37 Nerm (soft) Kurdi 66 75/64
>> 7: 344.53124 46 Segah of Ushshaq 83 11/9
>> 8: 394.92187 52 Segah 95 5/4
>> 9: 440.62500 58 Dik Nishabur 106 9/7
>> 10: 491.01562 65 Chargah 118 4/3
>> 11: 536.71875 71 Nim Hijaz 129 15/11
>> 12: 587.10937 78 Hijaz 141 7/5
>> 13: 657.42187 87 (Saba cluster) 158 19/13
>> 14: 707.81249 94 Neva 170 3/2
>> 15: 727.73437 96 (Sarp Neva) 175 32/21
>> 16: 778.12500 103 Bayyati 187 11/7
>> 17: 849.60937 113 (Hisar/Huzzam cluster) 204 18/11
>> 18: 899.99999 119 Huseyni 216 27/16
>> 19: 931.64062 123 (Sarp Huseyni) 224 12/7
>> 20: 982.03124 130 Koutchouk (little) Ajem 236 30/17
>> 21: 1040.62500 138 (Evdj cluster) 250 51/28
>> 22: 1091.01562 145 Evdj 262 15/8
>> 23: 1149.60937 152 Mahurek 276 64/33
>> 24: 1200.00000 159 Gerdaniye 288 2/1
>>
>>
>
> [Your reply]
>
>> Since there are no more steps between rast and shuri, you should
>> name the second degree as dik rast. The same goes for sarp dugah,
>> sarp neva, and sarp huseyni. They should all be dik in
>> parantheses. Also, you should name 154 cents as Zengule without the
>> paranthesis. 274 cents makes a very low kurdi, but kurdi
>> nonetheless. Similarly, 345 cents is could be a dik kurdi if you
>> prefer. 441 cents is buselik and nishabur at the same time since
>> there are no other options. I'd rather you called it a buselik. Saba
>> cluster should be plainly saba. You could drop one y from Bayyati to
>> make it more Turkish. 845 cents is plainly hisar without the
>> paranthesis. Little ajem is needless since we took kurdi
>> plainly. Let's call it simply ajem. Evdj cluster is a misnomer here,
>> it should be a dik ajem. Mahurek ought to be mahur.
>
> This does make a lot of sense to me: if we are looking at this
> temperament as a system in itself, then if there is only one possible
> tuning for a given perde, why not just say "ajem" or "mahur," etc.?
> Your point about the spelling of Bayati may be especially fitting
> because I've noticed that at least according to Marcus, the modern
> Arab system doesn't have a perde Bayati (or Bayyati)!
>
> As for "Evdj cluster" at 1041 cents, I realize that my concept of a
> "neutral seventh" might better fit the Arab perde Awj, although your
> "Evdj kumesi" at around 11:6 or 1049 cents might fit a typical Arab
> tuning better than 1041 cents. The difference is that whereas Awj is
> typically some kind of middle seventh, Turkish Evdj is evidently at
> around 15/8 or 4096/2187 above Rast -- or 48 commas, rather than
> somewhere around 46 or 47 commas for Arab perde Awj. This might be a
> difference related to Segah at around 17 commas (5/4 or 8192/6561)
> above Rast in Maqam Rast as understood in Turkey, but 16 commas or so
> in a typical Egyptian interpretation (e.g. something like Zalzal's
> 27/22, which you have mentioned as one practice).
>
>> But you see, you cannot play a decent Hijaz with this setup, because
>> you are either consigned to a very low kurdi or a very high dik
>> kurdi.
>
> Certainly we agree that for Maqam Hijaz starting on Dugah,
> 0-70-383-504 cents or C*-Db*-E*-F* (Dugah-Kurdi-Hijaz-Neva) would > place
> the second step of the tetrachord "very low," contrary to the usual
> understanding that the lowest step should be smaller than the
> highest (here 70-313-121). However, at least if the first step of
> Hijaz is understood to be a middle second, I might guess that a "very
> high dik kurdi" could be fairly idiomatic:
>
>
> Dugah Dik Kurdi Hijaz Neva Huseyni Ajem Gerdaniye > Muhayyer
> C* D E* F* G* Ab* > Bb* C*
> 0 141 383 504 696 778 > 996 1200
> 141 242 121 192 82 218 204
>
> Here the Hijaz tetrachord is quite close to Tala`i's 140-240-120, and
> the first step is larger than the highest. Flexibly we might state
> this conception of Hijaz as something around 6-11-5 commas. Of
> course, for the classic kind of Hijaz described by Shirazi at
> 12/11x7/6x22/21 (a permutation of Ptolemy's Intense Chromatic), or
> 151-267-81, we would have to transpose elsewhere, for example, using a
> spelling with some enharmonic equivalents (in Zest-24, F#=Gb, etc., in
> each 12-note circle):
>
> Beyati Dik Huseyni Rast Shuri Kurdi Segah Nim Hijaz Tiz > Beyati
> F#* G# A#* B* C#* D E F#*
> 0 154 422 505 696 766 959 1200
> 154 268 83 191 70 192 241
>
> This classic type of Hijaz is more like 6.5-12-3.5 commas.
>
> With many thanks,
>
> Margo
> mschulter@...
>
>
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🔗Margo Schulter <mschulter@...>

Hello, everyone.

The topic of Marchettus (or Marchetto) of Padua is a perenially
popular one, and with good reasons. The early 14th century is an
incredibly creative and productive era in European musical practice
and theory, as likewise in the Islamic world in this and the
immediately previous epochs, as with Safi al-Din al-Urmawi and Qutb
al-Din al-Shirazi.

While the quantitative details of the fivefold division of the tone
proposed by Marchettus are fascinating, I would start with his
practical musical intent: to describe and guide the intonation of
singers performing polyphonic music. He is not addressing, for
example, the tuning of keyboard instruments, but the rather the art of
that most flexible instrument: the human voice.

What he seems to be describing is a system of Pythagorean intonation,
or something very like it, which however may be and indeed should be
modified in directed polyphonic progressions by contrary motion which
involve certain accidental alterations. These alterations make major
third expanding to fifths or major sixths expanding to octaves larger,
and ascending melodic semitones smaller, than in regular Pythagorean
intonation.

Exactly how much smaller or larger, as George Secor and I and many
others have asked, is the more problematic question. Since we're
dealing with voices, by nature variable, it might be a moot question.
Indeed Christopher Page, who gives great attention to this business of
"stretching" major thirds and sixths at apt locations in various
French and Italian pieces composed in the 14th century, prefers to use
singers alone in order to let them use their own good judgement
without interference from fixed-pitch instruments.

Note that while Marchettus seems to take Pythagorean intonation as the
basis for his system, this important modification moves in the
opposite direction from the 5-limit tendencies which Walter Odington
indeed reflects around the same era -- thus revealing that then, as
now, local practices and tastes may vary! Page, by the way, also
recognizes that Odington's treatise points to a likely "mollification"
of major and minor thirds in English music, in contrast both to usual
Pythagorean tuning and its accentuated variants as described by
Marchettus.

It would seem that the ideal of "one size fits all" does not hold for
musical intervals such as major and minor thirds -- in 1318 or 2008!

While the mathematics of Marchettus may not always be clear, and some
of the measurements he invokes can be mutually contradictory, the
musical intent seems much clearer to me: extra-wide major thirds and
sixths in cadences involving what we would now term sharps. He says
that the regular semitone or limma, 256:243 or 90 cents in Pythagorean
tuning, is equal to "two parts" of a tune; but that in these
progressions, an especially small interval called a "diesis" is used
including only "one" of the five parts of a tone.

If these "parts" are approximately equal, as _some_ of his theory
might fairly suggest, then an interpretation like this might result:

<http://www.bestII.com/~mschulter/PythEnharImprov01.mp3>

A more nuanced interpretation is that maybe the extra-wide major
thirds and sixth have approximately septimal ratios around 9:7 and
12:7, which might lead to something like this:

<http://www.bestII.com/~mschulter/coop001.ogg>
<http://www.bestII.com/~mschulter/coop020.mid>

In the moderate view, using a system of approximately nine commas to a
9:8 tone at 204 cents, the metrics in practice of the intervals
discussed by Marchettus might be as follows:

----------------------------------------------------------------------
Interval commas cents notation
----------------------------------------------------------------------
diesis, cadential modification of limma 3 67 C#-D, G#-A
limma, "enharmonic" or regular semitone 4 90 B-C, A-Bb
apotome, "diatonic" semitone 5 114 Bb-B
chroma, cadential chromatic step 6 137 C-C#, G-G#
-----------------------------------------------------------------------

As George Secor and I have noted -- both with a great liking for this
kind of septimal flavor -- one could arrive at this kind of reading by
taking a certain interpretation of a comment by Marchettus that the
wide cadential major sixth is at an equal distance from the fifth and
the octave. If this means an arithmetic division on a monochord, we
get 12:8:6 for the division of the octave into a lower fifth and upper
fourth; and then 8:7:6 for the division of the upper fourth into two
equal lengths, so that the cadential major sixth has a ratio of about
12:7 (933 cents). Thus see:

<http://www.medieval.org/emfaq/harmony/marchetmf.html>
<http://xenharmony.wikispaces.com/space/showimage/17puzzle.pdf>

In my view this reading is both consistent with the general spirit of
this treatise, and musically effective -- but a more dramatic reading
is also possible, and in line with some other mathematical cues.

When Marchettus says that the cadential major sixth differs by "six
dieses" from either a fifth or an octave, that suggests comparably
sized units. If so, then some expansion of the usual Pythagorean major
third and sixth by something rather more than a comma -- closer to two
commas -- seems indicated. A comparatively moderate reading arguably
fitting this general model could suggest a division like 12:11 for the
chroma and 33:32 for the diesis (151 and 53 cents), or about 6-1/2 and
2-1/2 commas.

If one wants a theoretically attractive model realizable on a
monochord, Jay Rahn's seems about as good as any:

diaschisma diaschisma comma diaschisma diaschisma
|------------|-----------|-----|----------|-----------|
81 79 77 76 74 72
43 44 23 46 47
|------------------------|
limma = 2 parts -- 87
|------------------------------|
apotome = 3 parts -- 110
|-----------------------------------------|------------|
chroma = 4 part s -- 156 diesis = 47

The advantage of this scheme is that it agrees with the classic
doctrine of Philolaus, as reported by Boethius, that the tone may be
divided into four diaschismata (each equal to half of a limma, or
about 45.11 cents) plus a comma (531441:524288, or 23.46 cents).
Note that this is only one meaning of the term "diaschisma" (plural
"diaschismata"), and "demi-limma" would be a good synonym. If the
comma is the middle of the "five parts" at 77:76, its size is slightly
smaller than 23 cents, and indeed, as Scala notes, very close to 1/53
octave (Ozan's "Holdrian comma").

If a comma is taken as about half of a diaschisma, or a quarter of a
limma, then this "fivefold division" of a tone can be treated as an
approximate "ninefold division," with two commas to each of the four
diaschismata, plus another comma to complete the tone.

The main problem with this admirably elegant solution is that it
requires greatly unequal dieses -- in contrast to the description of a
cadential major sixth as "six dieses" from either a fifth or an
octave, which might imply a more even unit of measurement. However,
the "six dieses" observation would remain valid under this solution if
we reasonably take "six dieses" to mean "a regular 9:8 tone or five
dieses, plus a sixth diesis equal to a diaschisma or demi-limma -- in
other words, roughly 11 commas." Since a fourth consists of about 22
commas, and marks the distance from a 3:2 fifth to a 2:1 octave, the
cadential major sixth at about 11 commas from either (say about 951
cents, or 26:15 or 64:37 in JI/RI terms) would nicely fulfill this
description.

As you have pointed out, Ozan, a fivefold monochord division like
45:44:43:42:41:40 would also be possible -- and Rahn mentions the
possibility. Indeed, I have a 24-note tuning with two Pythagorean
chains at 459:448 apart, dividing a limma into steps of 48/42 cents.
If we take the diesis at 48 cents, very close to Rahn's reading, then
something like the "ninefold division" into near-equal commas results.
However, we can also use a 42-cent diesis -- interestingly, very close
to 41:40 (42.749 cents). This division thus could result in something
not too far from 29-EDO like:

----------------------------------------------------------------------
Interval commas cents notation
----------------------------------------------------------------------
diesis, cadential modification of limma 2- 43 C#-D, G#-A
limma, "enharmonic" or regular semitone 3.5 79 B-C, A-Bb
apotome, "diatonic" semitone 5.5 125 Bb-B
chroma, cadential chromatic step 7+ 161 C-C#, G-G#
-----------------------------------------------------------------------

The 18:17:16 division mentioned by Marchettus seems a curious
digression when taken in the context of the musical as well
mathematical setting sketched out by Marchettus. This division is
advocated as a practical and desirable one by the author Quintillian
in classic Rome. Boethius seems to describe it more as an example of
how it is impossible to divide a superparticular ratio such as 9:8
into two equal (and rational, one might add) ratios, than as an actual
standard by which the tone should be divided.

Of course, while the mathematical intricacies and ambiguities are
fascinating in their own right, Marchettus is focusing on a problem
just as relevant in 2008 as in 1318 -- how should we tune those
cadential semitones?

With modern tuning systems, there are a number of possibilities for
music in a 14th-century or derivative 21st-century style. That's an
essay in itself, and the term "Marchettan" has become a buzzword or
motto that can be used to describe almost any kind of temperament or
intonation where major intervals are notably larger than Pythagorean
and minor intervals (including cadential minor seconds or semitones)
smaller.

Strictly speaking, the term "Marchettan" might most properly be
applied to a system where there is a contrast between a usual minor
second somewhere around 70-90 cents, say, and a smaller cadential
step. Freely, however, the term might be applied to tunings such as
17-EDO or 22-EDO with only one size of smaller or minor semitone, but
notably or even impressively narrower than Pythagorean. It can also
apply to irregular temperaments where especially narrow semitones --
along with notably wide major thirds and sixths approaching or
possibly quite surpassing septimal ratios -- occur at certain
positions.

Possibly the very ambiguity of the mathematics in Marchettus is part
of the charm: are septimal ratios intended, or something like 29-EDO,
or a monochord approximation of the "four diaschismata plus a comma"
division which could lead to the concept of nine approximately equal
commas in a 9:8 tone? I tend to agree with Page that for singers,
artistic intuition coupled with a general understanding of the nature
of accentuated Pythagorean intonation and of period style may be the
best guide.

For fixed-pitch instruments, Marchettus has inspired a variety of
neomedieval tunings -- rather as ancient Greek writings on music
inspired the opera and oratorio around 1600. It bears emphasis that
Marchettus does not, to my best knowledge, discuss temperament, or
keyboard instruments -- but his intriguing theory and evocative
description of the musical values behind it are a wonderful entre to a
new century of possibilities.

With many thanks,

Margo Schulter
mschulter@...

🔗Ozan Yarman <ozanyarman@...>

Margo, this was inspirational. Thank you!

Oz.

On Aug 17, 2008, at 11:19 AM, Margo Schulter wrote:

> Hello, everyone.
>
> The topic of Marchettus (or Marchetto) of Padua is a perenially
> popular one, and with good reasons. The early 14th century is an
> incredibly creative and productive era in European musical practice
> and theory, as likewise in the Islamic world in this and the
> immediately previous epochs, as with Safi al-Din al-Urmawi and Qutb
> al-Din al-Shirazi.
>
> While the quantitative details of the fivefold division of the tone
> proposed by Marchettus are fascinating, I would start with his
> practical musical intent: to describe and guide the intonation of
> singers performing polyphonic music. He is not addressing, for
> example, the tuning of keyboard instruments, but the rather the art of
> that most flexible instrument: the human voice.
>
> What he seems to be describing is a system of Pythagorean intonation,
> or something very like it, which however may be and indeed should be
> modified in directed polyphonic progressions by contrary motion which
> involve certain accidental alterations. These alterations make major
> third expanding to fifths or major sixths expanding to octaves larger,
> and ascending melodic semitones smaller, than in regular Pythagorean
> intonation.
>
> Exactly how much smaller or larger, as George Secor and I and many
> others have asked, is the more problematic question. Since we're
> dealing with voices, by nature variable, it might be a moot question.
> Indeed Christopher Page, who gives great attention to this business of
> "stretching" major thirds and sixths at apt locations in various
> French and Italian pieces composed in the 14th century, prefers to use
> singers alone in order to let them use their own good judgement
> without interference from fixed-pitch instruments.
>
> Note that while Marchettus seems to take Pythagorean intonation as the
> basis for his system, this important modification moves in the
> opposite direction from the 5-limit tendencies which Walter Odington
> indeed reflects around the same era -- thus revealing that then, as
> now, local practices and tastes may vary! Page, by the way, also
> recognizes that Odington's treatise points to a likely "mollification"
> of major and minor thirds in English music, in contrast both to usual
> Pythagorean tuning and its accentuated variants as described by
> Marchettus.
>
> It would seem that the ideal of "one size fits all" does not hold for
> musical intervals such as major and minor thirds -- in 1318 or 2008!
>
> While the mathematics of Marchettus may not always be clear, and some
> of the measurements he invokes can be mutually contradictory, the
> musical intent seems much clearer to me: extra-wide major thirds and
> sixths in cadences involving what we would now term sharps. He says
> that the regular semitone or limma, 256:243 or 90 cents in Pythagorean
> tuning, is equal to "two parts" of a tune; but that in these
> progressions, an especially small interval called a "diesis" is used
> including only "one" of the five parts of a tone.
>
> If these "parts" are approximately equal, as _some_ of his theory
> might fairly suggest, then an interpretation like this might result:
>
> <http://www.bestII.com/~mschulter/PythEnharImprov01.mp3>
>
> A more nuanced interpretation is that maybe the extra-wide major
> thirds and sixth have approximately septimal ratios around 9:7 and
> 12:7, which might lead to something like this:
>
> <http://www.bestII.com/~mschulter/coop001.ogg>
> <http://www.bestII.com/~mschulter/coop020.mid>
>
> In the moderate view, using a system of approximately nine commas to a
> 9:8 tone at 204 cents, the metrics in practice of the intervals
> discussed by Marchettus might be as follows:
>
>
> ----------------------------------------------------------------------
> Interval commas cents notation
> ----------------------------------------------------------------------
> diesis, cadential modification of limma 3 67 C#-D, G#-A
> limma, "enharmonic" or regular semitone 4 90 B-C, A-Bb
> apotome, "diatonic" semitone 5 114 Bb-B
> chroma, cadential chromatic step 6 137 C-C#, G-G#
> -----------------------------------------------------------------------
>
> As George Secor and I have noted -- both with a great liking for this
> kind of septimal flavor -- one could arrive at this kind of reading by
> taking a certain interpretation of a comment by Marchettus that the
> wide cadential major sixth is at an equal distance from the fifth and
> the octave. If this means an arithmetic division on a monochord, we
> get 12:8:6 for the division of the octave into a lower fifth and upper
> fourth; and then 8:7:6 for the division of the upper fourth into two
> equal lengths, so that the cadential major sixth has a ratio of about
> 12:7 (933 cents). Thus see:
>
> <http://www.medieval.org/emfaq/harmony/marchetmf.html>
> <http://xenharmony.wikispaces.com/space/showimage/17puzzle.pdf>
>
> In my view this reading is both consistent with the general spirit of
> this treatise, and musically effective -- but a more dramatic reading
> is also possible, and in line with some other mathematical cues.
>
> When Marchettus says that the cadential major sixth differs by "six
> dieses" from either a fifth or an octave, that suggests comparably
> sized units. If so, then some expansion of the usual Pythagorean major
> third and sixth by something rather more than a comma -- closer to two
> commas -- seems indicated. A comparatively moderate reading arguably
> fitting this general model could suggest a division like 12:11 for the
> chroma and 33:32 for the diesis (151 and 53 cents), or about 6-1/2 and
> 2-1/2 commas.
>
> If one wants a theoretically attractive model realizable on a
> monochord, Jay Rahn's seems about as good as any:
>
> diaschisma diaschisma comma diaschisma diaschisma
> |------------|-----------|-----|----------|-----------|
> 81 79 77 76 74 72
> 43 44 23 46 47
> |------------------------|
> limma = 2 parts -- 87
> |------------------------------|
> apotome = 3 parts -- 110
> |-----------------------------------------|------------|
> chroma = 4 part s -- 156 diesis = 47
>
> The advantage of this scheme is that it agrees with the classic
> doctrine of Philolaus, as reported by Boethius, that the tone may be
> divided into four diaschismata (each equal to half of a limma, or
> about 45.11 cents) plus a comma (531441:524288, or 23.46 cents).
> Note that this is only one meaning of the term "diaschisma" (plural
> "diaschismata"), and "demi-limma" would be a good synonym. If the
> comma is the middle of the "five parts" at 77:76, its size is slightly
> smaller than 23 cents, and indeed, as Scala notes, very close to 1/53
> octave (Ozan's "Holdrian comma").
>
> If a comma is taken as about half of a diaschisma, or a quarter of a
> limma, then this "fivefold division" of a tone can be treated as an
> approximate "ninefold division," with two commas to each of the four
> diaschismata, plus another comma to complete the tone.
>
> The main problem with this admirably elegant solution is that it
> requires greatly unequal dieses -- in contrast to the description of a
> cadential major sixth as "six dieses" from either a fifth or an
> octave, which might imply a more even unit of measurement. However,
> the "six dieses" observation would remain valid under this solution if
> we reasonably take "six dieses" to mean "a regular 9:8 tone or five
> dieses, plus a sixth diesis equal to a diaschisma or demi-limma -- in
> other words, roughly 11 commas." Since a fourth consists of about 22
> commas, and marks the distance from a 3:2 fifth to a 2:1 octave, the
> cadential major sixth at about 11 commas from either (say about 951
> cents, or 26:15 or 64:37 in JI/RI terms) would nicely fulfill this
> description.
>
> As you have pointed out, Ozan, a fivefold monochord division like
> 45:44:43:42:41:40 would also be possible -- and Rahn mentions the
> possibility. Indeed, I have a 24-note tuning with two Pythagorean
> chains at 459:448 apart, dividing a limma into steps of 48/42 cents.
> If we take the diesis at 48 cents, very close to Rahn's reading, then
> something like the "ninefold division" into near-equal commas results.
> However, we can also use a 42-cent diesis -- interestingly, very close
> to 41:40 (42.749 cents). This division thus could result in something
> not too far from 29-EDO like:
>
> ----------------------------------------------------------------------
> Interval commas cents notation
> ----------------------------------------------------------------------
> diesis, cadential modification of limma 2- 43 C#-D, G#-A
> limma, "enharmonic" or regular semitone 3.5 79 B-C, A-Bb
> apotome, "diatonic" semitone 5.5 125 Bb-B
> chroma, cadential chromatic step 7+ 161 C-C#, G-G#
> -----------------------------------------------------------------------
>
>
> The 18:17:16 division mentioned by Marchettus seems a curious
> digression when taken in the context of the musical as well
> mathematical setting sketched out by Marchettus. This division is
> advocated as a practical and desirable one by the author Quintillian
> in classic Rome. Boethius seems to describe it more as an example of
> how it is impossible to divide a superparticular ratio such as 9:8
> into two equal (and rational, one might add) ratios, than as an actual
> standard by which the tone should be divided.
>
> Of course, while the mathematical intricacies and ambiguities are
> fascinating in their own right, Marchettus is focusing on a problem
> just as relevant in 2008 as in 1318 -- how should we tune those
> cadential semitones?
>
> With modern tuning systems, there are a number of possibilities for
> music in a 14th-century or derivative 21st-century style. That's an
> essay in itself, and the term "Marchettan" has become a buzzword or
> motto that can be used to describe almost any kind of temperament or
> intonation where major intervals are notably larger than Pythagorean
> and minor intervals (including cadential minor seconds or semitones)
> smaller.
>
> Strictly speaking, the term "Marchettan" might most properly be
> applied to a system where there is a contrast between a usual minor
> second somewhere around 70-90 cents, say, and a smaller cadential
> step. Freely, however, the term might be applied to tunings such as
> 17-EDO or 22-EDO with only one size of smaller or minor semitone, but
> notably or even impressively narrower than Pythagorean. It can also
> apply to irregular temperaments where especially narrow semitones --
> along with notably wide major thirds and sixths approaching or
> possibly quite surpassing septimal ratios -- occur at certain
> positions.
>
> Possibly the very ambiguity of the mathematics in Marchettus is part
> of the charm: are septimal ratios intended, or something like 29-EDO,
> or a monochord approximation of the "four diaschismata plus a comma"
> division which could lead to the concept of nine approximately equal
> commas in a 9:8 tone? I tend to agree with Page that for singers,
> artistic intuition coupled with a general understanding of the nature
> of accentuated Pythagorean intonation and of period style may be the
> best guide.
>
> For fixed-pitch instruments, Marchettus has inspired a variety of
> neomedieval tunings -- rather as ancient Greek writings on music
> inspired the opera and oratorio around 1600. It bears emphasis that
> Marchettus does not, to my best knowledge, discuss temperament, or
> keyboard instruments -- but his intriguing theory and evocative
> description of the musical values behind it are a wonderful entre to a
> new century of possibilities.
>
> With many thanks,
>
> Margo Schulter
> mschulter@...
>
>
>
> ------------------------------------
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
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>
>

🔗Margo Schulter <mschulter@...>

(Part II)

--------------------------------------------
3. The world at 705 cents: neutral intervals
--------------------------------------------

In our 19-tET gamelan tuning, we used an augmented second [-5, 9] and
augmented sixth [-5, 10] to produce the small 253-cent minor third and
948-cent minor seventh in this shading of slendro. As the tuning
changes, however, these same chains of generators can prduce intervals
of quite another color, as they do very pleasantly at 705 cents.

In fact, augmented and diminished intervals are absolutely vital in
enjoying the diverse forms of a medieval and modern Near Eastern
tuning called Rast -- or, better stated, a family of tunings of which
one was evidently developed in the 8th century by Mansur Zalzal, a
musician in Baghdad who favored a fretting for the `ud (which would
become known in Europe as the lute) later reported by theorists such
as al-Farabi and Ibn-Sina. A 705-cent generator or beta opens the way
to one most agreeable shading of this scale:

[0, 0] [-1, 2] [-5, 9] [1, -1] [0, 1] [-4, 8] [2, -2] [1, 0]
C D D# F G G# Bb C
0 210 345 495 705 840 990 1200
210 135 150 210 135 150 210

The first thing to notice about this scale, for those not already
attuned to Near Eastern modes, is purely melodic: the distinctive
steps of 135 and 150 cents somewhere between a semitone and a whole
tone as used in European and Near Eastern music alike. These
intermediate steps are often called "neutral seconds," and the Turkish
musician and theorist Ozan Yarman calls them simply "middle seconds."
Whatever we call them, they give Near Eastern modal systems a flavor
and intricacy unknown in traditional European modality or tonality
based on tones and semitones as the only "standard" steps.

In either the maqam traditions of the Arab world, Turkey, and
Kurdistan, or the Persian dastgah system which has grown out of the
maqam system in recent centuries, the subtleties of pure melodic
development and modulation have been the main focus, rather than
harmony for two or more voices. However, musicians within these
traditions are now exploring a range of harmonic possibilities, while
the maqam and dastgah traditions can enrich the melodic and harmonic
resources of musicians coming from other backgrounds.

In maqam theory, Rast is at one level the name for a tetrachord, or
division of the fourth, with a major second; a neutral or middle third
somewhere between minor and major, and a fourth -- here 0-210-345-495
cents. In Zalzal's version of Rast, known in modern Arab theory as
Nirz Rast, there are two such tetrachords, here at C-D-D#-F and
F-G-G#-Bb. The fourth degree of the scale F, the top of the first
tetrachord, is also the lowest note of the second -- an arrangement of
what are termed _conjunct_ tetrachords. An octave consists of these
two tetrachords plus an upper major second or tone Bb-C.

Rast Rast tone
|-----------------------|-------------------------|........|
C D D# F G G# Bb C
0 210 345 495 705 840 990 1200
210 135 150 210 135 150 210

In the medieval and modern Near East, as in medieval Europe, tuning in
pure fifths and fourths is an important musical current of intonation:
thus al-Farabi gives some of the intervals in Zalzal's scale as
Pythagorean ratios built from primes 2-3 only. Above the lowest step,
these include the 9:8 tone C-D, the 4:3 fourth C-F, the 3:2 fifth C-G,
and the 16:9 minor seventh C-Bb (at 204, 498, 702, and 996 cents). In
our 705-cent tuning, the fifth is wide of pure by about 3 cents, and
we find more generally that regular major intervals are larger, and
minor intervals smaller, than their Pythagorean sizes.

Our special focus here is on the neutral or middle intervals of
Zalzal's scale, which in a rendition with a 705-cent generator often
take on sizes very close to simple JI ratios. Thus the neutral third
C-D# at 345 cents is very close to 11:9 (347 cents); while the neutral
sixth C-G# at 840 cents gives a virtually just 13:8, the "harmonic
sixth" based on the 13th partial.

As it happens, al-Farabi gives Zalzal's neutral third and sixth as the
ratios 27:22 and 18:11, or 355 and 853 cents -- somewhat larger and
"brighter" than in our 705-cent version on C. A different mapping
would give us intervals at 360 and 855 cents, quite close to
al-Farabi's version.

In the Zalzal tuning on C, our neutral third and sixth very close to
11:9 and 13:8 tell us that 705 cents is a good generator for the 11th
and 13th partials. We find these near-just approximation of 11 and 13
involved in the fine tuning of other intervals also. Thus the regular
minor third D-F at 285 cents is close to 13:11 (289 cents), and yet
closer to the more complex 33:28, also based on 11 prime, at 284
cents. A regular major third such as Bb-D at 420 cents is very close
to 14:11 (418 cents).

These regular thirds serve nicely, like their Pythagorean counterparts
at 81:64 and 32:27 (408 and 294 cents), as relatively concordant but
active intervals often resolving in directed progressions to stable
intervals such as fifths -- as we saw in the previous section. Thus
it's possible to use a generator of 705 cents, much like one of 702
cents, for much of the medieval European repertory, including the
complex part-writing of the 12th-14th centuries.

From this point of view, we can describe this territory as
"supra-Pythagorean" -- rather like Pythagorean, but involving a yet
larger fifth or generator, so that major intervals are wider and minor
ones narrower. Although much beautiful music can be made using only
the "regular diatonic" intervals known to standard medieval European
theory, the neutral intervals add a new element to the mix -- or
actually a very traditional one in Near Eastern music for the last
millennium and more.

Exploring a few progressions in our Zalzalian Rast scale will
highlight various sides of this exciting equation. Here we'll focus on
one characteristic form of cadence taken from a 13th-14th century
European style: a three-voice cadence where a third expands to a fifth
and a sixth to an octave, bringing us to a 2-3-4 chord. At 705 cents,
the 3:2 fifth is tempered about 3 cents wide, and the 4:3 fourth about
3 cents narrow -- a tempering about half again as great as in 12-tET,
but rather less than in many meantone tunings where 4-7 cents of
impurity may prevail.

Interestingly, this form of cadence on our resting note or final C
involves only the usual Pythagorean -- or supra-Pythagorean -- steps
and intervals:

Bb C
F G
D C

The minor third and sixth at 285 cents and 780 cents, the latter close
to the simple ratio of 11:7 and former, as we have noted, to 13:11 or
33:28, expand smoothly to the fifth and octave, each voice moving by
an ample whole tone of 210 cents, a bit larger than the 9:8 of
Pythagorean tuning at 204 cents. Whole-tone motion in all voices is
characteristic of many cadences favored during the 13th century in
Europe; by around 1300, there is an increasing desire that decisive
cadences should involve ascending or descending semitonal motion in
one or more of the parts, a policy that mostly remains in place until
the period around 1900.

To make a regular cadence on a given step of our scale in this
approach, we need to have a regular fifth available above it, so that
we can form a 2-3-4 chord as the goal of our cadence. Five of the
seven degrees of our Zalzal scale meet this test: C, F, G, G#, and Bb.
We found that the cadence on C nicely fits a conventional 13th-century
European style; the others each add something new, melodic or
harmonic, drawing on our neutral intervals.

The cadence on F introduces a close approximation of another just
interval:

D# F
Bb C
G F

Our opening chord has a minor third G-Bb at 285 cents plus our 13:8
neutral sixth G-D# at 840 cents -- actually an augmented fifth built
from eight fifths up less four octaves [-4, 8]. The two upper voices
thus form a "superfourth" or small tritone at 555 cents, very close to
the 11:8 (551 cents) based on the 11th partial. Melodically, the two
lower voices proceed as in the previous cadence on C, expanding
stepwise from minor third to fifth with each moving by a 210-cent
tone. The upper voice, however, ascends by a distinctive neutral
second at 150 cents -- a virtually just 12:11 (151 cents) also given
by al-Farabi as one of the melodic intervals in his version of
Zalzal's tuning.

Both the outer 13:8 sixth and the near-11:8 superfourth between the
upper voices (an excellent name suggested by David Keenan) give this
cadence a special flavor. Now let us move to the cadence on G:

F G
C D
G# G

Harmonically, this cadence introduces us to the intervals of the
diminished fourth G#-C [4, -8] at 360 cents, a virtually just 16:13
(359 cents), and the diminished seventh G#-F [-5, +9] at 855 cents,
quite close to al-Farabi's ratio of 18:11 or 853 cents in his version
of Zalzal's scale. The 16:13 diminished fourth or neutral third
between the lower voices expands to a fifth with the lowest voice
descending by a 135-cent chromatic semitone G#-G [-4, +7] or neutral
second rather close to 13:12 (139 cents), and the middle voice
ascending by a 210-cent tone. The outer voices likewise resolve from
the 855-cent neutral sixth to an octave, the upper voice like the
middle one ascending by a 210-cent tone.

Again, both the harmonic neutral intervals, and the neutral second
step in the lowest voice, provide radically new resources for those
not already accustomed to them, while illustrating how the same chain
of generators can have different musical uses as the tuning changes.
At 702 cents, the same diminished fourth and seventh G#-C-F would
produce a chord at 0-384-882 cents, with a "schismatic" major third
and sixth very close to a just 5:4 and 5:3 (0-386-884 cents); here, we
get neutral intervals very close to a just 16:13 and 18:11.

The cadence on G# brings into a play a delightful feature of scales
and modes with neutral steps which George Secor noted in 1978 when
exploring his 17-tone well-temperament, and which a regular
temperament at 705 cents likewise illustrates:

G G#
D D#
Bb G#

The opening chord is a typical Pythagorean or supra-Pythagorean
structure with a major third at 420 cents and major sixth at 915
cents, close to 14:11 and 56:33 (418 and 916 cents). In a 13th-14th
century European style, this chord often resolves to A-E-A, with the
lowest voice descending by a usual diatonic semitone (Bb-A) where the
upper voices each ascend by a tone. Here, however, the resolution is
quite different.

Each voice moved by some flavor of middle or neutral second: the
lowest descending by the 150-cent step Bb-G#, almost a just 12:11; and
the upper voices ascending by 135-cent steps, D-D# and G-G#. The
150-cent step is formed by a diminished third at ten generators down
plus six octaves [6, -10]; and the smaller 135-cent step by the
chromatic semitone, equal to seven generators up less four octaves
[-4, 7].

Secor has aptly suggested that this type of progression be called an
_equable_ cadence. The idea is that in order for a major third to
expand to a fifth (here Bb-D to G#-D#), or a major sixth to an octave
(Bb-G to G#-G#), the two voices involved must together move by the
total of a minor third -- here 285 cents, the difference between a
420-cent major third and a 705-cent fifth; or a 915-cent major sixth
and a 1200-cent octave.

In the equable cadence, this motion is divided "equably" between the
voices so that each moves by some size of neutral second: here by
steps of 150 and 135 cents. Secor's term was borrowed from the
"Equable Diatonic" tuning of Ptolemy, which divides a fourth into
string ratios of 12:11:10:9 (0-151-316-498 cents). In that tuning, a
minor third at 6:5 or 12:10 (316 cents) is divided into two neutral
second steps at 12:11 (151 cents) and 11:10 (165 cents).

Here the division of the smaller 285-cent third into steps of 135 and
150 cents is rather close to another just intonation (JI) division:
13:12:11, where a 13:11 third at 289 cents is divided into steps of
13:12 at 139 cents and 12:11 at 150 cents.

Musically, the equable cadence has a quality strikingly different from
anything in a mode based on tones and semitones alone. It is one
result of a "fusion" style combining Near Eastern modes and neutral
second steps with some general cadential principles coming from
medieval Europe.

Finally, to complete our tour of cadences, we a cadence to Bb, the
seventh degree of this Zalzalian Rast:

G# Bb
D# F
C Bb

Here the opening chord has a 345-cent neutral third plus an 840-cent
neutral sixth, very close as we have seen to 11:9 and 13:8; the lower
voice descends by a whole tone, while each upper voice ascends by a
150-cent neutral second.

This quick tour of cadences may suggest the diversity of colors and
subtle variations available in our 705-cent temperament. Neutral
second steps at 135 or 150 cents, neutral thirds at 345 or 360 cents,
and neutral sixths at 840 or 855 cents are largely interchangeable,
and yet subtly different. Fortunately, the Thummer's 19 notes per
octave provide room both for enjoying this diversity, and sometimes
for making musically significant choices.

For example, a mode or pattern known in the Arab world as Maqam
Bayyati and in the Persian dastgah system as Shur Dastgah, has a lower
tetrachord where a minor third above the resting note or final is
divided into two neutral seconds -- Secor's equable division. In both
Arab and Persian practice, there is a general preference that the
smaller neutral second step should precede or be placed below the
larger. To follow this convention, we can choose from among eight
positions with this ordering where a smaller 135-cent step precedes a
larger 150-cent one, for example:

[0, 0] [-4, 7] [2, -3] [1, -1] [0, 1] [3, -4] [2, -2] [1, 0]
C C# Eb F G Ab Bb C
0 135 285 495 705 780 990 1200
135 150 210 210 75 210 210

Interestingly, obtaining neutral intervals in our 705-cent tuning
based on often quite accurate approximations of primes 2-3-11-13
involves the same generator chains we would use around 702 cents or
Pythagoran tuning for "schismatic" intervals at or very close to
ratios of 2-3-5 (e.g. 5:4 and 6:5 thirds).

Like the Pythagorean region around 702 cents, our 705-cent tuning
offers some nice approximations of Archytan intervals based on primes
2-3-7 -- but uses long generator chains for this purpose. Fortunately,
these chains are slightly shorter at 705 cents; and we have the
additional advantage of being able to combine these 2-3-7 intervals
with a wealth of neutral intervals available here, but not at 702
cents, within a 19-note tuning set.

Let us quickly consider a beautiful tuning of the Persian philosopher
and music theorist Ibn Sina, who described it in the earlier 11th
century. According to John Chalmers, this tuning is based on a
tetrachord of 1/1-14/13-7/6-4/3 (0-128-139-498 cents), with the
smaller 14:13 neutral second below the larger 13:12 step. George Secor
has discussed a very similar tuning in which the larger 13:12 step
comes first, forming a tetrachord of 12:13:14:16 (0-139-267-498
cents). Our 705-cent tuning offers a solution which "splits the
difference" between these two versions:

[0, 0] [-4, 7] [-8, 14] [1, -1] [0, 1] [-4, 8] [-8, 15] [1, 0]
Db D D# Gb Ab A A# Db
0 135 270 495 705 840 975 1200
135 135 225 210 135 135 225

Here we have a special kind of equable division: the 270-cent minor
third, very close to a just 7:6 or 267 cents, is divided into two
identical 135-cent neutral seconds. This kind of precisely equal
division may be more typical of modern tuning systems than of the
classical understanding of a theorist such as Ptolemy or Ibn Sina,
where equable divisions often involve ratios of adjoining integers (or
also, partials, in modern harmonic series terms) such as 12-13-14,
which always produce somewhat unequal steps.

When Archytan or 2-3-7 and neutral intervals are combined, some
memorable cadences become available which would not otherwise be
possible. For example:

A# A
Ab A
D# D
Db D

The opening chord at 0-270-705-975 cents is not too far from a just
12:14:18:21 with a 7:6 minor third, 3:2 fifth, and 7:4 minor seventh
above the lowest note (0-267-702-969 cents). In this equable cadence,
the minor seventh contracts to a fifth and each minor third to a
unison, with each voice moving by an identical neutral second step of
135 cents!

Another directed two-voice resolution in this cadence might not be so
obvious: the expansion of the middle pair of voices from a large major
third D#-Ab at 435 cents to a fifth D-A, likewise with 135-cent steps
in each voice. This major third is at an almost perfectly just 9:7
(435.084 cents), which like the smaller neutral sixth at 840 cents (by
comparison to 13:8 at 840.528 cents) is a special attraction of the
705-cent temperament, quite apart from its easy arithmetic with
conveniently rounded interval sizes in cents.

To highlight this 9:7 major third, let's change the arrangement of the
voices:

Db D
A# A
Ab A
D# D

Now our opening chord is 0-435-705-930 cents or approximately
14:18:21:24, with the active and outgoing 9:7 third and the major
sixth quite close to 12:7 (933 cents) expanding to the fifth and
octave of a complete 2-3-4 chord (D-A-D).

This seven-note scale also includes within it a slendro tuning
featuring an interval of 480 cents, interestingly the same narrow
fourth occurring in 5-EDO, which might be taken as one starting point
for the subtly unequal gamelan tunings which obtain in practice:

[0, 0] [-1, 2] [-9, 16] [0, 1] [-8, 15] [1, 0]
Gb Ab A# Db D# Gb
0 210 480 705 975 1200
210 270 225 270 225

A just 2-3-7 prime version would be 1/1-9/8-21/16-3/2-7/4-2/1 or
0-204-471-702-969-1200 cents, formed by partials 16:18:21:24:28:32.
While the idea of tuning a gamelan based on just ratios is more an
aspect of recent international developments than of traditional
Javanese or Balinese practice, small fourths around 470-490 cents are
very typical of traditional slendro tunings.

Returning to seven-note scales, I should emphasize that while my
discussion has focused on a "neomedieval fusion" style combining
aspects of medieval European and Near Eastern practices, scales
combining neutral intervals with supra-Pythagorean or Archytan ones
are by no means limited to this application.

George Secor, in his article focusing on his own 17-tone
well-temperament, gives a number of scales which can be nicely
approximated using a 705-cent generator, describing their use for
example in a kind of "four-part xenharmony" in some ways curiously
analogous to 18th-19th century European tonality. Since his
presentation, published in _Xenharmonikon_ 18, is happily available on
the Web, I will here give a link, followed by some of his scales
discussed in the article and their generator chains at 705 cents.

<http://xenharmony.wikispaces.com/space/showimage/17puzzle.pdf>

The first scale is very similar to the Ibn Sina/Secor tuning above,
except that the 7/4 minor seventh at 969 cents is replaced by an 11/6
neutral seventh at 1049 cents -- or, in our tuning, the 975-cent minor
seventh Db-A# by a 1050-cent neutral seventh Db=B. Secor's just ratios
and their values in cents, which he approximates in the article with
his 17-note well-temperament or 17-WT for short, are shown above the
generator chains and interval sizes in our 705-cent tuning:

1/1 13/12 7/6 4/3 3/2 13/8 11/6 2/1
0 139 267 498 702 841 1049 1200
139 128 231 204 139 209 151

[0, 0] [-4, 7] [-8, 14] [1, -1] [0, 1] [-4, 8] [-5, 10] [1, 0]
Db D D# Gb Ab A B Db
0 135 270 495 705 840 1050 1200
135 135 225 210 135 210 150

Interestingly, the 11:6 neutral seventh results at 705 cents from a
chain of ten generators up -- the same chain which, around 1/4-comma
meantone (696.58 cents) or 31-tET (696.77 cents), yields a fine
approximation of a 7:4 minor seventh. See Secor's article at p. 69 for
some of the possibilities of this scale.

A closely related version shown by Secor at p. 71 has a neutral rather
than minor third:

1/1 13/12 11/9 4/3 3/2 13/8 11/6 2/1
0 139 347 498 702 841 1049 1200
139 209 151 204 139 209 151

[0, 0] [-4, 7] [-5, 9] [1, -1] [0, 1] [-4, 8] [-5, 10] [1, 0]
Db D E Gb Ab A B Db
0 135 345 495 705 840 1050 1200
135 210 150 210 135 210 150

Yet another of Secor's variations has an 11:8 superfourth or small
tritone at 551 cents in place of the perfect fourth in our previous
scales, an interval here approximated by a chain of 11 generators up
less six octaves at 555 cents:

1/1 13/12 11/9 11/8 3/2 13/8 11/6 2/1
0 139 347 551 702 841 1049 1200
139 209 204 151 139 209 151

[0, 0] [-4, 7] [-5, 9] [-6, 11] [0, 1] [-4, 8] [-5, 10] [1, 0]
Db D E F Ab A B Db
0 135 345 495 705 840 1050 1200
135 210 210 150 135 210 150

While the examples of part-writing in this presentation have focused
on chords in three or four voices, typical of a neomedieval style,
Secor often focuses on more complex structures closely approximating
partials in the harmonic series. An example which shows off these
impressively rich and "different" possibilities is the 6:7:9:11:13
chord, which Secor discusses at page 76 of his article in the setting
of his 17-WT system.

The 6:7:9:11:13 chord is an example of what is termed an "isoharmonic"
chord, where in a JI version there are equal differences between the
partial numbers of successive notes, as with the upper four notes at
partials 7-9-11-13, where this difference is 2. In 6-7-9-11-13, we
might in strict terms speak of a "semi-isoharmonic" chord, since the
first pair of partials 6-7 have a difference of 1.

In the 705-cent tuning, close approximations of this five-note chord
or "pentad" are available at 5 of the 19 locations. While all five
notes sound simultaneously as a chord, I find it easier to write out
the notes and intervals horizontally in their JI and tempered
versions:

1/1 7/6 3/2 11/6 13/6
0 267 702 1049 1339
7:6 9:7 11:9 13:11
267 435 347 289

[0, 0] [-8,14] [0, 1] [-5, 10] [-3, 7]
0 270 705 1050 1335
270 435 345 285
Db D# Ab B D
Eb E# Bb C# E
Gb G# Db E G
Ab A# Eb F# A
Cb C# Gb A C

Note that our 705-cent tuning is very accurate for these ratios of
2-3-7-11-13, but with the complications of some long chains of
generators (especially for ratios of 7), and intricate note spellings
and mappings for the keyboardist. In contrast, Secor's 17-WT
approximates ratios of 7 not quite so accurately but much more simply
in the nearer part of the 17-note circle: for example, a regular minor
third like D-F is quite close to 7:6; a major third like C-E to 9:7,
and a minor seventh like D-C to 7:4.

George Secor's informative article is highly recommended for an
understanding both of some of the scales and examples above using the
705-cent tuning, and of the versatile 17-WT system.

-------------
4. Conclusion
-------------

In looking at a slendro tuning at 19-tET and some regular Pythagorean
and Archytan or 2-3-7 sonorities at 702 cents (Part I), and here a
range possibilities at 705 cents involving partials 2-3-7-11-13, my
purpose has been to present some material which might enrich the
Thummer/Matrix paradigm.

Desiring to focus above all on musical possibilities rather than fine
points of theory, I would conclude with an open question.

To what degree can some of the generator chains and musical
applications shown above be helpfully assisted with special
temperament mappings -- e.g. for neutral thirds or ratios of 2-3-7 at
705 cents -- and to what degree does the intricate mixture of
intervals likely to be needed suggest that the keyboardist is still
often the best person to sort out the artistic possibilities "on the
fly"?

Experience may be the obvious basis for resulving such questions. I
wonder what George Secor, a premier user and advocate of the
generalized keyboard, might say about these issues.

With many thanks,

Margo Schulter
mschulter@...

🔗Dave Keenan <d.keenan@...>

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
> While the quantitative details of the fivefold division of the tone
> proposed by Marchettus are fascinating, I would start with his
> practical musical intent: to describe and guide the intonation of
> singers performing polyphonic music. He is not addressing, for
> example, the tuning of keyboard instruments, but the rather the art of
> that most flexible instrument: the human voice.

Yes indeed.

> What he seems to be describing is a system of Pythagorean intonation,
> or something very like it, which however may be and indeed should be
> modified in directed polyphonic progressions by contrary motion which
> involve certain accidental alterations. These alterations make major
> third expanding to fifths or major sixths expanding to octaves larger,
> and ascending melodic semitones smaller, than in regular Pythagorean
> intonation.

Agreed.

> Exactly how much smaller or larger, as George Secor and I and many
> others have asked, is the more problematic question. Since we're
> dealing with voices, by nature variable, it might be a moot question.

But we can have a lot of fun trying to figure out how large Marchetto
was trying to tell us it should be. It seems he had theorised
something fairly specific, even if he wasn't very good at explaining
it in terms that could be understood hundreds of years later.

> Note that while Marchettus seems to take Pythagorean intonation as the
> basis for his system, this important modification moves in the
> opposite direction from the 5-limit tendencies
...

Agreed.

> While the mathematics of Marchettus may not always be clear, and some
> of the measurements he invokes can be mutually contradictory, the
> musical intent seems much clearer to me: extra-wide major thirds and
> sixths in cadences involving what we would now term sharps.

Yes.

> He says
> that the regular semitone or limma, 256:243 or 90 cents in Pythagorean
> tuning, is equal to "two parts" of a tone;

I've only seen where he refers to an "enharmonic" semitone as being
"two parts" of a tone (out of his 5 unequal parts). Does he actually
refer to it as a "regular semitone" in any sense? Does he actually
refer to it as a limma, or as 256:243?

I still see no way to decide between about 68 cents and about 90 cents
for his "enharmonic" semitone. As explained in my previous post
responding to Monz.

> but that in these
> progressions, an especially small interval called a "diesis" is used
> including only "one" of the five parts of a tone.

I find it most likely that this diesis is in th erange 43 to 48 cents,
not the comma-sized "diesis", and not an equal 5-part division of the
tone (40 cents).

> If these "parts" are approximately equal, as _some_ of his theory
> might fairly suggest,

How is that suggested? What would be the point of first dividing the
tone into 9 (or 8) if the aim was an approximately equal 5-fold division?

> then an interpretation like this might result:
>
> <http://www.bestII.com/~mschulter/PythEnharImprov01.mp3>
>
> A more nuanced interpretation is that maybe the extra-wide major
> thirds and sixth have approximately septimal ratios around 9:7 and
> 12:7, which might lead to something like this:
>
> <http://www.bestII.com/~mschulter/coop001.ogg>
> <http://www.bestII.com/~mschulter/coop020.mid>

I don't see any evidence for a 7-limit interpretation either.

> In the moderate view, using a system of approximately nine commas to a
> 9:8 tone at 204 cents, the metrics in practice of the intervals
> discussed by Marchettus might be as follows:
>
>
> ----------------------------------------------------------------------
> Interval commas cents notation
> ----------------------------------------------------------------------
> diesis, cadential modification of limma 3 67 C#-D, G#-A
> limma, "enharmonic" or regular semitone 4 90 B-C, A-Bb
> apotome, "diatonic" semitone 5 114 Bb-B
> chroma, cadential chromatic step 6 137 C-C#, G-G#
> -----------------------------------------------------------------------

I can see that his enharmonic and diatonic may be 4 and 5 commas
respectively. They could also be 3 and 6 commas respectively. But I do
not agree that his chromatic semitone could be as small as 6 commas.
Doesn't he clearly state that his chromatic semitone corresponds to 4
of his dieses, and that there are 5 of his dieses in a whole-tone.

You would need to group the 9 commas _very_ unevenly into 5 "dieses",
to make 4 dieses encompass 6 commas. You would need to have 3 dieses
of a single comma, and two dieses of 3 commas, within the tone.

I think it's fairly clear that he groups the commas as 1, 2, 2, 2, 2.
At least I see no suggestion of 1, 1, 1, 3, 3 or 3, 1, 1, 1, 3. Nor do
I accept Jay Rahn's argument for 2, 2, 1, 2, 2.

So I find that his chromatic semitone is most likely to be 7 commas
(around 159 cents) and the remaining diesis 2 commas (around 45 cents).

>
> As George Secor and I have noted -- both with a great liking for this
> kind of septimal flavor -- one could arrive at this kind of reading by
> taking a certain interpretation of a comment by Marchettus that the
> wide cadential major sixth is at an equal distance from the fifth and
> the octave. If this means an arithmetic division on a monochord, we
> get 12:8:6 for the division of the octave into a lower fifth and upper
> fourth; and then 8:7:6 for the division of the upper fourth into two
> equal lengths, so that the cadential major sixth has a ratio of about
> 12:7 (933 cents). Thus see:
>
> <http://www.medieval.org/emfaq/harmony/marchetmf.html>
> <http://xenharmony.wikispaces.com/space/showimage/17puzzle.pdf>
>
> In my view this reading is both consistent with the general spirit of
> this treatise,

Yes. I agree that's a possible reading.

> and musically effective -- but a more dramatic reading
> is also possible, and in line with some other mathematical cues.
>
> When Marchettus says that the cadential major sixth differs by "six
> dieses" from either a fifth or an octave, that suggests comparably
> sized units.

Yes it does.

> If so, then some expansion of the usual Pythagorean major
> third and sixth by something rather more than a comma -- closer to two
> commas -- seems indicated.

Why not exactly two commas?

> A comparatively moderate reading arguably
> fitting this general model could suggest a division like 12:11 for the
> chroma and 33:32 for the diesis (151 and 53 cents), or about 6-1/2 and
> 2-1/2 commas.
>
> If one wants a theoretically attractive model realizable on a
> monochord, Jay Rahn's seems about as good as any:
>
> diaschisma diaschisma comma diaschisma diaschisma
> |------------|-----------|-----|----------|-----------|
> 81 79 77 76 74 72
> 43 44 23 46 47
> |------------------------|
> limma = 2 parts -- 87
> |------------------------------|
> apotome = 3 parts -- 110
> |-----------------------------------------|------------|
> chroma = 4 part s -- 156 diesis = 47
>
> The advantage of this scheme is that it agrees with the classic
> doctrine of Philolaus, as reported by Boethius, that the tone may be
> divided into four diaschismata (each equal to half of a limma, or
> about 45.11 cents) plus a comma (531441:524288, or 23.46 cents).
> Note that this is only one meaning of the term "diaschisma" (plural
> "diaschismata"), and "demi-limma" would be a good synonym. If the
> comma is the middle of the "five parts" at 77:76, its size is slightly
> smaller than 23 cents, and indeed, as Scala notes, very close to 1/53
> octave (Ozan's "Holdrian comma").
> If a comma is taken as about half of a diaschisma, or a quarter of a
> limma, then this "fivefold division" of a tone can be treated as an
> approximate "ninefold division," with two commas to each of the four
> diaschismata, plus another comma to complete the tone.

This is all well and good, except that I believe Marchetto puts the
single-comma "diesis" at one end, not in the middle, based on that
passage that Monz and I have been discussing.

That doesn't invalidate the sizes suggested here (within +-3 cents)
but it does upen up the possibility of the enharmonic being 3 commas
and the diatonic being 6.

> The main problem with this admirably elegant solution is that it
> requires greatly unequal dieses --

But Marchettos _says_ they are unequal. Doesn't he?

> in contrast to the description of a
> cadential major sixth as "six dieses" from either a fifth or an
> octave, which might imply a more even unit of measurement. However,
> the "six dieses" observation would remain valid under this solution if
> we reasonably take "six dieses" to mean "a regular 9:8 tone or five
> dieses, plus a sixth diesis equal to a diaschisma or demi-limma -- in
> other words, roughly 11 commas." Since a fourth consists of about 22
> commas, and marks the distance from a 3:2 fifth to a 2:1 octave, the
> cadential major sixth at about 11 commas from either (say about 951
> cents, or 26:15 or 64:37 in JI/RI terms) would nicely fulfill this
> description.

Aha! Yes. Well done. Does this agree with my chromatic semitone of 7
commas (about 159 cents) and its remnant of a 2-comma diesis (about 45
cents)?

> The 18:17:16 division mentioned by Marchettus seems a curious
> digression when taken in the context of the musical as well
> mathematical setting sketched out by Marchettus. This division is
> advocated as a practical and desirable one by the author Quintillian
> in classic Rome. Boethius seems to describe it more as an example of
> how it is impossible to divide a superparticular ratio such as 9:8
> into two equal (and rational, one might add) ratios, than as an actual
> standard by which the tone should be divided.

Thanks for that. I'm currently thinking of it as an example, from
Marchetto, of what not to do.

-- Dave Keenan

🔗Ozan Yarman <ozanyarman@...>

Margo, the 705-cent tuning yields terrific results. I also notice that 63-tone equal octave temperament comes very close to representing this tuning, giving us close-to-excellent 13-limit Just Intonation.

Oz.

On Aug 18, 2008, at 10:18 AM, Margo Schulter wrote:

> (Part II)
>
> --------------------------------------------
> 3. The world at 705 cents: neutral intervals
> --------------------------------------------
>
> In our 19-tET gamelan tuning, we used an augmented second [-5, 9] and
> augmented sixth [-5, 10] to produce the small 253-cent minor third and
> 948-cent minor seventh in this shading of slendro. As the tuning
> changes, however, these same chains of generators can prduce intervals
> of quite another color, as they do very pleasantly at 705 cents.
>
> In fact, augmented and diminished intervals are absolutely vital in
> enjoying the diverse forms of a medieval and modern Near Eastern
> tuning called Rast -- or, better stated, a family of tunings of which
> one was evidently developed in the 8th century by Mansur Zalzal, a
> musician in Baghdad who favored a fretting for the `ud (which would
> become known in Europe as the lute) later reported by theorists such
> as al-Farabi and Ibn-Sina. A 705-cent generator or beta opens the way
> to one most agreeable shading of this scale:
>
> [0, 0] [-1, 2] [-5, 9] [1, -1] [0, 1] [-4, 8] [2, -2] [1, 0]
> C D D# F G G# Bb C
> 0 210 345 495 705 840 990 1200
> 210 135 150 210 135 150 210
>
> The first thing to notice about this scale, for those not already
> attuned to Near Eastern modes, is purely melodic: the distinctive
> steps of 135 and 150 cents somewhere between a semitone and a whole
> tone as used in European and Near Eastern music alike. These
> intermediate steps are often called "neutral seconds," and the Turkish
> musician and theorist Ozan Yarman calls them simply "middle seconds."
> Whatever we call them, they give Near Eastern modal systems a flavor
> and intricacy unknown in traditional European modality or tonality
> based on tones and semitones as the only "standard" steps.
>
> In either the maqam traditions of the Arab world, Turkey, and
> Kurdistan, or the Persian dastgah system which has grown out of the
> maqam system in recent centuries, the subtleties of pure melodic
> development and modulation have been the main focus, rather than
> harmony for two or more voices. However, musicians within these
> traditions are now exploring a range of harmonic possibilities, while
> the maqam and dastgah traditions can enrich the melodic and harmonic
> resources of musicians coming from other backgrounds.
>
> In maqam theory, Rast is at one level the name for a tetrachord, or
> division of the fourth, with a major second; a neutral or middle third
> somewhere between minor and major, and a fourth -- here 0-210-345-495
> cents. In Zalzal's version of Rast, known in modern Arab theory as
> Nirz Rast, there are two such tetrachords, here at C-D-D#-F and
> F-G-G#-Bb. The fourth degree of the scale F, the top of the first
> tetrachord, is also the lowest note of the second -- an arrangement of
> what are termed _conjunct_ tetrachords. An octave consists of these
> two tetrachords plus an upper major second or tone Bb-C.
>
> Rast Rast tone
> |-----------------------|-------------------------|........|
> C D D# F G G# Bb C
> 0 210 345 495 705 840 990 1200
> 210 135 150 210 135 150 210
>
> In the medieval and modern Near East, as in medieval Europe, tuning in
> pure fifths and fourths is an important musical current of intonation:
> thus al-Farabi gives some of the intervals in Zalzal's scale as
> Pythagorean ratios built from primes 2-3 only. Above the lowest step,
> these include the 9:8 tone C-D, the 4:3 fourth C-F, the 3:2 fifth C-G,
> and the 16:9 minor seventh C-Bb (at 204, 498, 702, and 996 cents). In
> our 705-cent tuning, the fifth is wide of pure by about 3 cents, and
> we find more generally that regular major intervals are larger, and
> minor intervals smaller, than their Pythagorean sizes.
>
> Our special focus here is on the neutral or middle intervals of
> Zalzal's scale, which in a rendition with a 705-cent generator often
> take on sizes very close to simple JI ratios. Thus the neutral third
> C-D# at 345 cents is very close to 11:9 (347 cents); while the neutral
> sixth C-G# at 840 cents gives a virtually just 13:8, the "harmonic
> sixth" based on the 13th partial.
>
> As it happens, al-Farabi gives Zalzal's neutral third and sixth as the
> ratios 27:22 and 18:11, or 355 and 853 cents -- somewhat larger and
> "brighter" than in our 705-cent version on C. A different mapping
> would give us intervals at 360 and 855 cents, quite close to
> al-Farabi's version.
>
> In the Zalzal tuning on C, our neutral third and sixth very close to
> 11:9 and 13:8 tell us that 705 cents is a good generator for the 11th
> and 13th partials. We find these near-just approximation of 11 and 13
> involved in the fine tuning of other intervals also. Thus the regular
> minor third D-F at 285 cents is close to 13:11 (289 cents), and yet
> closer to the more complex 33:28, also based on 11 prime, at 284
> cents. A regular major third such as Bb-D at 420 cents is very close
> to 14:11 (418 cents).
>
> These regular thirds serve nicely, like their Pythagorean counterparts
> at 81:64 and 32:27 (408 and 294 cents), as relatively concordant but
> active intervals often resolving in directed progressions to stable
> intervals such as fifths -- as we saw in the previous section. Thus
> it's possible to use a generator of 705 cents, much like one of 702
> cents, for much of the medieval European repertory, including the
> complex part-writing of the 12th-14th centuries.
>
> From this point of view, we can describe this territory as
> "supra-Pythagorean" -- rather like Pythagorean, but involving a yet
> larger fifth or generator, so that major intervals are wider and minor
> ones narrower. Although much beautiful music can be made using only
> the "regular diatonic" intervals known to standard medieval European
> theory, the neutral intervals add a new element to the mix -- or
> actually a very traditional one in Near Eastern music for the last
> millennium and more.
>
> Exploring a few progressions in our Zalzalian Rast scale will
> highlight various sides of this exciting equation. Here we'll focus on
> one characteristic form of cadence taken from a 13th-14th century
> European style: a three-voice cadence where a third expands to a fifth
> and a sixth to an octave, bringing us to a 2-3-4 chord. At 705 cents,
> the 3:2 fifth is tempered about 3 cents wide, and the 4:3 fourth about
> 3 cents narrow -- a tempering about half again as great as in 12-tET,
> but rather less than in many meantone tunings where 4-7 cents of
> impurity may prevail.
>
> Interestingly, this form of cadence on our resting note or final C
> involves only the usual Pythagorean -- or supra-Pythagorean -- steps
> and intervals:
>
> Bb C
> F G
> D C
>
> The minor third and sixth at 285 cents and 780 cents, the latter close
> to the simple ratio of 11:7 and former, as we have noted, to 13:11 or
> 33:28, expand smoothly to the fifth and octave, each voice moving by
> an ample whole tone of 210 cents, a bit larger than the 9:8 of
> Pythagorean tuning at 204 cents. Whole-tone motion in all voices is
> characteristic of many cadences favored during the 13th century in
> Europe; by around 1300, there is an increasing desire that decisive
> cadences should involve ascending or descending semitonal motion in
> one or more of the parts, a policy that mostly remains in place until
> the period around 1900.
>
> To make a regular cadence on a given step of our scale in this
> approach, we need to have a regular fifth available above it, so that
> we can form a 2-3-4 chord as the goal of our cadence. Five of the
> seven degrees of our Zalzal scale meet this test: C, F, G, G#, and Bb.
> We found that the cadence on C nicely fits a conventional 13th-century
> European style; the others each add something new, melodic or
> harmonic, drawing on our neutral intervals.
>
> The cadence on F introduces a close approximation of another just
> interval:
>
> D# F
> Bb C
> G F
>
> Our opening chord has a minor third G-Bb at 285 cents plus our 13:8
> neutral sixth G-D# at 840 cents -- actually an augmented fifth built
> from eight fifths up less four octaves [-4, 8]. The two upper voices
> thus form a "superfourth" or small tritone at 555 cents, very close to
> the 11:8 (551 cents) based on the 11th partial. Melodically, the two
> lower voices proceed as in the previous cadence on C, expanding
> stepwise from minor third to fifth with each moving by a 210-cent
> tone. The upper voice, however, ascends by a distinctive neutral
> second at 150 cents -- a virtually just 12:11 (151 cents) also given
> by al-Farabi as one of the melodic intervals in his version of
> Zalzal's tuning.
>
> Both the outer 13:8 sixth and the near-11:8 superfourth between the
> upper voices (an excellent name suggested by David Keenan) give this
> cadence a special flavor. Now let us move to the cadence on G:
>
> F G
> C D
> G# G
>
> Harmonically, this cadence introduces us to the intervals of the
> diminished fourth G#-C [4, -8] at 360 cents, a virtually just 16:13
> (359 cents), and the diminished seventh G#-F [-5, +9] at 855 cents,
> quite close to al-Farabi's ratio of 18:11 or 853 cents in his version
> of Zalzal's scale. The 16:13 diminished fourth or neutral third
> between the lower voices expands to a fifth with the lowest voice
> descending by a 135-cent chromatic semitone G#-G [-4, +7] or neutral
> second rather close to 13:12 (139 cents), and the middle voice
> ascending by a 210-cent tone. The outer voices likewise resolve from
> the 855-cent neutral sixth to an octave, the upper voice like the
> middle one ascending by a 210-cent tone.
>
> Again, both the harmonic neutral intervals, and the neutral second
> step in the lowest voice, provide radically new resources for those
> not already accustomed to them, while illustrating how the same chain
> of generators can have different musical uses as the tuning changes.
> At 702 cents, the same diminished fourth and seventh G#-C-F would
> produce a chord at 0-384-882 cents, with a "schismatic" major third
> and sixth very close to a just 5:4 and 5:3 (0-386-884 cents); here, we
> get neutral intervals very close to a just 16:13 and 18:11.
>
> The cadence on G# brings into a play a delightful feature of scales
> and modes with neutral steps which George Secor noted in 1978 when
> exploring his 17-tone well-temperament, and which a regular
> temperament at 705 cents likewise illustrates:
>
> G G#
> D D#
> Bb G#
>
> The opening chord is a typical Pythagorean or supra-Pythagorean
> structure with a major third at 420 cents and major sixth at 915
> cents, close to 14:11 and 56:33 (418 and 916 cents). In a 13th-14th
> century European style, this chord often resolves to A-E-A, with the
> lowest voice descending by a usual diatonic semitone (Bb-A) where the
> upper voices each ascend by a tone. Here, however, the resolution is
> quite different.
>
> Each voice moved by some flavor of middle or neutral second: the
> lowest descending by the 150-cent step Bb-G#, almost a just 12:11; and
> the upper voices ascending by 135-cent steps, D-D# and G-G#. The
> 150-cent step is formed by a diminished third at ten generators down
> plus six octaves [6, -10]; and the smaller 135-cent step by the
> chromatic semitone, equal to seven generators up less four octaves
> [-4, 7].
>
> Secor has aptly suggested that this type of progression be called an
> _equable_ cadence. The idea is that in order for a major third to
> expand to a fifth (here Bb-D to G#-D#), or a major sixth to an octave
> (Bb-G to G#-G#), the two voices involved must together move by the
> total of a minor third -- here 285 cents, the difference between a
> 420-cent major third and a 705-cent fifth; or a 915-cent major sixth
> and a 1200-cent octave.
>
> In the equable cadence, this motion is divided "equably" between the
> voices so that each moves by some size of neutral second: here by
> steps of 150 and 135 cents. Secor's term was borrowed from the
> "Equable Diatonic" tuning of Ptolemy, which divides a fourth into
> string ratios of 12:11:10:9 (0-151-316-498 cents). In that tuning, a
> minor third at 6:5 or 12:10 (316 cents) is divided into two neutral
> second steps at 12:11 (151 cents) and 11:10 (165 cents).
>
> Here the division of the smaller 285-cent third into steps of 135 and
> 150 cents is rather close to another just intonation (JI) division:
> 13:12:11, where a 13:11 third at 289 cents is divided into steps of
> 13:12 at 139 cents and 12:11 at 150 cents.
>
> Musically, the equable cadence has a quality strikingly different from
> anything in a mode based on tones and semitones alone. It is one
> result of a "fusion" style combining Near Eastern modes and neutral
> second steps with some general cadential principles coming from
> medieval Europe.
>
> Finally, to complete our tour of cadences, we a cadence to Bb, the
> seventh degree of this Zalzalian Rast:
>
> G# Bb
> D# F
> C Bb
>
> Here the opening chord has a 345-cent neutral third plus an 840-cent
> neutral sixth, very close as we have seen to 11:9 and 13:8; the lower
> voice descends by a whole tone, while each upper voice ascends by a
> 150-cent neutral second.
>
> This quick tour of cadences may suggest the diversity of colors and
> subtle variations available in our 705-cent temperament. Neutral
> second steps at 135 or 150 cents, neutral thirds at 345 or 360 cents,
> and neutral sixths at 840 or 855 cents are largely interchangeable,
> and yet subtly different. Fortunately, the Thummer's 19 notes per
> octave provide room both for enjoying this diversity, and sometimes
> for making musically significant choices.
>
> For example, a mode or pattern known in the Arab world as Maqam
> Bayyati and in the Persian dastgah system as Shur Dastgah, has a lower
> tetrachord where a minor third above the resting note or final is
> divided into two neutral seconds -- Secor's equable division. In both
> Arab and Persian practice, there is a general preference that the
> smaller neutral second step should precede or be placed below the
> larger. To follow this convention, we can choose from among eight
> positions with this ordering where a smaller 135-cent step precedes a
> larger 150-cent one, for example:
>
>
> [0, 0] [-4, 7] [2, -3] [1, -1] [0, 1] [3, -4] [2, -2] [1, 0]
> C C# Eb F G Ab Bb C
> 0 135 285 495 705 780 990 1200
> 135 150 210 210 75 210 210
>
> Interestingly, obtaining neutral intervals in our 705-cent tuning
> based on often quite accurate approximations of primes 2-3-11-13
> involves the same generator chains we would use around 702 cents or
> Pythagoran tuning for "schismatic" intervals at or very close to
> ratios of 2-3-5 (e.g. 5:4 and 6:5 thirds).
>
> Like the Pythagorean region around 702 cents, our 705-cent tuning
> offers some nice approximations of Archytan intervals based on primes
> 2-3-7 -- but uses long generator chains for this purpose. Fortunately,
> these chains are slightly shorter at 705 cents; and we have the
> additional advantage of being able to combine these 2-3-7 intervals
> with a wealth of neutral intervals available here, but not at 702
> cents, within a 19-note tuning set.
>
> Let us quickly consider a beautiful tuning of the Persian philosopher
> and music theorist Ibn Sina, who described it in the earlier 11th
> century. According to John Chalmers, this tuning is based on a
> tetrachord of 1/1-14/13-7/6-4/3 (0-128-139-498 cents), with the
> smaller 14:13 neutral second below the larger 13:12 step. George Secor
> has discussed a very similar tuning in which the larger 13:12 step
> comes first, forming a tetrachord of 12:13:14:16 (0-139-267-498
> cents). Our 705-cent tuning offers a solution which "splits the
> difference" between these two versions:
>
> [0, 0] [-4, 7] [-8, 14] [1, -1] [0, 1] [-4, 8] [-8, 15] [1, 0]
> Db D D# Gb Ab A A# Db
> 0 135 270 495 705 840 975 1200
> 135 135 225 210 135 135 225
>
> Here we have a special kind of equable division: the 270-cent minor
> third, very close to a just 7:6 or 267 cents, is divided into two
> identical 135-cent neutral seconds. This kind of precisely equal
> division may be more typical of modern tuning systems than of the
> classical understanding of a theorist such as Ptolemy or Ibn Sina,
> where equable divisions often involve ratios of adjoining integers (or
> also, partials, in modern harmonic series terms) such as 12-13-14,
> which always produce somewhat unequal steps.
>
> When Archytan or 2-3-7 and neutral intervals are combined, some
> memorable cadences become available which would not otherwise be
> possible. For example:
>
> A# A
> Ab A
> D# D
> Db D
>
> The opening chord at 0-270-705-975 cents is not too far from a just
> 12:14:18:21 with a 7:6 minor third, 3:2 fifth, and 7:4 minor seventh
> above the lowest note (0-267-702-969 cents). In this equable cadence,
> the minor seventh contracts to a fifth and each minor third to a
> unison, with each voice moving by an identical neutral second step of
> 135 cents!
>
> Another directed two-voice resolution in this cadence might not be so
> obvious: the expansion of the middle pair of voices from a large major
> third D#-Ab at 435 cents to a fifth D-A, likewise with 135-cent steps
> in each voice. This major third is at an almost perfectly just 9:7
> (435.084 cents), which like the smaller neutral sixth at 840 cents (by
> comparison to 13:8 at 840.528 cents) is a special attraction of the
> 705-cent temperament, quite apart from its easy arithmetic with
> conveniently rounded interval sizes in cents.
>
> To highlight this 9:7 major third, let's change the arrangement of the
> voices:
>
> Db D
> A# A
> Ab A
> D# D
>
> Now our opening chord is 0-435-705-930 cents or approximately
> 14:18:21:24, with the active and outgoing 9:7 third and the major
> sixth quite close to 12:7 (933 cents) expanding to the fifth and
> octave of a complete 2-3-4 chord (D-A-D).
>
> This seven-note scale also includes within it a slendro tuning
> featuring an interval of 480 cents, interestingly the same narrow
> fourth occurring in 5-EDO, which might be taken as one starting point
> for the subtly unequal gamelan tunings which obtain in practice:
>
> [0, 0] [-1, 2] [-9, 16] [0, 1] [-8, 15] [1, 0]
> Gb Ab A# Db D# Gb
> 0 210 480 705 975 1200
> 210 270 225 270 225
>
> A just 2-3-7 prime version would be 1/1-9/8-21/16-3/2-7/4-2/1 or
> 0-204-471-702-969-1200 cents, formed by partials 16:18:21:24:28:32.
> While the idea of tuning a gamelan based on just ratios is more an
> aspect of recent international developments than of traditional
> Javanese or Balinese practice, small fourths around 470-490 cents are
> very typical of traditional slendro tunings.
>
> Returning to seven-note scales, I should emphasize that while my
> discussion has focused on a "neomedieval fusion" style combining
> aspects of medieval European and Near Eastern practices, scales
> combining neutral intervals with supra-Pythagorean or Archytan ones
> are by no means limited to this application.
>
> George Secor, in his article focusing on his own 17-tone
> well-temperament, gives a number of scales which can be nicely
> approximated using a 705-cent generator, describing their use for
> example in a kind of "four-part xenharmony" in some ways curiously
> analogous to 18th-19th century European tonality. Since his
> presentation, published in _Xenharmonikon_ 18, is happily available on
> the Web, I will here give a link, followed by some of his scales
> discussed in the article and their generator chains at 705 cents.
>
> <http://xenharmony.wikispaces.com/space/showimage/17puzzle.pdf>
>
> The first scale is very similar to the Ibn Sina/Secor tuning above,
> except that the 7/4 minor seventh at 969 cents is replaced by an 11/6
> neutral seventh at 1049 cents -- or, in our tuning, the 975-cent minor
> seventh Db-A# by a 1050-cent neutral seventh Db=B. Secor's just ratios
> and their values in cents, which he approximates in the article with
> his 17-note well-temperament or 17-WT for short, are shown above the
> generator chains and interval sizes in our 705-cent tuning:
>
> 1/1 13/12 7/6 4/3 3/2 13/8 11/6 2/1
> 0 139 267 498 702 841 1049 1200
> 139 128 231 204 139 209 151
>
> [0, 0] [-4, 7] [-8, 14] [1, -1] [0, 1] [-4, 8] [-5, 10] [1, 0]
> Db D D# Gb Ab A B Db
> 0 135 270 495 705 840 1050 1200
> 135 135 225 210 135 210 150
>
> Interestingly, the 11:6 neutral seventh results at 705 cents from a
> chain of ten generators up -- the same chain which, around 1/4-comma
> meantone (696.58 cents) or 31-tET (696.77 cents), yields a fine
> approximation of a 7:4 minor seventh. See Secor's article at p. 69 for
> some of the possibilities of this scale.
>
> A closely related version shown by Secor at p. 71 has a neutral rather
> than minor third:
>
> 1/1 13/12 11/9 4/3 3/2 13/8 11/6 2/1
> 0 139 347 498 702 841 1049 1200
> 139 209 151 204 139 209 151
>
> [0, 0] [-4, 7] [-5, 9] [1, -1] [0, 1] [-4, 8] [-5, 10] [1, 0]
> Db D E Gb Ab A B Db
> 0 135 345 495 705 840 1050 1200
> 135 210 150 210 135 210 150
>
> Yet another of Secor's variations has an 11:8 superfourth or small
> tritone at 551 cents in place of the perfect fourth in our previous
> scales, an interval here approximated by a chain of 11 generators up
> less six octaves at 555 cents:
>
> 1/1 13/12 11/9 11/8 3/2 13/8 11/6 2/1
> 0 139 347 551 702 841 1049 1200
> 139 209 204 151 139 209 151
>
> [0, 0] [-4, 7] [-5, 9] [-6, 11] [0, 1] [-4, 8] [-5, 10] [1, 0]
> Db D E F Ab A B Db
> 0 135 345 495 705 840 1050 1200
> 135 210 210 150 135 210 150
>
> While the examples of part-writing in this presentation have focused
> on chords in three or four voices, typical of a neomedieval style,
> Secor often focuses on more complex structures closely approximating
> partials in the harmonic series. An example which shows off these
> impressively rich and "different" possibilities is the 6:7:9:11:13
> chord, which Secor discusses at page 76 of his article in the setting
> of his 17-WT system.
>
> The 6:7:9:11:13 chord is an example of what is termed an "isoharmonic"
> chord, where in a JI version there are equal differences between the
> partial numbers of successive notes, as with the upper four notes at
> partials 7-9-11-13, where this difference is 2. In 6-7-9-11-13, we
> might in strict terms speak of a "semi-isoharmonic" chord, since the
> first pair of partials 6-7 have a difference of 1.
>
> In the 705-cent tuning, close approximations of this five-note chord
> or "pentad" are available at 5 of the 19 locations. While all five
> notes sound simultaneously as a chord, I find it easier to write out
> the notes and intervals horizontally in their JI and tempered
> versions:
>
> 1/1 7/6 3/2 11/6 13/6
> 0 267 702 1049 1339
> 7:6 9:7 11:9 13:11
> 267 435 347 289
>
> [0, 0] [-8,14] [0, 1] [-5, 10] [-3, 7]
> 0 270 705 1050 1335
> 270 435 345 285
> Db D# Ab B D
> Eb E# Bb C# E
> Gb G# Db E G
> Ab A# Eb F# A
> Cb C# Gb A C
>
> Note that our 705-cent tuning is very accurate for these ratios of
> 2-3-7-11-13, but with the complications of some long chains of
> generators (especially for ratios of 7), and intricate note spellings
> and mappings for the keyboardist. In contrast, Secor's 17-WT
> approximates ratios of 7 not quite so accurately but much more simply
> in the nearer part of the 17-note circle: for example, a regular minor
> third like D-F is quite close to 7:6; a major third like C-E to 9:7,
> and a minor seventh like D-C to 7:4.
>
> George Secor's informative article is highly recommended for an
> understanding both of some of the scales and examples above using the
> 705-cent tuning, and of the versatile 17-WT system.
>
>
> -------------
> 4. Conclusion
> -------------
>
> In looking at a slendro tuning at 19-tET and some regular Pythagorean
> and Archytan or 2-3-7 sonorities at 702 cents (Part I), and here a
> range possibilities at 705 cents involving partials 2-3-7-11-13, my
> purpose has been to present some material which might enrich the
> Thummer/Matrix paradigm.
>
> Desiring to focus above all on musical possibilities rather than fine
> points of theory, I would conclude with an open question.
>
> To what degree can some of the generator chains and musical
> applications shown above be helpfully assisted with special
> temperament mappings -- e.g. for neutral thirds or ratios of 2-3-7 at
> 705 cents -- and to what degree does the intricate mixture of
> intervals likely to be needed suggest that the keyboardist is still
> often the best person to sort out the artistic possibilities "on the
> fly"?
>
> Experience may be the obvious basis for resulving such questions. I
> wonder what George Secor, a premier user and advocate of the
> generalized keyboard, might say about these issues.
>
> With many thanks,
>
> Margo Schulter
> mschulter@...
>
>

🔗Margo Schulter <mschulter@...>

[Please forgive me for posting this quickly without adequate proofing;
I wanted to share my excitement at the result of the 6/3 comma
division for usual apotome and limma suggested as one possible reading
by Dave Keenan, coupled with a 7/2 division for chroma and diesis, as
developed in what follows.]

Hello, Dave, and thank you for an outstanding response which, as I'll
acknowledge in more detali below, has lots of very perceptive
questions and musically telling points. Really, reading your remarks
was one of the high points of my week so far.

>> Exactly how much smaller or larger, as George Secor and I and
>> many others have asked, is the more problematic question. Since
>> we're dealing with voices, by nature variable, it might be a moot
>> question.

> But we can have a lot of fun trying to figure out how large
> Marchetto was trying to tell us it should be. It seems he had
> theorised something fairly specific, even if he wasn't very good
> at explaining it in terms that could be understood hundreds of
> years later.

Indeed! As someone myself addicted to this pastime, I know it's a
fascinating mixture of historical grounding and creative
interpretations to fit with one's own tuning practices or taste -- at
least for those of us who get involved in the medieval/neomedieval
arena. Your points lend a healthy balance on this last point.

>> Note that while Marchettus seems to take Pythagorean intonation as
>> the basis for his system, this important modification moves in the
>> opposite direction from the 5-limit tendencies ...

> Agreed.

For the sake of others who may be reading this, I might just add an
amusing demonstration of the complications that might result from a
5-limit interpretation of what might be considered quite unfortunate
language from a 20th-century perspective in calling the apotome the
"diatonic" semitone. I tend to think that Marchetto mixed concepts or
language in a curious way, and happened to arrive at a term that would
serve ideally to mislead people in search of "5-limit" readings in
early music. As you well observe, Ptolemy and others have lots of
diatonics: why not, as I recall you commented elsewhere, 12:11 as in
the Equable or Even Diatonic, if we take this line of reasoning?

What I would rely on is the logic of Marchetto's notation and
examples, and the style of early 14th-century Italian music (including
the pieces he is known or suspected to have composed). This isn't the
English musical dialect of _Sumer is icumen in_, where a 5-limit
reading would be stylistically persuasive even if we didn't have
Theinred of Dover and Walter Odington telling us how close the
theoretical ratios for thirds are to a simple 5:4 and 6:5. It isn't
the early 15th century as reflected by the Faenza Codex from Italy or
the style of the young Dufay or the older layers of the Buxheim Organ
Book, where "schismatic" thirds from sharps tuned as Pythagorean flats
are implied by the style and documented by some remarks and debates in
the literature (including Prosdocimus and Ugolino, who appear to
advocate a 17-note tuning in order to preserve rather than to alter
the regular sizes of thirds -- a sign that others _are_ altering them
in a way that leads to meantone temperaments by around mid-century).

For my quick demonstration to confirm our agreed point, I'd invite
people to consider this two-voice progression of Marchetto:

C C# D
F E D

Here we have a 3:2 fifth followed by a cadential major sixth resolving
to a 2:1 octave, to that the total expansion or motion in both voices
should add up to 4:3, or 498 cents. Further, Marchettus defines a tone
as 9:8, and doesn't discuss using two sizes of tones. So taking the
ratios which he gives of 2:1, 3:2, 4:3, and 9:8 as a basis, we have:

C ? C# ? D
F ? E -204 D

As it happens, there's no apotome in this progression -- it would
occur in Bb - B, for example. However, the other three sizes are in
evidence: the "usual" F-E in the lowest part preceding the 9/8 tone;
the chroma C-C#; and the diesis C#-D.

According to Marchettus, it would seem that C-D should, like E-D, be a
9:8 tone. Suppose we accept a division something like what you have
proposed, around 7-2 commas, or 158-46 cents. Thus:

C +158 C# +46 D
F ? E -204 D

Then F-E should be 90 cents, a Pythagorean limma or about 4 commas.

C +158 C# +46 D
F -90 E -204 D

Of course, as you point it, that usual or "enharmonic" semitone could
actually be smaller, which might mean that singers would produce usual
minor thirds rather smaller than 32:27 or 294 cents. Other world
traditions where semitones a bit smaller than 90 cents, where tuning
seems based on "essentially pure" fifths and fourths, provide a very
credible precedent.

However, if Marchettus really means that the apotome is the "usual"
semitone, that would raise some complications. Specifically,
Marchettus indicates that Bb-B is an apotome, the usual understanding,
so that the "mi-sign" as I'll call it (like a natural or "square-B"
sign) alters by an apotome, as does the flat or "fa-sign." This
implies that call it the "diatonic" semitone although he does, it's
not the usual or regular semitone step like A-Bb or E-F (routine
mi-fa).

If it were, we might expect either more intricate accidentals to show
alterations of a limma, or signs showing tones reduced by a comma (as
happen in medieval Near Eastern and modern Turkish music, for
example). Marchettus, indeed, refers to thirds and sixths as
"tolerable dissonances" -- in contrast to Odington's remarks that
singers produce them so that they are "fully concordant."

Suppose that the "diatonic semitone" of "three parts" means that this
is the usual step at E-F, B-C, A-Bb, etc. Then we might have:

C +150 C# +30 D
F -114 E -204 D

Here I have made the chroma C-C# equal to only 150 cents rather than
158 or 159 cents. Note that if E-F is really 114 cents, then the lower
voice's descent of F-D is in all 32768:19683 or 318 cents, which plus
the 150-cent chroma (around 12:11, my arbitrary value, not any
specification of Marchetto!) is 468 cents, leaving only 30 cents for
the diesis!

Anyway, I need to check some sources and confirm, if possible, that
the limma is indeed the "enharmonic semitone" -- a curious wording,
since I'd associate the enharmonic genus with some kind of ditone plus
two dieses, although at least one of them might be a large as a small
semitone which could be used in the other genera also.

> I've only seen where he refers to an "enharmonic" semitone as being
> "two parts" of a tone (out of his 5 unequal parts). Does he
> actually refer to it as a "regular semitone" in any sense? Does he
> actually refer to it as a limma, or as 256:243?

I'll need to check my sources: but I definitely recall that he
identifies the "two parts" as the usual limma; and I stated this in a
paper that I wrote in 2001 and linked to my previous post, so let's
hope I'm right <grin>. The apotome and limma stand out very clearly to
me, and also to people like Jan Herlinger who notes that in the
Renaissance, people attracted to Marchetto changed the definition so
that the major semitone would be the regular one, fitting meantone (or
5-limit JI).

You have an important point that equating the limma or usual semitone
with "two parts" -- or more generally the _minor_ semitone -- doesn't
necessarily imply 256:243. That I should check also.

[Going back to the Latin text used for Jan Herlinger's translation, I
see that Marchetto indeed repeatedly makes it clear that the
"enharmonic" semitone is the usual one, and identifies it with Plato's
_lima_, but evidently not with 256:243. He does make it clear that it
is less than half of a tone, but invokes 18:17 as its ratio --
unlikely to be a practical value, as we agree.]

> I still see no way to decide between about 68 cents and about 90 cents
> for his "enharmonic" semitone. As explained in my previous post
> responding to Monz.

The 68-cent hypothesis is fun! That's about absolutely optimal by
George Secor's standards, and with a 9:8 tone would yield a regular
minor third very close to 7:6! We'd get a usual major third around 430
cents -- not taking into account questions of whether or how the 9:8
tone might get affected in size, etc. -- this is flexible vocal
intonation rather than a regular temperament, so I guess it could get
quite involved.

>> but that in these
>> progressions, an especially small interval called a "diesis" is used
>> including only "one" of the five parts of a tone.

> I find it most likely that this diesis is in th erange 43 to 48 cents,
> not the comma-sized "diesis", and not an equal 5-part division of the
> tone (40 cents).

Yes, if we follow the "one part" in the most natural reading, as
opposed to devising clever interpretations to obtain a larger size (a
game I've played), I'd agree that 43-48 cents is a logical
estimate. Something like 1/31 octave or 128:125 seems a bit small --
fine for Vicentino in a 16th-century meantone setting, where as far as
I know the step is used as a "sliding" of the voice rather than a
directed cadential "semitone" or surrogate.

>> If these "parts" are approximately equal, as _some_ of his theory
>> might fairly suggest,

> How is that suggested? What would be the point of first dividing
> the tone into 9 (or 8) if the aim was an approximately equal
> 5-fold division?

Actually "suggested" might be well modified to "inferred by some
readers," as your just query duly alerts us! Prosdocimus did infer
this and take Marchetto to task for proposing that a superparticular
ratio like 9:8 be divided into five or any number of "equal" parts --
possibly showing mainly how he can be misinterpreted in an authentic
early 15th-century manner, for those who care to follow this tradition
<grin>. The "six dieses" of distance between a cadential major sixth
and either an octave or a fifth might play into the hand of someone
who wants to find a 14th-century version of 29-EDO -- but only because
that hand could be overly eager to grasp at any EDO paradigm.

The point about the 9-fold division is also exactly to the
issue. Somewhat more than a century later, Ugolino isn't shy about how
to divide a limma into two near-equal dieses for the sake of
completeness, thus making his 17-note keyboard tuning a 19-note tuning
to accommodate the manner of the ancients as he describes it: just do
an arithmetic division of 256:243, i.e. 512:499:486, at E-F and B-C.

> I can see that his enharmonic and diatonic may be 4 and 5 commas
> respectively. They could also be 3 and 6 commas respectively.

The 3/6 hypothesis for the limma and apotome is really fascinating;
the corresponding regular tuning would be 17-EDO, but of course this
is flexible vocal intonation. Here 17-EDO or George's 17=WT is of
interest mainly in showing how pleasant this kind of arrangement might
be for lots of 14th-century music.

> But I do
> not agree that his chromatic semitone could be as small as 6 commas.

> Doesn't he clearly state that his chromatic semitone corresponds to 4
> of his dieses, and that there are 5 of his dieses in a whole-tone.

Yes, precisely.

> You would need to group the 9 commas _very_ unevenly into 5 "dieses",
> to make 4 dieses encompass 6 commas. You would need to have 3 dieses
> of a single comma, and two dieses of 3 commas, within the tone.

True, and I admit that this is the kind of kludge that I've posted
here -- but it's quite distinct from the simplest and most elegant
reading of the text, which is what you have summed up.

> I think it's fairly clear that he groups the commas as 1, 2, 2, 2, 2.
> At least I see no suggestion of 1, 1, 1, 3, 3 or 3, 1, 1, 1, 3. Nor do
> I accept Jay Rahn's argument for 2, 2, 1, 2, 2.

Rahn at least has the sizes right. Maybe he and I somehow like the
idea of the "four diaschismata plus a comma" (Boethius, not Marchetto)
with the single comma in the middle, a kind of symmetry. However, I
agree that Marchetto's emphasis on using odd numbers would very much
fit a sequence of 1-3-5-7-9 commas, your 1-2-2-2-2, or a division of
81:80:78:76:74:72. The main hazard of this natural reading in the
tuning community is that the opening 81:80 could be taken as another
cue to infer a 5-limit orientation where it's unlikely to apply.

Yes, a theorist such as Jacobus of Liege in this same epoch of
European music is fully aware of the 81:80 and 64:63 commas in
discussing the theoretical possibility of using ratios such as 5:1,
5:4, 7:1, and 7:6, finding them likely somewhat concordant but
impractical in the prevailing tuning system, rather like more recent
theorists dismissing the seventh partial because it is impractical in
12-EDO. However, I would regard 81:80 in Marchetto's division as
simply a coincidence; this is on the basis of considering his
presentation in full context, and also the style he is addressing.

> So I find that his chromatic semitone is most likely to be 7 commas
> (around 159 cents) and the remaining diesis 2 commas (around 45 cents).

That's a very fair conclusion.

[On the possible reading of an 8:7:6 division of the fourth to define
the cadential major sixth as "equally distant" from 3:2 or 2:1, and
thus at 12:7 on a monochord]

> Yes. I agree that's a possible reading.

However, I must confirm that indeed the 7-2 comma reading of the "five
parts" as divided between chroma and diesis is the obvious one, as you
have demonstrated, and that that would also produce a cadential major
sixth about "midway between" fifth and octave (more in a geometric
way).

>> When Marchettus says that the cadential major sixth differs by "six
>> dieses" from either a fifth or an octave, that suggests comparably
>> sized units.

> Yes it does.

Or, to be fair on my part, "It _could_ suggest equal dieses."

>> If so, then some expansion of the usual Pythagorean major
>> third and sixth by something rather more than a comma -- closer to two
>> commas -- seems indicated.

> Why not exactly two commas?

I agree -- why not indeed? As Jan Herlinger and Geoerge Secor have
pointed out, one consequence is a cadential major sixth about halfway
between a Pythagorean major sixth and minor seventh (906 and 996
cents), with Herlinger simply finding this unlikely and Secor
observing that "ambiguity" in the categorization of such an interval
might result. However, that doesn't mean it might not happen; and I
find it both practical and thrilling, even if leaning often toward a
7-based reading (not stated by Marchettus, I'd emphasize).

[On a 2-2-1-2-2 scheme]

> This is all well and good, except that I believe Marchetto puts the
> single-comma "diesis" at one end, not in the middle, based on that
> passage that Monz and I have been discussing.

Yes, and the emphasis on odd numbers supports your reading also, as I
seem to recall you may have pointed out in that discussion.

> That doesn't invalidate the sizes suggested here (within +-3 cents)
> but it does upen up the possibility of the enharmonic being 3 commas
> and the diatonic being 6.

Yes, I'm just now catching the connection between this proposal and
the fivefold/nine-comma division. If we take the "first part" or 1,
and the second or 1-3, as the "two parts" of the limma, we indeed get
three commas -- leaving six for the apotome. This is really exciting,
both refreshingly different _and_ a perfectly reasonable reading of
the division, however we tie it in with other clues.

>> The main problem with this admirably elegant solution is that it
>> requires greatly unequal dieses --

> But Marchettos _says_ they are unequal. Doesn't he?

True, if one reads the 1, 1-3, 3-5, 5-7, 7-9 passage naturally --
rather than assuming that some EDO is involved, or the geometrically
equal division that Prosdocimus assumes and refutes.

>> in contrast to the description of a cadential major sixth as "six
>> dieses" from either a fifth or an octave, which might imply a more
>> even unit of measurement. However, the "six dieses" observation
>> would remain valid under this solution if we reasonably take "six
>> dieses" to mean "a regular 9:8 tone or five dieses, plus a sixth
>> diesis equal to a diaschisma or demi-limma -- in other words,
>> roughly 11 commas." Since a fourth consists of about 22 commas, and
>> marks the distance from a 3:2 fifth to a 2:1 octave, the cadential
>> major sixth at about 11 commas from either (say about 951 cents, or
>> 26:15 or 64:37 in JI/RI terms) would nicely fulfill this
>> description.

>> Aha! Yes. Well done. Does this agree with my chromatic semitone of 7
>> commas (about 159 cents) and its remnant of a 2-comma diesis (about 45
>> cents)?

Let's try it. For simplicity or "starting from the familiar," I'll
assume for the moment that in a two-voice cadence, the lower voice
descends by a 9:8 tone, and that the upper voice ascends by about 45
cents. We're expanding from some kind of wide major sixth to 2:1 or
1200 cents, the the sixth must be 1200 less 204 less 45 or 951 cents.
Going back to our formula cited above given by Marchetto:

C 159 C# 45 D
F 90 E 204 D

702 951 1200

I wonder how this might be modified with the 6/3 comma division for
apotome and limma, or roughly 136/68 cents. If we assume Marchetto's
ratios of 2:1, 3:2, 4:3, and 9:8, there might be contradictions; I
tend to get something like this to avoid them:

Just for fun, let's assume that the 6/3 division of apotome/limma
applies rather a 5/4 division as above. The apotome isn't in evidence
here, but the limma is (F-E), so we have (taking your 68 cents):

C 159 C# 45 D
F 68 E 226 D

702 929 1200

My logic would be like this. If we agree that the upper voice ascends
with its two steps of 7/2 commas by a 9:8 tone in all, then to move
from a pure fifth to a pure octave the other voice must contribute
another 32:27 or 294 cents of motion in the opposite or descending
direction. If its first step is 68 cents, our 3-comma limma, this
means a larger tone of 226 cents -- essentially identical to the
extended Pythagoren steps of 227 cents and 67 cents, the first equal
to two apotomes (the _tonus maior_ of Jacobus of Liege around 1325),
and the second to the classic 256:243 limma less a comma.

The interesting result is that our cadential major sixth is now 929
cents -- a 12:7 less the septimal schisma, essentially. I didn't
anticipate this curious side-effect of the 6/3 division, coupled to
7/2 for chroma/diesis: the cadential major sixth is in the 12:7
neighborhood, rather than around 951 cents.

In essence this is 2-3-7 JI, as in Archytas or Ptolemy's Tonic
Diatonic, with tones at either 9:8 or 8:7 (putting aside the septimal
schisma of ~3.80 cents), and likewise a limma of 28:27. It's curious
that while we agree that a "7-based" reading is mostly in the eyes of
a creative beholder and interpreter, your hypothesis seems to bring
much the same result. I'd emphasize, of course, that you raised 6-3
commas simply as a possible reading, not necessarily as the intended
one.

By the way, in a different kind of system, we get a rather similar
result in Zest-24 at one location, albeit with a yet wider "chroma" at
a bit larger than 54:49 or 168 cents:

Db 171 D* 46 Eb
Gb 70 F 217 Eb

696 937 1200

While the major sixth F-D* is slightly larger than 12:7, it's still in
that general vicinity -- although elsewhere there are intervals of 946
and 950 cents illustrating the size often read in Marchetto, if one
assumes a 5/4 apotome/limma division plus a 7/2 chroma/diesis
division.

>> The 18:17:16 division mentioned by Marchettus seems a curious
>> digression when taken in the context of the musical as well
>> mathematical setting sketched out by Marchettus. This division is
>> advocated as a practical and desirable one by the author Quintillian
>> in classic Rome. Boethius seems to describe it more as an example of
>> how it is impossible to divide a superparticular ratio such as 9:8
>> into two equal (and rational, one might add) ratios, than as an actual
>> standard by which the tone should be divided.

> Thanks for that. I'm currently thinking of it as an example, from
> Marchetto, of what not to do.

Yes, or maybe a curious digression mentioning a division cited in
Boethius as an academic exercise. An English organ treatise of 1373
does recommend this division for organ pipes: to find the length of a
pipe for adding an accidental semitone (maybe other than Bb, a part of
the regular gamut which may have been calculated as part of the
regular chain of fifths down from F), take the average of the two
pipes forming the tone you wish to divide, thus C-C#-D at 18:17:16.

This might have the advantage both of being easy to calculate, and of
producing notably milder tunings for common major thirds like A-C# at
around 393 cents (294 + 99, or A-C-C#), although it leads to two
notably impure fifths rather than one Wolf in a 12-note tuning. Mark
Lindley discusses this, as well as the use of a regular 12-note chain
of fifths at Gb-B in early 15th-century Europe that does result in
some schismatic thirds involving sharps of the kind that Ozan has
discussed in Safi al-Din al-Urmawi and in a maqam like modern Turkish
Rast -- e.g. A-C# tuned A-Db at 8192:6561 or 384 cents on an early
15th-century European keyboard.

Anyway, getting back to Marchetto and similar styles, I should mention
that I've been re-exploring a regular temperament that I was very
excited about in the year 2000, and seems to me to fill a special
niche not irrelevant to this general discussion, although, of course,
distinct from Marchetto's own intended vocal tuning.

This regular 24-note tuning is based on a fifth of 704.61 cents, about
the mildest temperament of the fifth that can yield fairly close
7-based ratios from a chain of 13, 14, 15, or 16 _regular_ fifths for
9:7, 7:6, 7:4, and 21:16 respectively (here 440, 264, 969, and 473
cents or so). We have 7:4 and also 11:8 virtually just. albeit with
9:7 not so accurate, at about 5 cents wide (a very mild "Marchettan"
touch? <grin>).

Playing this, I'm tempted to describe it as about the heaviest
tempering of the fifth I'd be comfortable using in something meant to
be "not too far" from Pythagorean -- I recall your suggestion that
something like 2.7 cents might define a "near-just" range for the
fifth, and here it's 2.65 cents wide or so.

Anyway, we have some interesting step sizes:

Tone: 209 cents
Apotome: 132 cents
Limma: 77 cents
Chroma: 154 cents
Diesis: 55 cents

I really, really, like this. It's not as faithful to superparticular
ratios like 11:12:13:14 for neutral seconds as Peppermint 24 is, not
to mention the heavier tempering of the regular fifths (2.65 rather
than 2.14 cents); the main advantage is having 23 regular fifths
(G#-D# gets stretched to around 714 cents in Peppermint), and the
simplicity of a single chain. It may be a bit simple having only two
sizes of neutral seconds, for example, but I find it one pleasant
place on the continuum.

Here we get a tempered version of Marchetto's progression as follows:

C* 154 D 55 D
F* 77 E* 209 D

705 936 1200

Anyway, I like this -- while recognizing that it might actually rather
more "conservative" in its degree of chroma/limma contrast than the
7/2 division quite naturally read in Marchetto, as you have shown.

With many thanks,

Margo

🔗Margo Schulter <mschulter@...>

> Margo, the Yarman24 rendition of maqam Segah would be:
> 0: 1/1 0.000 unison, perfect prime
> 1: 16/15 111.731 minor diatonic semitone
> 2: 6/5 315.641 minor third
> 3: 501.956 cents 501.956
> 4: 3/2 701.955 perfect fifth
> 5: 8/5 813.686 minor sixth
> 6: 9/5 1017.596 just minor seventh, BP seventh
> 7: 1105.866 cents 1105.866
> 8: 2/1 1200.000 octave

Dear Ozan,

Please let me say what a pleasure it is to be discussing the maqamat
or makamlar with you: as always, both educational and enlightening.
Is there a movement afoot to revive the learning of the Mutazilah Era?

> Bear in mind that the 0th step is perde segah, which is 5/4 of Yarman24:

This is a very interesting point, because the procedure for finding
Maqam Segah or Sikah is the same for Arab as for Turkish flavors, at
least for the most part, but with somewhat different results. A fine
point I would guess from your Segah is that the upper tetrachord might
be Kurdi when descending, but Hijaz when ascending, as in Maqam
Nihavend (or Nahawand):

Kurdi 9:8 Hijaz
Ascending: |-----------------------|.......|--------------------------|
Segah Chargah Neva Huseyni Evdj Gerdaniye S�nb�le Tiz Segah
1/1 16/15 6/5 501.956 3/2 8/5 1105.866 2/1
0 112 316 502 702 814 1106 1200
112 204 186 200 112 292 94

Kurdi 9:8 Kurdi
Descending: |-----------------------|.......|--------------------------|
Segah Chargah Neva Huseyni Evdj Gerdaniye Muhayyer Tiz Segah
1/1 16/15 6/5 501.956 3/2 8/5 9/5 2/1
0 112 316 502 702 814 1018 1200
112 204 186 200 112 204 182

I hope that I have most of the perde names correct, based on looking
at some of the papers on your Web site.

Yarman24 is indeed clever, and has a feature that I tried to
emulate. Why don't I include your listing at the end of this post for
people who might find it convenient, along with a Scala file.

The interesting thing I notice is that in this flavor of Segah, perde
segah and perde huseyni form a fourth close to a just 4:3. Thinking of
these as the third and sixth steps of Maqam Rast in a mostly 5-limit
flavor, I realize that segah might incline toward 5/4 and huseyni
toward 27/16 above rast -- but here with enough "tempering" so that
the fourth remains near 4:3. In the 79-MOS, for example on a qanun, I
would guess that you might use the wide or meantone fourth at about
505 cents, thus something like 392-898 cents for these perdeler --
indeed, I am hardly guessing, just reading and checking your papers!

In the tuning I'm now exploring, a regular 24-note system (much less
clever than Yarman24) with fifths at 704.607 cents, Segah is formed a
bit differently, as also in at least some Arab traditions where Rast
has a tuning more like that of al-Farabi's version of Zalzal's scale
for the `ud or lute. Here segah might be somewhere around 27/22 above
rast. In this tuning, placing perde rast for the moment at B, we have:

Maqam Rast

Rast 9:8 Rast
|-----------------|.....|-------------------|
Rast Dugah Segah Chargah Neva Huseyni Evdj Gerdaniye
B C# Eb E F# G# Bb B
0 209 363 495 705 914 1068 1200
209 154 132 209 209 154 132

Interestingly, some of the intervals aren't so far from flavors found
in Ozan24, such as 363 cents (a large 16/13 or small 100/81 or 121/98)
and 1068 cents (not too narrow of 13/7); I might call this a submajor
flavor of Rast, with evdj (Arabic awj) at 1068 cents notably wider
than 81/44 (1057 cents) in al-Farabi's version of Zalzal's tuning.

When we take the same perdeler, one flavor of what is often known in
Arab theory as the "fundamental scale" based on perde rast, with the
1/1 on perde segah (or in Arabic, sikah), we get this very interesting
result, with a curious intonational twist in this tuning:

Maqam Segah (or Sikah)

Rast 9:8 Rast
|-------------------|......|-------------------------|
Segah Chargah Neva Huseyni Evdj Gerdaniye Muhayyer Tik Segah
Eb E F# G# Bb B C# Eb
0 132 342 551 705 837 1046 1200
132 210 209 154 132 209 154

In this version of Segah, the neutral third of Rast at perde segah and
the major sixth at perde huseyni form a small tritone or
"superfourth," as Dave Keenan has well named it, which in this tuning
is at a virtually just 11/8, formed by the augmented third Eb-G#. This
is the one location in the tuning where a perfect fourth is not
available -- but the augmented third is just what we want for this
flavor of Segah, and makes it possible to play these basic notes of
the maqam on a single keyboard, although obviously inflections or
modulations will draw on more notes.

The sweet polyphonic flavors of this version include a most charming
final cadence:

C# Eb
G# Bb
E Eb

The major third and sixth of E-G#-C# at 0-418-914 cents resolve to the
fifth and octave, with the lowest voice descending by 132 cents and
the upper ones ascending by 154 cents. This is deliciously a "middle
second" flavor, what George Secor has termed an equable cadence.

Anyway, looking at some of your tunings, I noted the feature that two
different flavors of neutral intervals such as thirds and sixths may
be available above a single perde or step such as rast, as happens
delightfully with middle thirds and sevenths in Ozan24, and of course
with many shadings in 79-MOS. Why don't I give Ozan24 here as you
listed it, along with a Scala file which I hope is correct:

0: 1/1 0.000 unison, perfect prime
1: 84.360 cents 84.360
2: 38/35 142.373
3: 192.180 cents 192.180
4: 9/8 203.910 major whole tone
5: 292.180 cents 292.180
6: 17/14 336.130 supraminor third
7: 16/13 359.472 tridecimal neutral third
8: 5/4 386.314 major third
9: 19/15 409.244 undevicesimal ditone
10: 4/3 498.045 perfect fourth
11: 584.079 cents 584.079
12: 36/25 631.283 classic diminished fifth
13: 696.090 cents 696.090
14: 3/2 701.955 perfect fifth
15: 788.270 cents 788.270
16: 18/11 852.592 undecimal neutral sixth
17: 888.270 cents 888.270
18: 27/16 905.865 Pythagorean major sixth
19: 16/9 996.090 Pythagorean minor seventh
20: 20/11 1034.996 large minor seventh
21: 13/7 1071.702 16/3-tone
22: 15/8 1088.269 classic major seventh
23: 21/11 1119.463 undecimal major seventh
24: 2/1 1200.000 octave

! Yarman24.scl
!
Ozan Yarman's tuning including Rameau circle (TL #76333)
24
!
84.36000
38/35
192.18000
9/8
292.18000
17/14
16/13
5/4
19/15
4/3
584.07900
36/25
696.09000
3/2
788.27000
18/11
888.27000
27/16
16/9
20/11
13/7
15/8
21/11
2/1

In my 704.607-cent temperament, I thus arrived at this 23-note subset
-- not so much that it would be used in practice rather than the full
24 notes, but because it has a special property:

! perdeler23-symmetrical.scl
!
Maqam/Dastgah tuning, mirror symmetry, two flavors of neutral 2nd-3rd-6th-7th
23
!
76.96546
132.24835
153.93092
209.21381
286.17928
341.46217
363.14474
418.42763
495.39309
550.67598
572.35855
627.64145
649.32401
704.60691
781.57237
836.85526
858.53783
913.82072
990.78618
1046.06908
1067.75165
1123.03454
2/1

One might say that a starting point for this set is a 17-note system
(which, however, unlike Yarman17, cannot circulate in this temperament):
we then add four perdeler or notes so as to have _two_ versions of
neutral seconds, thirds, sixths, and sevenths. Two more steps are
added so that we can have two pairs of tritones: the middle ones
closer to 600 cents fitting the regular chain of fifths, and the
smallest and largest at a virtually just 11/8 and 16/11. Thus we have
six additional perdeler or steps in all, and 17 plus 6 or 23 notes.

The special property of this system is that the notes and intervals
have a "mirror" pattern on the two sides of the "axis" at 600 cents,
not itself represented by a note. If we count 1/1 and 2/1 as distinct
steps, then we have 24 steps in all within and including an octave:
and these form 12 pairs, each adding up to 1200 cents: thus

1/1 (0) 2/1 (1200)
77 1123
132 1067
154 1046
.... ....
572 628

This degree of temperament is quite close to the 705-cent system we
have discussed, or 63-EDO which is very similar, as you observed. A
compromise of this slightly milder tempering of the fifth is that
11/9, 13/8, 11/6, and especially 9/7 will be less accurate. A possible
benefit is that we have virtually just versions of 7/4, 11/8, and also
23/16 (the regular tritone), a ratio mentioned in your 79-MOS
presentations. An open question is whether 132 cents or 68/63 is low
enough for a dik kurdi (as in Yarnan24) and yet high enough for a very
low segah in Maqam Huseyni or Ushshaq (or some related flavors).

Of course, as should be made plain, the full 24-note tuning is meant
to be used where ratios of 2-3-7-11-13 are desired, but does not
really represent ratios of 5, although a few odd meantone-size
intervals are present involving very long chains of fifths. Thus it
offers some, but not all, maqam flavors.

! eb24.scl
!
Regular "e-based" tuning, Blackwood's R or tone/limma at e, ~2.71828
24
!
55.28289
132.24835
187.53125
209.21382
264.49671
286.17928
341.46217
418.42763
473.71052
495.39309
550.67598
627.64145
682.92434
704.60691
759.88980
836.85526
892.13815
913.82072
969.10362
990.78618
1046.06908
1123.03454
1178.31743
2/1

With many thanks,

Margo

🔗Dave Keenan <d.keenan@...>

🔗Margo Schulter <mschulter@...>

Dear Dave,

Thank you for such an encouraging reply, and for some specific
suggestions which, as I'll show, may have intriguing consequences.

> Thanks for sharing your excitement. It certainly is fun doing such
> detective work collectively. Whatever would we do if the old texts
> were always clear and precise.

That is well stated: ambiguity does have its charms.

> And thanks for clearly setting out the consequences of my suggested
> readings.

> I agree that the possible appearance of the syntonic comma, as the
> smallest of Marchetto's dieses, contains no implications of 5-limit
> intervals. This is because it seems likely to only occur in
> combination with one or more of its neighbouring commas in the
> 80:79:78:77:76:75:74:73:72 division.

This may tie in with one of your themes, that "JI," in a more aural
and possibly useful concept, should relate not only to integer ratios
or more specifically superparticular ones in isolation, but to the
musical context. That's another discussion, and I'll resist the
tendency to wander.

> I tend to think that Marchetto's cadential 3rd and 6th are likely
> to be Merciful rather than Just, for maximum contrast. However this
> doesn't help narrow the possibilities as there are MI intervals on
> either side of the 7-limit JI interpretations (9:7 and 12:7) that
> you and George have considered. These correspond well enough to two
> possible interpretations of Marchetto. They are (in cents) either
> 422 and 923 or 448 and 943.

From reading Marchetto about how 3rds and 6ths in directed
progressions "strive" toward stable concords, I would say that you
have a strong argument. Indeed, maybe MI might stand not only for
Merciful Intonation or Metastable Intonation, but also Marchettan
Intonation, with a fair historical basis.

Of course, what singers might do in practice is an open question: for
some, a tuning like 7:9:12 for E-G#-C# before D-A-D might be "active
enough" -- but full MI is certainly a fair guess. Now comes the fun
part.

First, while 422 and 923 cents would be quite reasonable -- and I can
easily singers making such adjustments, only about half a septimal or
64:63 comma, or 7/12 Pythagorean comma (say 14-15 cents) -- I suspect
we'll agree that the division of Marchetto fits better with 448/943.

Now starting with the 943-cent major sixth, let's consider this
example of his illustrating the division of tone into chroma and
diesis, assuming Pythagorean intonation in the lower voice:

+151 +53
C ------------- C# -------------- D
702 943 1200
F ------------- E -------------- D
-90 -204

Curiously, we have what amounts to a virtually precise division of the
9:8 tone in the upper voice into steps at 12:11 (151 cents) and 33:32
(53 cents)! I would hasten to add that seems a coincidence, rather
than an attempt to emulate Zalzal and al-Farabi, since Marchetto does
not elsewhere discuss neutral intervals (other than the chroma itself,
which he does not to my knowledge categorize as a "neutral" or middle
interval).

How about the 448-cent major third? Let's take another, and rather
analogour, Marchettan progression where a unison expands to a
fifth. again assuming regular Pythagorean sizes for steps in the lower
voice, here a minor third at 32:27 or 294 cents followed by a 9:8 tone
at 204 cents:

+154 +50
G -------------- G# --------------- A
0 448 702
G -------------- E --------------- D
-294 -204

Here the chroma is very slightly wide at 154 cents, and the diesis
slightly narrow at 50 cents -- but not too far from our previous
results with the 943-cent major sixth. An interesting nuance is that
the dieses are a bit wider than the monochord readings of the
"ninefold division into five parts" might suggest. Personally I very
much like small semitones around 50-60 cents, which could also be
described as quartertones -- but again, medieval or modern
interpreters of Marchetto need not be bound by my tastes <grin>.

Also, if one happens to be seeking a regular temperament that has both
regular semitones and special steps to realize these "Merciful" or MI
progressions, one candidate was, curiously, announced on this list
eight years ago early next month, and let to lots of responses from
both of us.

If a generator of 704.096 cents is chosen, then we get these values
which can all be made available at a reasonable number of positions in
a tuning of 24 or more notes per octave:

tone: 208.19
limma: 79.52
apotome: 128.67
chroma: 159.04
diesis: 49.15

The defining property of this "Noble Fifth" tuning proposed by Keenan
Pepper is that the tone and apotome has logarithmic sizes (as
meansured for example in cents) standing in the ratio of Phi, or
about 1.61834. By a curious coincidence, the augmented fifth
(e.g. C-G#) at 832.76 cents is very close to the interval ratio of Phi
(833.090 cents).

As you would rightly have me emphasize, these are two quite different
uses of Phi, as a comparison between logarithmic ratios, and as an
interval ratio, which should not be confused!

With this tuning, here is how our MI interpretations of Marchetto come
out:

+159 +49
C ------------- C# -------------- D
704 943 1200
F ------------- E -------------- D
-80 -208

The MI major sixth (from 14 fifths down or fourths up) is 942.661
cents. On a 24-note keyboard layout, this might be F*-C* to E*-D
to D*-D* -- with the major sixth equal to a regular minor seventh
at 992 cents less the 49-cent diesis.

Now for the major third progression to a fifth:

+159 +49
G -------------- G# --------------- A
0 447 704
G -------------- E --------------- D
-288 -208

The MI major third from 13 fourths up at 446.757 cents is very close
to theoretical MI. Here we might have, on a 24-note keyboard
arrangement, G*-G* to the major third E*-A to D*-A* -- with the major
third equal to a 496-cent fourth less the 49-cent diesis.

Thus in addition to illustrating Phi as a logarithmic ratio, and
closely approximating Phi as an interval ratio with its augmented
fifth, the Pepper tuning (or Wilson/Pepper, if we take into account
Erv WIlson's earlier indication of this temperament on the Scale
Tree), seems also for its major thirds and sixths derived from a chain
of 13 or 14 generators to approximate the Phi-based MI concept, where
the region of maximum complexity between two simple ratios is
estimated using a Phi-weighted mediant.

To be on the safe side, I'll emphasize that temperaments (regular or
otherwise) are a modern digression or diversion rather than a topic
presented by Marchetto himself -- but it's fun to see how the original
Pepper tuning might be considered "near-optimal" from the viewpoint of
a modern regular system approaching the MI model you've suggested. You
might consider this a kind of tribute to your "chains-of-fifths" papers,
from an angle that has very interestingly developed over the last eight
years as to MI or the like.

> I've been meaning to ask. Can you please explain the different usages
> of "Marchetto" and "Marchettus"?

As far as I know, they're simply Italian and Latin versions of the
same name for the same person -- so either is correct, and the choice
is a matter of one's style or mood at the moment. Here I've tended to
favor Marchetto because that's the version that you and others are
mostly using. I would guess that it is a nickname or affectionate
diminutive for Marcus or Marco -- just as the great composer Perotin
from the era around 1200 is known by a diminutive for French Pierre (or
in Latin, Petrus), here, as with Marchetto/Marchettus, the "standard"
version of the name, and in Latin Perotinus.

Another theorist of the era around 1300, Jacobus of Liege or Jacobus
de Montibus (if, as seems likely, he's the author referred to by the
latter name in a source called the Berkeley Manuscript or Paris
Anonymous from around 1375, with some of it carrying this date) is
likewise sometimes called Jacques de Liege, or even "James of Liege."

With many thanks,

Margo

🔗Margo Schulter <mschulter@...>

🔗robert thomas martin <robertthomasmartin@...>

🔗Mike Battaglia <battaglia01@...>

🔗Margo Schulter <mschulter@...>

🔗Ozan Yarman <ozanyarman@...>

Hello Margo,

Again, I am sorry for the late reply. My comments are below:

On Aug 21, 2008, at 8:27 AM, Margo Schulter wrote:

>> Margo, the Yarman24 rendition of maqam Segah would be:
>> 0: 1/1 0.000 unison, perfect prime
>> 1: 16/15 111.731 minor diatonic semitone
>> 2: 6/5 315.641 minor third
>> 3: 501.956 cents 501.956
>> 4: 3/2 701.955 perfect fifth
>> 5: 8/5 813.686 minor sixth
>> 6: 9/5 1017.596 just minor seventh, BP seventh
>> 7: 1105.866 cents 1105.866
>> 8: 2/1 1200.000 octave
>
> Dear Ozan,
>
> Please let me say what a pleasure it is to be discussing the maqamat
> or makamlar with you: as always, both educational and enlightening.
> Is there a movement afoot to revive the learning of the Mutazilah Era?
>

Ah, your sentiments are much refreshing. Indeed, we seem to have
revived the Medieval spirit of knowledge and cultural exchange between
East and West in this list! I too have learned much in our discourses.

>> Bear in mind that the 0th step is perde segah, which is 5/4 of
>> Yarman24:
>
> This is a very interesting point, because the procedure for finding
> Maqam Segah or Sikah is the same for Arab as for Turkish flavors, at
> least for the most part, but with somewhat different results.

That is because, perde segah or sikah (note that I make it a habit not
to capitalize perde names) is the third degree of the natural
diatonical scale that is Rast.

> A fine
> point I would guess from your Segah is that the upper tetrachord might
> be Kurdi when descending, but Hijaz when ascending, as in Maqam
> Nihavend (or Nahawand):
>
> Kurdi 9:8 Hijaz
> Ascending:
> |-----------------------|.......|--------------------------|
> Segah Chargah Neva Huseyni Evdj Gerdaniye S¸nb¸le
> Tiz Segah
> 1/1 16/15 6/5 501.956 3/2 8/5 > 1105.866 2/1
> 0 112 316 502 702 814 1106
> 1200
> 112 204 186 200 112 292 94
>

The perde that is a fourth above segah is dik hisar or hisarek, not
huseyni. Also, the first tetrachord is called Segah. The Kurdi
tetrachord would require 256/243 or 20/19 as the second degree. But
you got the other perde names and tetrachord correct!

> Kurdi 9:8 Kurdi
> Descending:
> |-----------------------|.......|--------------------------|
> Segah Chargah Neva Huseyni Evdj Gerdaniye Muhayyer
> Tiz Segah
> 1/1 16/15 6/5 501.956 3/2 8/5
> 9/5 2/1
> 0 112 316 502 702 814 1018
> 1200
> 112 204 186 200 112 204 182
>

Again, mind perde dik hisar, which is characteristic of Maqam Segah,
and the segah tetrachords instead of Kurdi. When the first half tone
is the apotome, we have a Segah tetrachord.

> I hope that I have most of the perde names correct, based on looking
> at some of the papers on your Web site.
>

You've done quite well!

> Yarman24 is indeed clever, and has a feature that I tried to
> emulate. Why don't I include your listing at the end of this post for
> people who might find it convenient, along with a Scala file.
>

Why not indeed?

> The interesting thing I notice is that in this flavor of Segah, perde
> segah and perde huseyni form a fourth close to a just 4:3.

segah-dik hisar makes a perfect fourth. Huseyni is always about a
comma higher than dik hisar.

> Thinking of
> these as the third and sixth steps of Maqam Rast in a mostly 5-limit
> flavor, I realize that segah might incline toward 5/4 and huseyni
> toward 27/16 above rast -- but here with enough "tempering" so that
> the fourth remains near 4:3.

But huseyni is already a pure 27/16 above rast. In retrospect, dik
hisar comes very close to 5/3.

> In the 79-MOS, for example on a qanun, I
> would guess that you might use the wide or meantone fourth at about
> 505 cents, thus something like 392-898 cents for these perdeler --
> indeed, I am hardly guessing, just reading and checking your papers!
>

Ah, but you see, in 79 MOS 2deg159-tloET (I hereby instert an "l" to
signify logarithmic, and "o" to imply octave), I have tempered both
segah and huseyni to achieve a perfect fourth between them. I admit it
is a stretch and actual segah will incline to a lower relative pitch
while huseyni will tend to a higher one.

> In the tuning I'm now exploring, a regular 24-note system (much less
> clever than Yarman24) with fifths at 704.607 cents, Segah is formed a
> bit differently, as also in at least some Arab traditions where Rast
> has a tuning more like that of al-Farabi's version of Zalzal's scale
> for the `ud or lute. Here segah might be somewhere around 27/22 above
> rast. In this tuning, placing perde rast for the moment at B, we have:
>
> Maqam Rast
>
> Rast 9:8 Rast
> |-----------------|.....|-------------------|
> Rast Dugah Segah Chargah Neva Huseyni Evdj Gerdaniye
> B C# Eb E F# G# Bb B
> 0 209 363 495 705 914 1068 1200
> 209 154 132 209 209 154 132
>

segah appears to be closer to 21/17 than 27/22, which is agreeable
with the Turkish tradition. Ditto for evdj at 63/64. So far so good.

> Interestingly, some of the intervals aren't so far from flavors found
> in Ozan24, such as 363 cents (a large 16/13 or small 100/81 or 121/98)
> and 1068 cents (not too narrow of 13/7); I might call this a submajor
> flavor of Rast, with evdj (Arabic awj) at 1068 cents notably wider
> than 81/44 (1057 cents) in al-Farabi's version of Zalzal's tuning.
>

More often than not, Rast is executed in that manner in Turkish maqammusic.

> When we take the same perdeler, one flavor of what is often known in
> Arab theory as the "fundamental scale" based on perde rast, with the
> 1/1 on perde segah (or in Arabic, sikah), we get this very interesting
> result, with a curious intonational twist in this tuning:
>
> Maqam Segah (or Sikah)
>
> Rast 9:8 Rast
> |-------------------|......|-------------------------|
> Segah Chargah Neva Huseyni Evdj Gerdaniye Muhayyer Tik Segah
> Eb E F# G# Bb B C# Eb
> 0 132 342 551 705 837 1046 1200
> 132 210 209 154 132 209 154
>
> In this version of Segah, the neutral third of Rast at perde segah and
> the major sixth at perde huseyni form a small tritone or
> "superfourth," as Dave Keenan has well named it, which in this tuning
> is at a virtually just 11/8, formed by the augmented third Eb-G#. This
> is the one location in the tuning where a perfect fourth is not
> available

A wild twist! Strangely enough, the Segah character of the scale is
not lost even with the introduction of the superfourth instead of a
pure one. I make note of this peculiar situation. Is 11/8 as consonant
as 4/3 for Segah, I wonder?

> -- but the augmented third is just what we want for this
> flavor of Segah, and makes it possible to play these basic notes of
> the maqam on a single keyboard, although obviously inflections or
> modulations will draw on more notes.
>
> The sweet polyphonic flavors of this version include a most charming
> final cadence:
>
> C# Eb
> G# Bb
> E Eb
>

A splendid Segah cadence!

> The major third and sixth of E-G#-C# at 0-418-914 cents resolve to the
> fifth and octave, with the lowest voice descending by 132 cents and
> the upper ones ascending by 154 cents. This is deliciously a "middle
> second" flavor, what George Secor has termed an equable cadence.
>

I admit it is very attractive.

> Anyway, looking at some of your tunings, I noted the feature that two
> different flavors of neutral intervals such as thirds and sixths may
> be available above a single perde or step such as rast, as happens
> delightfully with middle thirds and sevenths in Ozan24, and of course
> with many shadings in 79-MOS. Why don't I give Ozan24 here as you
> listed it, along with a Scala file which I hope is correct:
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 84.360 cents 84.360
> 2: 38/35 142.373
> 3: 192.180 cents 192.180
> 4: 9/8 203.910 major whole tone
> 5: 292.180 cents 292.180
> 6: 17/14 336.130 supraminor third
> 7: 16/13 359.472 tridecimal neutral third
> 8: 5/4 386.314 major third
> 9: 19/15 409.244 undevicesimal ditone
> 10: 4/3 498.045 perfect fourth
> 11: 584.079 cents 584.079
> 12: 36/25 631.283 classic diminished fifth
> 13: 696.090 cents 696.090
> 14: 3/2 701.955 perfect fifth
> 15: 788.270 cents 788.270
> 16: 18/11 852.592 undecimal neutral sixth
> 17: 888.270 cents 888.270
> 18: 27/16 905.865 Pythagorean major sixth
> 19: 16/9 996.090 Pythagorean minor seventh
> 20: 20/11 1034.996 large minor seventh
> 21: 13/7 1071.702 16/3-tone
> 22: 15/8 1088.269 classic major seventh
> 23: 21/11 1119.463 undecimal major seventh
> 24: 2/1 1200.000 octave
>
>
> ! Yarman24.scl
> !
> Ozan Yarman's tuning including Rameau circle (TL #76333)
> 24
> !
> 84.36000 nim zengule
> 38/35 zengule
> 192.18000 dik zengule
> 9/8 dugah
> 292.18000 kurdi
> 17/14 dik kurdi
> 16/13 nerm segah
> 5/4 segah
> 19/15 buselik
> 4/3 tschargah
> 584.07900 nim hijaz
> 36/25 hijaz
> 696.09000 dik hijaz
> 3/2 neva
> 788.27000 nim hisar
> 18/11 hisar
> 888.27000 dik hisar
> 27/16 huseyni
> 16/9 ajem
> 20/11 dik ajem
> 13/7 nerm evdj
> 15/8 evdj
> 21/11 mahur
> 2/1 gerdaniye
>

Yes, that is correct. I have added the perde names above.

>
> In my 704.607-cent temperament, I thus arrived at this 23-note subset
> -- not so much that it would be used in practice rather than the full
> 24 notes, but because it has a special property:
>
> ! perdeler23-symmetrical.scl
> !
> Maqam/Dastgah tuning, mirror symmetry, two flavors of neutral
> 2nd-3rd-6th-7th
> 23
> !
> 76.96546
> 132.24835
> 153.93092
> 209.21381
> 286.17928
> 341.46217
> 363.14474
> 418.42763
> 495.39309
> 550.67598
> 572.35855
> 627.64145
> 649.32401
> 704.60691
> 781.57237
> 836.85526
> 858.53783
> 913.82072
> 990.78618
> 1046.06908
> 1067.75165
> 1123.03454
> 2/1
>

I notice, that this might be classified as a subset of 109-tloET

>
> One might say that a starting point for this set is a 17-note system
> (which, however, unlike Yarman17, cannot circulate in this
> temperament):
> we then add four perdeler or notes so as to have _two_ versions of
> neutral seconds, thirds, sixths, and sevenths. Two more steps are
> added so that we can have two pairs of tritones: the middle ones
> closer to 600 cents fitting the regular chain of fifths, and the
> smallest and largest at a virtually just 11/8 and 16/11. Thus we have
> six additional perdeler or steps in all, and 17 plus 6 or 23 notes.
>

I notice with satisfaction that you have provided a 837 cent relative
pitch that can account for a Karjighar flavour better than Yarman24
can render. This allows for a hijaz tetrachord that is 132+231+132
cents.

> The special property of this system is that the notes and intervals
> have a "mirror" pattern on the two sides of the "axis" at 600 cents,
> not itself represented by a note. If we count 1/1 and 2/1 as distinct
> steps, then we have 24 steps in all within and including an octave:
> and these form 12 pairs, each adding up to 1200 cents: thus
>
> 1/1 (0) 2/1 (1200)
> 77 1123
> 132 1067
> 154 1046
> .... ....
> 572 628
>

Interesting symmetry!

> This degree of temperament is quite close to the 705-cent system we
> have discussed, or 63-EDO which is very similar, as you observed. A
> compromise of this slightly milder tempering of the fifth is that
> 11/9, 13/8, 11/6, and especially 9/7 will be less accurate. A possible
> benefit is that we have virtually just versions of 7/4, 11/8, and also
> 23/16 (the regular tritone), a ratio mentioned in your 79-MOS
> presentations. An open question is whether 132 cents or 68/63 is low
> enough for a dik kurdi (as in Yarman24) and yet high enough for a very
> low segah in Maqam Huseyni or Ushshaq (or some related flavors).
>

I think 132 cents is just right for a dik kurdi suitable for Nihavend,
Ushshaq and Hijaz maqams.

> Of course, as should be made plain, the full 24-note tuning is meant
> to be used where ratios of 2-3-7-11-13 are desired, but does not
> really represent ratios of 5, although a few odd meantone-size
> intervals are present involving very long chains of fifths. Thus it
> offers some, but not all, maqam flavors.
>
> ! eb24.scl
> !
> Regular "e-based" tuning, Blackwood's R or tone/limma at e, ~2.71828
> 24
> !
> 55.28289
> 132.24835
> 187.53125
> 209.21382
> 264.49671
> 286.17928
> 341.46217
> 418.42763
> 473.71052
> 495.39309
> 550.67598
> 627.64145
> 682.92434
> 704.60691
> 759.88980
> 836.85526
> 892.13815
> 913.82072
> 969.10362
> 990.78618
> 1046.06908
> 1123.03454
> 1178.31743
> 2/1
>

I still find it important that the simplest 5-limit JI can be
represented on some keys, and a 12-tone cycle is achieved for common-
practice footing.

>
> With many thanks,
>
> Margo
>

Cordially,
Oz.

🔗Mike Battaglia <battaglia01@...>

🔗Margo Schulter <mschulter@...>

>> Please let me say what a pleasure it is to be discussing the maqamat
>> or makamlar with you: as always, both educational and enlightening.
>> Is there a movement afoot to revive the learning of the Mutazilah Era?

> Ah, your sentiments are much refreshing. Indeed, we seem to have
> revived the Medieval spirit of knowledge and cultural exchange between
> East and West in this list! I too have learned much in our discourses.

Dear Ozan (and Mike),

Thank you for these inspiring words, which made my day as well as Mike
Battaglia's. Mike, thank you for your encouragement to Ozan and me,
and please feel free to ask questions or offer comments on anything in
these conversations which might be unclear or a bit specialized. I
know that I can easily write things which are clearest "when already
understood," as a computer programming saying might have it -- so
trying to explain things, or place them in a fuller context, can be a
great learning experience for me as well as anyone else involved.

For now, because I am a bit tired, and also because maybe the mood of
this dialogue calls for a moment of celebration in between the
episodes of more detailed technical analysis, I might mention some
very pleasant musical experiences that your ideas have brought about,
Ozan. Also, I will share a humorous story and try briefly to respond
to a well-taken point you make, which possibly suggests a larger truth
about tuning systems.

One very exciting experience for me with the 704.607-cent regular
temperament I am again exploring (almost identical to 109-tloET, as
you observed) was when I sought out a Segah with a 4/3 above perde
segah. This was quite a delightful "discovery":

Descending

Segah tone Segah
|----------------------|.......|--------------------------|
segah chargah neva hisar evdj gerdaniye muhayyer tik segah
Bb B C# Eb F F# G# Bb
0 132 341 495 705 837 1046 1200
132 209 154 209 132 209 154

For the ascending version, I hit on the obvious Hijaz tetrachord with
perde kurdi:

Ascending

Segah tone Hijaz
|----------------------|.......|--------------------------|
segah chargah neva hisar evdj gerdaniye s�nb�le tik segah
Bb B C# Eb F F# A Bb
0 132 341 495 705 837 1123 1200
132 209 154 209 132 286 77

However, reading over the fine points of your post, I realized at some
point that an additional flavor for the ascending version of Segah
might be possible in a different transposition:

Ascending

Segah tone Hijaz
|----------------------|.......|--------------------------|
dik tik
segah chargah neva hisar evdj gerdaniye muhayyer segah
Bb* B* C#* Eb* F* F#* A Bb*
0 132 341 495 705 837 1068 1200
132 209 154 209 132 231 132

Your ideas about the 132-231-132 tetrachord can lead in some
fascinating directions, hwoever suited or otherwise for a given maqam;
what I want to share here is mainly the sheer excitement of exploring
these different flavors.

Also, I was intrigued by the remarks you made about Karjighar, which
led me to learn a little about its tetrachords, at least, and to find
a version in this temperament which had me delighted with its flavors.
This is an incentive actually to learn the seyir and understand how
those flavors combine to form a maqam. I am keeping in mind that one
learns a maqam not by merely considering the intervals in cents, but
by listening and playing taksim.

Another post of yours from August 9, which somehow I only found on
Yahoo this week -- my apologies for failing to notice it earlier! --
reminded me of a related point. This is the humorous story promised
above.

It is said in chess that one should not make a move without knowing
what the next move is likely to be -- and the one after that!

Similarly, before attempting to discuss the tuning of a maqam such as
Suz-i Dilara as that name is understood in Turkey, I would be wise to
learn something about its seyir! When I learned from your post the
difference between Mahur and Suz-i Dilara, the latter an intricate
terkib, this lesson was really driven home. By the way, my last
sentence certainly is not meant to imply that any maqam is other than
"intricate" -- especially to those who know it well.

Now for the philosophical reflection, much helped by reviewing some of
your earlier posts over the last few years which help me to put the
question of Rast/Mahur/Suz-i-Dilara in better perspective.

If I write what is to follow well, by the way, I hope that some of it
may be of interest even to those who might not be so concerned with
the specific technical questions we are addressing.

> I still find it important that the simplest 5-limit JI can be
> represented on some keys, and a 12-tone cycle is achieved for
> common- practice footing.

Please let me hasten to agree that for the musical styles and
situations you address, these requirements are indeed vital: the
79-MOS addresses them superbly, and Ozan24 shows how they can be
addressed quite ingeniously with much more modest resources. Indeed I
might regard the 79-MOS as a palatial estate filled with many fragrant
gardens, each offering a different musical flavor from some region or
style of the Maqam/Dastgah world.

In contrast, the 704.607-cent or "e-based" tuning is like a small and
quiet garden with curious plants and flowers that may offer some
special colors and flavors. It is meant to be a charming sample from
the vast world of Maqam/Dastgah music -- but by no means a complete or
even representative one!

Also, the "footing" for this tuning is different: to support a certain
pleasant but limited range of Maqam/Dastgah flavors, plus a reasonable
if modern version of the European "common practice," as I might call
it, of the 13th-14th centuries, in part coinciding with the late
Mutazilah era. For this latter purpose, each keyboard presents a
tempered variation on the 12-note Pythagorean tuning (Eb-G#) which
seems to have been very common on 14th-century keyboards.

A regular temperament with fifths tempered by about 2.65 cents might
be at about the maximum degree of impurity for 3:2 and 4:3 that I
would be comfortable using when seeking something not too far from a
medieval European sound. The other regular diatonic intervals for
13th-14th century European purposes are somewhat accentuated -- major
intervals wider, minor ones narrower -- which can be both congenial
and musically exciting. Special cadences like those of Marchettus of
Padua using yet more accentuated intervals are also available.

As someone whose musical formation and training focus largely on
medieval European music, I feel right at home. However, here it might
not be out of place to quote the proverb of a 16th-century English
musician named Thomas Morley -- "Such lips, such lettuce," said while
watching a donkey eating thistles!

In a setting where it is important either to represent the full range
of Maqam/Dastgah colors, or to support music on a "common practice"
footing in the usual sense of major/minor tonality, it is very easy
to show why this tuning will fail, Ozan24 will succeed with the
same number of notes per octave, and the 79-MOS on the qanun or
another instrument will, of course, succeed brilliantly.

From that perspective, I might say that my regular tuning fails in
some of the same ways as AEU -- requiring a broken chain of fifths for
a reasonable Rast, having C# always higher than Db -- but may possibly
fail in somewhat more interesting ways <grin>. Instead of almost
nothing being acknowledged as a middle or neutral interval, we might
slightly exaggerate by saying that almost everything might seem to be
a middle interval or to produce one, including the apotome of 132
cents, almost 6 commas, or not quite 9 yarmans (each equal to about
2/3 of a Holdrian comma, although the size of a yarman may minutely
vary, like the secor, depending on the implementation of the 79-MOS).

Thus if one seeks a 5-limit Rast, with AEU one will have the right
interval sizes but in the wrong places for an unbroken chain of fifths
and regular note spellings for this maqam.

With the e-based tuning, one will have the wrong interval sizes in the
wrong places <grin>. From another point of view, we indeed have the
right sizes for another flavor of Rast which you have mentioned as
used in Turkey: 363 and 1068 cents, close to 21/17 and 63/34, and also
for Ajemli Rast a sixth at 859 cents, close to 23/14. However, these
are not 5-limit intervals, nor are they based on an unbroken chain of
fifths permitting regular diatonic spellings. Rather, they result from
precisely the same types of spellings that produce near-pure 5-limit
intervals in AEU: e.g. F#-Bb, F#-Eb, and F#-F at 363/859/1068 cents
here and 384/882/1086 cents in AEU.

The strategy of this 704.607-cent temperament is in some ways a very
modest one: the aim is specifically to support a certain subset of
Maqam/Dastgah flavors, plus 13th-14th century European common practice
(quite different from the 18th-19th century "common practice" usually
identified with that phrase). It is meant, as mentioned above, to be a
small and modest garden, another musical perspective which might serve
to highlight the much vaster territory and range of maqam structures
and flavors covered by a system like the 79-MOS.

> Cordially,
> Oz.

With many thanks,

Margo

🔗Ozan Yarman <ozanyarman@...>

A late response (with all due apologies):

On Aug 31, 2008, at 11:17 AM, Margo Schulter wrote:

>
>>> Please let me say what a pleasure it is to be discussing the maqamat
>>> or makamlar with you: as always, both educational and enlightening.
>>> Is there a movement afoot to revive the learning of the Mutazilah
>>> Era?
>
>> Ah, your sentiments are much refreshing. Indeed, we seem to have
>> revived the Medieval spirit of knowledge and cultural exchange
>> between
>> East and West in this list! I too have learned much in our
>> discourses.
>
> Dear Ozan (and Mike),
>
> Thank you for these inspiring words, which made my day as well as Mike
> Battaglia's. Mike, thank you for your encouragement to Ozan and me,
> and please feel free to ask questions or offer comments on anything in
> these conversations which might be unclear or a bit specialized. I
> know that I can easily write things which are clearest "when already
> understood," as a computer programming saying might have it -- so
> trying to explain things, or place them in a fuller context, can be a
> great learning experience for me as well as anyone else involved.
>
> For now, because I am a bit tired, and also because maybe the mood of
> this dialogue calls for a moment of celebration in between the
> episodes of more detailed technical analysis, I might mention some
> very pleasant musical experiences that your ideas have brought about,
> Ozan. Also, I will share a humorous story and try briefly to respond
> to a well-taken point you make, which possibly suggests a larger truth
> about tuning systems.
>
> One very exciting experience for me with the 704.607-cent regular
> temperament I am again exploring (almost identical to 109-tloET, as
> you observed) was when I sought out a Segah with a 4/3 above perde
> segah. This was quite a delightful "discovery":
>
> Descending
>
> Segah tone Segah
> |----------------------|.......|--------------------------|
> segah chargah neva hisar evdj gerdaniye muhayyer tik segah
> Bb B C# Eb F F# G# Bb
> 0 132 341 495 705 837 1046 1200
> 132 209 154 209 132 209 154
>

This is a very nice rendition. But you see, the perfect fourth above
segah is always dik hisar, not hisar. After sunbule comes tiz segah,
not tik.

> For the ascending version, I hit on the obvious Hijaz tetrachord with
> perde kurdi:
>
> Ascending
>
> Segah tone Hijaz
> |----------------------|.......|--------------------------|
> segah chargah neva hisar evdj gerdaniye s¸nb¸le tik segah
> Bb B C# Eb F F# A Bb
> 0 132 341 495 705 837 1123 1200
> 132 209 154 209 132 286 77
>

This is also very nice, but mind the dik hisar. Once more, after
sunbule comes tiz segah, not tik.

> However, reading over the fine points of your post, I realized at some
> point that an additional flavor for the ascending version of Segah
> might be possible in a different transposition:
>
> Ascending
>
> Segah tone Hijaz
> |----------------------|.......|--------------------------|
> dik tik
> segah chargah neva hisar evdj gerdaniye muhayyer segah
> Bb* B* C#* Eb* F* F#* A Bb*
> 0 132 341 495 705 837 1068 1200
> 132 209 154 209 132 231 132
>

Again, mind the dik hisar. Also, the acute fifth between neva and what
you call dik muhayyer is puzzling. If you call it nim sunbule instead
of dik muhayyer, that would hit the spot.

I have reconsidered using huseyni in place of dik hisar, and came up
with three Yarman24 scales for maqam Segah:

(ascending =>) Ed 112 F 204 G 204 A 182 Bd 112 C 204 D 182 Ed

(ascending =>) Eb 112 F 204 G 204 A 182 Bd 112 C 292 D# 94 Ed

Eb 112 F 204 G 204 A 90 Bb 204 C 204 D 88 Eb (<= descending)

note ratio cent PERDE
Ed 5/4 386 segah
F 4/3 498 tschargah
G 3/2 702 neva
A 27/16 906 huseyni
Bb 16/9 996 ajem
Bd 15/8 1088 evdj
C 2/1 1200 gerdaniye
D 9/4 1404 muhayyer
D# 1492 sunbule
Ed 5/2 1586 tiz segah

I think using 27/16 in place of 5/3 (keeping huseyni-muhayyer a pure
fourth) is more characteristic of Segah. Usage of 5/3 with 16/9 would
be reminiscent of a descending Huzzam scale.

> Your ideas about the 132-231-132 tetrachord can lead in some
> fascinating directions, hwoever suited or otherwise for a given maqam;
> what I want to share here is mainly the sheer excitement of exploring
> these different flavors.
>

I'm pleased you find the mujannab-tanini-mujannab (JTJ) tetrachord
useful.

> Also, I was intrigued by the remarks you made about Karjighar, which
> led me to learn a little about its tetrachords, at least, and to find
> a version in this temperament which had me delighted with its flavors.
> This is an incentive actually to learn the seyir and understand how
> those flavors combine to form a maqam. I am keeping in mind that one
> learns a maqam not by merely considering the intervals in cents, but
> by listening and playing taksim.
>

Indeed so.

> Another post of yours from August 9, which somehow I only found on
> Yahoo this week -- my apologies for failing to notice it earlier! --
> reminded me of a related point. This is the humorous story promised
> above.
>
> It is said in chess that one should not make a move without knowing
> what the next move is likely to be -- and the one after that!
>
> Similarly, before attempting to discuss the tuning of a maqam such as
> Suz-i Dilara as that name is understood in Turkey, I would be wise to
> learn something about its seyir! When I learned from your post the
> difference between Mahur and Suz-i Dilara, the latter an intricate
> terkib, this lesson was really driven home. By the way, my last
> sentence certainly is not meant to imply that any maqam is other than
> "intricate" -- especially to those who know it well.
>
> Now for the philosophical reflection, much helped by reviewing some of
> your earlier posts over the last few years which help me to put the
> question of Rast/Mahur/Suz-i-Dilara in better perspective.
>

Glad to know my meager knowledge was useful.

> If I write what is to follow well, by the way, I hope that some of it
> may be of interest even to those who might not be so concerned with
> the specific technical questions we are addressing.
>
>> I still find it important that the simplest 5-limit JI can be
>> represented on some keys, and a 12-tone cycle is achieved for
>> common- practice footing.
>
> Please let me hasten to agree that for the musical styles and
> situations you address, these requirements are indeed vital: the
> 79-MOS addresses them superbly, and Ozan24 shows how they can be
> addressed quite ingeniously with much more modest resources.

I think it best to call it Yarman24. Sorting out Scala scales that I
came up with during my presence on the tuning list is still in the
back of my mind.

Thank you for your acknowledgment of my points above!

> Indeed I
> might regard the 79-MOS as a palatial estate filled with many fragrant
> gardens, each offering a different musical flavor from some region or
> style of the Maqam/Dastgah world.
>

Oh that is a very charming description. However I tend to think it is
more like a jungle!

> In contrast, the 704.607-cent or "e-based" tuning is like a small and
> quiet garden with curious plants and flowers that may offer some
> special colors and flavors. It is meant to be a charming sample from
> the vast world of Maqam/Dastgah music -- but by no means a complete or
> even representative one!
>

Despite all this modesty, I am inclined to admit that the 705 cent
tuning produces much agreeable results from the perspective of maqam
theory.

> Also, the "footing" for this tuning is different: to support a certain
> pleasant but limited range of Maqam/Dastgah flavors, plus a reasonable
> if modern version of the European "common practice," as I might call
> it, of the 13th-14th centuries, in part coinciding with the late
> Mutazilah era. For this latter purpose, each keyboard presents a
> tempered variation on the 12-note Pythagorean tuning (Eb-G#) which
> seems to have been very common on 14th-century keyboards.
>

Charming!

> A regular temperament with fifths tempered by about 2.65 cents might
> be at about the maximum degree of impurity for 3:2 and 4:3 that I
> would be comfortable using when seeking something not too far from a
> medieval European sound.

That is indeed about as much impurity one can tolerate in the building
of maqam tetrachords and pentachords too.

> The other regular diatonic intervals for
> 13th-14th century European purposes are somewhat accentuated -- major
> intervals wider, minor ones narrower -- which can be both congenial
> and musically exciting. Special cadences like those of Marchettus of
> Padua using yet more accentuated intervals are also available.
>

Delightful.

> As someone whose musical formation and training focus largely on
> medieval European music, I feel right at home. However, here it might
> not be out of place to quote the proverb of a 16th-century English
> musician named Thomas Morley -- "Such lips, such lettuce," said while
> watching a donkey eating thistles!
>

LOL! That is a cute proverb, but I feel it does grave injustice to
your attire. Your tuning does indeed seem to reflect the musical
spirit of Medieval Europe as well as house many maqam/dastgah flavours.

> In a setting where it is important either to represent the full range
> of Maqam/Dastgah colors, or to support music on a "common practice"
> footing in the usual sense of major/minor tonality, it is very easy
> to show why this tuning will fail, Ozan24 will succeed with the
> same number of notes per octave, and the 79-MOS on the qanun or
> another instrument will, of course, succeed brilliantly.
>

Very nicely stated. Conversely, Yarman24 and 79 MOS 2deg159-tET will
surely fail in representing Medieval cadences that you have
accentuated many times in our correspondences.

> From that perspective, I might say that my regular tuning fails in
> some of the same ways as AEU -- requiring a broken chain of fifths for
> a reasonable Rast, having C# always higher than Db -- but may possibly
> fail in somewhat more interesting ways <grin>.

Duely acknowledged.

> Instead of almost
> nothing being acknowledged as a middle or neutral interval, we might
> slightly exaggerate by saying that almost everything might seem to be
> a middle interval or to produce one, including the apotome of 132
> cents, almost 6 commas, or not quite 9 yarmans (each equal to about
> 2/3 of a Holdrian comma, although the size of a yarman may minutely
> vary, like the secor, depending on the implementation of the 79-MOS).
>

How nice of you to dub the 2/3 Holderian commas as a yarman unit, and
how pleasing to see that you noticed the variation of its size
depending on which approach one uses to derive 79 MOS 2deg159-tET.

> Thus if one seeks a 5-limit Rast, with AEU one will have the right
> interval sizes but in the wrong places for an unbroken chain of fifths
> and regular note spellings for this maqam.
>
> With the e-based tuning, one will have the wrong interval sizes in the
> wrong places <grin>. From another point of view, we indeed have the
> right sizes for another flavor of Rast which you have mentioned as
> used in Turkey: 363 and 1068 cents, close to 21/17 and 63/34, and also
> for Ajemli Rast a sixth at 859 cents, close to 23/14. However, these
> are not 5-limit intervals, nor are they based on an unbroken chain of
> fifths permitting regular diatonic spellings. Rather, they result from
> precisely the same types of spellings that produce near-pure 5-limit
> intervals in AEU: e.g. F#-Bb, F#-Eb, and F#-F at 363/859/1068 cents
> here and 384/882/1086 cents in AEU.
>
> The strategy of this 704.607-cent temperament is in some ways a very
> modest one: the aim is specifically to support a certain subset of
> Maqam/Dastgah flavors, plus 13th-14th century European common practice
> (quite different from the 18th-19th century "common practice" usually
> identified with that phrase). It is meant, as mentioned above, to be a
> small and modest garden, another musical perspective which might serve
> to highlight the much vaster territory and range of maqam structures
> and flavors covered by a system like the 79-MOS.
>

And your tuning does a splendid job too.

>> Cordially,
>> Oz.
>
> With many thanks,
>
> Margo

Oz.

🔗Margo Schulter <mschulter@...>

Re: [tuning] Re: For Ozan: Segah flavors

> A late response (with all due apologies):

Dear Ozan,

Please let me emphasize that no apologies are necessary, and that to
discuss the maqamat in this way is a delightful dialogue and
education. I am reminded of some of the dialogues in classic European
counterpoint texts such as Morley (1597) and Fux (1725). Like Fux you
have a good teacher's patience and love of the art; and, like Morley,
you have a forthrightness and sense of humor which helps bring the
lesson home.

[Indeed it is I who must beg your indulgence, for if lines of text were
cents, there are enough lines in this reply to take us from rast to a
pleasant segah.]

> On Aug 31, 2008, at 11:17 AM, Margo Schulter wrote:

> Descending
>
> Segah tone Segah
> |----------------------|.......|--------------------------|
> segah chargah neva hisar evdj gerdaniye muhayyer tik segah
> Bb B C# Eb F F# G# Bb
> 0 132 341 495 705 837 1046 1200
> 132 209 154 209 132 209 154

> This is a very nice rendition. But you see, the perfect fourth above
> segah is always dik hisar, not hisar. After sunbule comes tiz segah,
> not tik.

Ah, I see that I must have confused tik, an Arab term which may
correspond to dik (meaning a "steep" or elevated version of a given
perde within the octave), with the intended tiz to show the higher
octave. That mistake is noted so that I can avoid it in the future!

As to segah and dik hisar rather than hisar forming a perfect fourth,
I will take note and correct the example accordingly:

Descending

Segah tone Segah
|----------------------|......,.|---------------------------|
segah chargah neva dik hisar evdj gerdaniye muhayyer tiz segah
Bb B C# Eb F F# G# Bb
0 132 341 495 705 837 1046 1200
132 209 154 209 132 209 154

This raises another point also: in my attempt to name the perdeler of
this 704.607-cent tuning, I had thought of 837 cents from rast as nerm
hisar, and 859 cents as hisar. If 859 cents is by definition dik hisar
as the perfect fourth of segah at 363 cents, then might 837 cents best
be called simply hisar? Here, taking F# as rast, this note would be
located at D*, or 473 cents above segah, a narrow fourth very close to
21:16, and smaller than perfect by a comma.

> This is also very nice, but mind the dik hisar. Once more, after
> sunbule comes tiz segah, not tik.

This must have again been my confusion of Arabic tik (=dik) with the
correct tiz for a higher octave. Perhaps my repeated mistake was not
so unfortunate, for it gives an opportunity to learn the proper form
more vividly through repeated practice.

> However, reading over the fine points of your post, I realized at some
> point that an additional flavor for the ascending version of Segah
> might be possible in a different transposition:
>

> Ascending
>
> Segah tone Hijaz
> |----------------------|.......|--------------------------|
> dik tik
> segah chargah neva hisar evdj gerdaniye muhayyer segah
> Bb* B* C#* Eb* F* F#* A Bb*
> 0 132 341 495 705 837 1068 1200
> 132 209 154 209 132 231 132
>

> Again, mind the dik hisar. Also, the acute fifth between neva and
> what you call dik muhayyer is puzzling. If you call it nim sunbule
> instead of dik muhayyer, that would hit the spot.

Please let me correct my example accordingly, and also emend the 'tik
segah' to tiz segah:

> Ascending
>
> Segah tone Hijaz
> |----------------------|.......|--------------------------|
> nim tiz
> segah chargah neva hisar evdj gerdaniye s�nb�le segah
> Bb* B* C#* Eb* F* F#* A Bb*
> 0 132 341 495 705 837 1068 1200
> 132 209 154 209 132 231 132

This matter of nim s�nb�le rather than dik muhayyer as the apt name
here raises a more general question. As a beginner, I was unsure about
the best name for a step at 231 cents or 8:7 above rast (or its octave
gerdaniye) when I looked at your names in one listing of the 79-MOS:
dik dugah at 211 cents (the next step above dugah); sarp dugah at 226
cents (the closest to 8:7); and nim kurdi at 242 cents. Now I know
that nim kurdi, or nim s�nb�le, is the best choice.

Maybe I could best show how I tried to reason out the problem like
this:

limma
|------------------ 77 -----------------|
|----- 22 ----|---------- 55 -----------|
G#* comma A diesis A*
dugah ? kurdi
209 231 286

Here the basic perdeler, dugah and kurdi, are a regular limma apart,
or 77 cents, a bit smaller than the Pythagorean limma of 4 commas. In
this tuning system, I might guess that dik or possibly sarp could mean
"a 22-cent comma higher," and nim "a 55-cent diesis lower." Here I
have now learned that nim kurdi or "a diesis below kurdi" is the right
answer, rather than dik dugah or possibly sarp dugah, "a comma above
dugah."

My beginner's curiosity leads me to wonder if there is any general rule
for naming a step in this sort of situation -- but looking at your
article on 34-tone, 79-tone, and other systems, I might conclude that
the intermediate steps between basic perde a limma apart like rast-shuri,
dugah-kurdi, buselik-chargah, neva-beyati, and huseyni-ajem might each
have their own rules or local customs, so to speak. This might reflect
the wealth of maqamat and nuances of modulation.

Maybe like someone learning a language, I might better learn by lots
of practice with the guidance of a good native speaker, rather than
trying to understand all of the fine points of grammar at once!

> I have reconsidered using huseyni in place of dik hisar, and came up
> with three Yarman24 scales for maqam Segah:

> (ascending =>) Ed 112 F 204 G 204 A 182 Bd 112 C 204 D 182 Ed

> (ascending =>) Eb 112 F 204 G 204 A 182 Bd 112 C 292 D# 94 Ed

A very small point: here I suspect that in the second tuning, Ed
rather than Eb (perde kurdi) is meant, as the 112-cent step to F
confirms. In other words, these two versions have identical lower
pentachords, with only the upper tetrachords varying. In commas, I
might write these upper tetrachords as 5-9-8 (rather like Kurdi,
4-9-9) and around 5-13-4 (a kind of Hijaz).

> Eb 112 F 204 G 204 A 90 Bb 204 C 204 D 88 Eb (<= descending)

Here I wonder if the tonic or final is meant to be Ed rather than Eb,
as the 112-cent step up to F again suggests -- and whether, given the
the first step of the descent at 88 cents, the highest note should be,
rather than Eb, a D# or s�nb�le. I am very ready to be corrected!

386 498 702 906 996 0 204 292
5/4 4/3 3/2 27/16 16/9 1/1 9/8 ~32/27
> Ed 112 F 204 G 204 A 90 Bb 204 C 204 D 88 D# (<= descending)
0 112 316 520 610 814 1018 1106
1/1 16/15 6/5 27/20 64/45 8/5 9/5 ~256/135
or 36/19

> note ratio cent PERDE
> Ed 5/4 386 segah
> F 4/3 498 tschargah
> G 3/2 702 neva
> A 27/16 906 huseyni
> Bb 16/9 996 ajem
> Bd 15/8 1088 evdj
> C 2/1 1200 gerdaniye
> D 9/4 1404 muhayyer
> D# 1492 sunbule
> Ed 5/2 1586 tiz segah

If my guess happens to be right, then I might take as an example of
lowering both the fifth and the octave above the final by a limma or
so in order to add impetus to the descent. Here I think of the Persian
Shur Dastgah, where the fifth above the final is often a motaqayyer,
being lowered in descending passages by a koron equal to something
like 50-70 cents, or a bit less than the usual limma.

> I think using 27/16 in place of 5/3 (keeping huseyni-muhayyer a pure
> fourth) is more characteristic of Segah. Usage of 5/3 with 16/9
> would be reminiscent of a descending Huzzam scale.

And thus the 27:20 wide fourths at Ed-A -- as also in a 5-limit
version of Maqam Rast; in the 79-MOS, we might choose the compromise
of having segah at 392 cents and huseyni at 898 cents, a meantone
fourth apart; but I am not sure how desirable this would be for Maqam
Segah.

>> Your ideas about the 132-231-132 tetrachord can lead in some
>> fascinating directions, hwoever suited or otherwise for a given maqam;
>> what I want to share here is mainly the sheer excitement of exploring
>> these different flavors.

> I'm pleased you find the mujannab-tanini-mujannab (JTJ) tetrachord
> useful.

Yes, I have seen that notation in some notes to a CD offering a
performance of some musical examples from Safi al-Din al-Urmawi. Here
a point is that T or tanini, while it often means 9:8 or 9 commas (a
usual whole tone), may also mean 8:7 or 10 commas.

Generally, I understand that JTJ may be called `Iraq.

A tetrachord of 132-231-132 or roughly 6-10-6 commas, is what I would
call Buzurg, after the lower tetrachord of a tuning by that name given
by Qutb al-Din al-Shirazi as 1/1-14/13-16/13-4/3-56/39-3/2 or
14/13x8/7x13/12x14/13x117/112 (128-231-139-128-76). Of course, I know
that this name means many different things to many different people.

Here the difference between 14:13 and 13:12, or 169:168, is tempered
out, so that 132 cents may stand for either.

>> Please let me hasten to agree that for the musical styles and
>> situations you address, these requirements are indeed vital: the >
>> 79-MOS addresses them superbly, and Ozan24 shows how they can be >
>> addressed quite ingeniously with much more modest resources.

> I think it best to call it Yarman24. Sorting out Scala scales that I
> came up with during my presence on the tuning list is still in the
> back of my mind.

Noted! I will remember: Yarman24.

> Thank you for your acknowledgment of my points above!

Reading some of your discussions about recent Turkish theory and the
79-MOS has helped me to get a better perspective.

>> Indeed I might regard the 79-MOS as a palatial estate filled with
>> many fragrant gardens, each offering a different musical flavor
>> from some region or style of the Maqam/Dastgah world.

> Oh that is a very charming description. However I tend to think it
> is more like a jungle!

At times, I now realize through a bit of consideration, the need to
get as many intervals and practical chains of fifths as possible into
even a 79-note tuning may require some compromises, as I found when
trying to locate some medieval European progressions you mention
below.

From one perspective, a full 159-EDO might be ideal; from another, the
artistry of a 79-tone qanun such as your wonderful innovation shows
that the art of the intricate but possible is not to be disdained!

With the regular 704.607-cent tuning, also, there are some real
compromises even in covering a narrower ground: for example, the use
of a single 132-cent size for both 14/13 and 13/12. Curiously, both
this tuning and the 79-MOS involve some compromises here, although at
some locations you have a virtually just 14:13 at 127-129 cents
(varying minutely in size among these locations).

>> In contrast, the 704.607-cent or "e-based" tuning is like a small
>> and quiet garden with curious plants and flowers that may offer
>> some special colors and flavors. It is meant to be a charming
>> sample from the vast world of Maqam/Dastgah music -- but by no
>> means a complete or even representative one! >

> Despite all this modesty, I am inclined to admit that the 705 cent
> tuning produces much agreeable results from the perspective of maqam
> theory.

This is most welcome feedback. What I would love to do is to explore
this system with some guidance on your part, for I relish our
dialogues, and wonder if the process may possibly lead to any new
insights or refinements as to maqam music in a kind of 24-note system
which may not be of the most familiar type either for European music
or for al-Musiqa al-Sharqiyya ("Eastern Music") as it is often called
in Arabic, I now know.

>> A regular temperament with fifths tempered by about 2.65 cents
>> might be at about the maximum degree of impurity for 3:2 and 4:3
>> that I would be comfortable using when seeking something not too
>> far from a medieval European sound.

> That is indeed about as much impurity one can tolerate in the
> building of maqam tetrachords and pentachords too.

Happily, it seems to be just sufficient, when tempering in the wide
direction, to obtain some reasonable septimal flavors from chains of
13, 14, or 15 generators for 9:7, 7:6, and 7:4.

>> As someone whose musical formation and training focus largely on
>> medieval European music, I feel right at home. However, here it
>> might not be out of place to quote the proverb of a 16th-century
>> English musician named Thomas Morley -- "Such lips, such lettuce,"
>> said while watching a donkey eating thistles!

> LOL! That is a cute proverb, but I feel it does grave injustice to
> your attire. Your tuning does indeed seem to reflect the musical
> spirit of Medieval Europe as well as house many maqam/dastgah
> flavours.

Now the challenge is for me to learn some of those flavors, with your
fine examples of qanun music for inspiration.

>> In a setting where it is important either to represent the full range
>> of Maqam/Dastgah colors, or to support music on a "common practice"
>> footing in the usual sense of major/minor tonality, it is very easy
>> to show why this tuning will fail, Ozan24 will succeed with the
>> same number of notes per octave, and the 79-MOS on the qanun or
>> another instrument will, of course, succeed brilliantly.

> Very nicely stated. Conversely, Yarman24 and 79 MOS 2deg159-tET will
> surely fail in representing Medieval cadences that you have
> accentuated many times in our correspondences.

Here I found, upon considering the 79-MOS closely, that indeed one
will often need to use tempered rather than pure fourths and fifths in
such cadences. This is one necessary compromise involved in making
available the chains of tempered fifths required for the smooth
management of maqam flavors and modulations.

>> Instead of almost nothing being acknowledged as a middle or neutral
>> interval, we might slightly exaggerate by saying that almost
>> everything might seem to be a middle interval or to produce one,
>> including the apotome of 132 cents, almost 6 commas, or not quite 9
>> yarmans (each equal to about 2/3 of a Holdrian comma, although the
>> size of a yarman may minutely vary, like the secor, depending on
>> the implementation of the 79-MOS).

> How nice of you to dub the 2/3 Holderian commas as a yarman unit,
> and how pleasing to see that you noticed the variation of its size
> depending on which approach one uses to derive 79 MOS 2deg159-tET.

Yes, that is a great subtlety of the tuning, as when I noticed at one
location an approximation of 14:13 at not quite 127 cents, and then
used Scala to find that a range of sizes from 127-129 cents or so are
present from different perdeler.

[...]

> And your tuning does a splendid job too.

The challenge is for me to learn to use it well.

>> Cordially,
>> Oz.

With many thanks,

Margo

🔗Ozan Yarman <ozanyarman@...>

My replies are below:

On Sep 12, 2008, at 7:33 AM, Margo Schulter wrote:

> Re: [tuning] Re: For Ozan: Segah flavors
>
>> A late response (with all due apologies):
>
> Dear Ozan,
>
> Please let me emphasize that no apologies are necessary, and that to
> discuss the maqamat in this way is a delightful dialogue and
> education.

Very pleased to know it.

> I am reminded of some of the dialogues in classic European
> counterpoint texts such as Morley (1597) and Fux (1725). Like Fux you
> have a good teacher's patience and love of the art; and, like Morley,
> you have a forthrightness and sense of humor which helps bring the
> lesson home.
>

Ah, you are very kind. If only I was as capable as Morley and Fux!

> [Indeed it is I who must beg your indulgence, for if lines of text
> were
> cents, there are enough lines in this reply to take us from rast to a
> pleasant segah.]
>

That is a very charming analogy.

>> On Aug 31, 2008, at 11:17 AM, Margo Schulter wrote:
>
>
>> Descending
>>
>> Segah tone Segah
>> |----------------------|.......|--------------------------|
>> segah chargah neva hisar evdj gerdaniye muhayyer tik segah
>> Bb B C# Eb F F# G# Bb
>> 0 132 341 495 705 837 1046 1200
>> 132 209 154 209 132 209 154
>
>
>> This is a very nice rendition. But you see, the perfect fourth above
>> segah is always dik hisar, not hisar. After sunbule comes tiz segah,
>> not tik.
>
> Ah, I see that I must have confused tik, an Arab term which may
> correspond to dik (meaning a "steep" or elevated version of a given
> perde within the octave), with the intended tiz to show the higher
> octave. That mistake is noted so that I can avoid it in the future!
>

For you see, Arabs have borrowed the Turkish dik and transformed it to
their tik! I suspect a similar thing occured with qaba => qarar.

> As to segah and dik hisar rather than hisar forming a perfect fourth,
> I will take note and correct the example accordingly:
>
>
> Descending
>
> Segah tone Segah
> |----------------------|......,.|---------------------------|
> segah chargah neva dik hisar evdj gerdaniye muhayyer tiz
> segah
> Bb B C# Eb F F# G# Bb
> 0 132 341 495 705 837 1046 1200
> 132 209 154 209 132 209 154
>

Yes, this is correct.

> This raises another point also: in my attempt to name the perdeler of
> this 704.607-cent tuning, I had thought of 837 cents from rast as nerm
> hisar, and 859 cents as hisar. If 859 cents is by definition dik hisar
> as the perfect fourth of segah at 363 cents, then might 837 cents best
> be called simply hisar? Here, taking F# as rast, this note would be
> located at D*, or 473 cents above segah, a narrow fourth very close to
> 21:16, and smaller than perfect by a comma.

Ordinarily the distance between segah and dik hisar must always be a
perfect fourth. But now I see the problem. Your segah is way too low.
In that case, dik hisar is pulled down along with segah to what would
by default be the position of our classical hisar at 18/11. In its
stead, hisar now occupies the position a comma lower at 837 cents.

>
>
>> For the ascending version, I hit on the obvious Hijaz tetrachord with
>> perde kurdi:
>>
>> Ascending
>>
>> Segah tone Hijaz
>> |----------------------|.......|--------------------------|
>> segah chargah neva hisar evdj gerdaniye s¬∏nb¬∏le
>> tik segah
>> Bb B C# Eb F F# A Bb
>> 0 132 341 495 705 837 1123 1200
>> 132 209 154 209 132 286 77
>>
>
>> This is also very nice, but mind the dik hisar. Once more, after
>> sunbule comes tiz segah, not tik.
>
> This must have again been my confusion of Arabic tik (=dik) with the
> correct tiz for a higher octave. Perhaps my repeated mistake was not
> so unfortunate, for it gives an opportunity to learn the proper form
> more vividly through repeated practice.

Your dedication in this is much inspiring!

>
>
>> However, reading over the fine points of your post, I realized at
>> some
>> point that an additional flavor for the ascending version of Segah
>> might be possible in a different transposition:
>>
>
>> Ascending
>>
>> Segah tone Hijaz
>> |----------------------|.......|--------------------------|
>> dik tik
>> segah chargah neva hisar evdj gerdaniye muhayyer segah
>> Bb* B* C#* Eb* F* F#* A Bb*
>> 0 132 341 495 705 837 1068 1200
>> 132 209 154 209 132 231 132
>>
>
>> Again, mind the dik hisar. Also, the acute fifth between neva and
>> what you call dik muhayyer is puzzling. If you call it nim sunbule
>> instead of dik muhayyer, that would hit the spot.
>
> Please let me correct my example accordingly, and also emend the 'tik
> segah' to tiz segah:
>
>> Ascending
>>
>> Segah tone Hijaz
>> |----------------------|.......|--------------------------|
>> nim tiz
>> segah chargah neva hisar evdj gerdaniye s¸nb¸le segah
>> Bb* B* C#* Eb* F* F#* A Bb*
>> 0 132 341 495 705 837 1068 1200
>> 132 209 154 209 132 231 132
>

Yes this is correct. But why write sunbule as s,nb,le?

> This matter of nim s¸nb¸le rather than dik muhayyer as the apt name
> here raises a more general question. As a beginner, I was unsure about
> the best name for a step at 231 cents or 8:7 above rast (or its octave
> gerdaniye) when I looked at your names in one listing of the 79-MOS:
> dik dugah at 211 cents (the next step above dugah); sarp dugah at 226
> cents (the closest to 8:7); and nim kurdi at 242 cents. Now I know
> that nim kurdi, or nim s¸nb¸le, is the best choice.
>
> Maybe I could best show how I tried to reason out the problem like
> this:
>
> limma
> |------------------ 77 -----------------|
> |----- 22 ----|---------- 55 -----------|
> G#* comma A diesis A*
> dugah ? kurdi
> 209 231 286
>
> Here the basic perdeler, dugah and kurdi, are a regular limma apart,
> or 77 cents, a bit smaller than the Pythagorean limma of 4 commas. In
> this tuning system, I might guess that dik or possibly sarp could mean
> "a 22-cent comma higher," and nim "a 55-cent diesis lower." Here I
> have now learned that nim kurdi or "a diesis below kurdi" is the right
> answer, rather than dik dugah or possibly sarp dugah, "a comma above
> dugah."
>

Exactly! Because you have chosen an acute fifth as the generator of
your tuning, the commas and diaschismas have grown in size to dieses,
which you must take into account in the naming of perdeler instead of
the 22-cent comma (Ptolemy's comma?)

> My beginner's curiosity leads me to wonder if there is any general
> rule
> for naming a step in this sort of situation -- but looking at your
> article on 34-tone, 79-tone, and other systems, I might conclude that
> the intermediate steps between basic perde a limma apart like rast-
> shuri,
> dugah-kurdi, buselik-chargah, neva-beyati, and huseyni-ajem might each
> have their own rules or local customs, so to speak. This might reflect
> the wealth of maqamat and nuances of modulation.
>

I am inclined to say that almost all Middle Eastern traditions agree
on the size of the limma as being somewhere between 80-100 cents. In
naming perdes that are narrower, you have to keep track of not only
their distance from the previous and next perde, but also the
relationships with other distant perdes as I have tried to demonstrate
in the example of Segah.

> Maybe like someone learning a language, I might better learn by lots
> of practice with the guidance of a good native speaker, rather than
> trying to understand all of the fine points of grammar at once!

You might have a point there.

>
>
>> I have reconsidered using huseyni in place of dik hisar, and came up
>> with three Yarman24 scales for maqam Segah:
>
>> (ascending =>) Ed 112 F 204 G 204 A 182 Bd 112 C 204 D 182 Ed
>
>> (ascending =>) Eb 112 F 204 G 204 A 182 Bd 112 C 292 D# 94 Ed
>
> A very small point: here I suspect that in the second tuning, Ed
> rather than Eb (perde kurdi) is meant, as the 112-cent step to F
> confirms. In other words, these two versions have identical lower
> pentachords, with only the upper tetrachords varying.

That is right.

> In commas, I
> might write these upper tetrachords as 5-9-8 (rather like Kurdi,
> 4-9-9) and around 5-13-4 (a kind of Hijaz).
>

Like Kurdi, but is not. We call that a Segah or Iraq tetrachord. A
kind of Hijaz is correct. In AEU theory, the Hijaz tetrachord varies
in size from 5-12-5 to 5-13-4 to 4-13-5 commas.

>> Eb 112 F 204 G 204 A 90 Bb 204 C 204 D 88 Eb (<= descending)
>
> Here I wonder if the tonic or final is meant to be Ed rather than Eb,
> as the 112-cent step up to F again suggests -- and whether, given the
> the first step of the descent at 88 cents, the highest note should be,
> rather than Eb, a D# or s¸nb¸le. I am very ready to be corrected!
>

Uh oh. I made a typographical mistake. That should have been, as you
have noticed, Ed not Eb in the bottom note. However, the top note is
correct. It IS an Eb (sunbule) that is equivalent to a D# but spelled
as Eb due to the diatonical requirements of the maqam. Notice that D#
(not shown) is the leading tone exactly an octave below sunbule! Here
we have D#-Eb as the octave. Curious, is it not?

> 386 498 702 906 996 0 204 292
> 5/4 4/3 3/2 27/16 16/9 1/1 9/8 ~32/27
>> Ed 112 F 204 G 204 A 90 Bb 204 C 204 D 88 D# (<= descending)
> 0 112 316 520 610 814 1018 1106
> 1/1 16/15 6/5 27/20 64/45 8/5 9/5 ~256/135
> or 36/19
>

Your top ratios are correct. I cannot check the bottom ones, but I
assume them to be correct too.

>> note ratio cent PERDE
>> Ed 5/4 386 segah
>> F 4/3 498 tschargah
>> G 3/2 702 neva
>> A 27/16 906 huseyni
>> Bb 16/9 996 ajem
>> Bd 15/8 1088 evdj
>> C 2/1 1200 gerdaniye
>> D 9/4 1404 muhayyer
>> D# 1492 sunbule
>> Ed 5/2 1586 tiz segah
>
> If my guess happens to be right, then I might take as an example of
> lowering both the fifth and the octave above the final by a limma or
> so in order to add impetus to the descent. Here I think of the Persian
> Shur Dastgah, where the fifth above the final is often a motaqayyer,
> being lowered in descending passages by a koron equal to something
> like 50-70 cents, or a bit less than the usual limma.

Here, as the characteristic of Segah's descent, we lower the fifth and
the octave of the finalis by exactly a limma. You are a keen observer!

>
>
>> I think using 27/16 in place of 5/3 (keeping huseyni-muhayyer a pure
>> fourth) is more characteristic of Segah. Usage of 5/3 with 16/9
>> would be reminiscent of a descending Huzzam scale.
>
> And thus the 27:20 wide fourths at Ed-A -- as also in a 5-limit
> version of Maqam Rast; in the 79-MOS, we might choose the compromise
> of having segah at 392 cents and huseyni at 898 cents, a meantone
> fourth apart; but I am not sure how desirable this would be for Maqam
> Segah.

You can always reassign segah to 377 cents to fix matters straight. In
fact, I was quite undecided even after the acceptance of my doctorate
thesis and still am about the Segah business. Ordinarily, I wanted to
keep segah-huseyni a perfect fourth, but I see now that this approach
lends itself to certain abuses within historical context! For you see,
perdes segah and huseyni have opposite poles of attraction.
Fortunately, the huge resolution of 79 MOS 2deg159-tET takes care of matters by itself depending on how you make use of it.

>
>
>>> Your ideas about the 132-231-132 tetrachord can lead in some
>>> fascinating directions, hwoever suited or otherwise for a given
>>> maqam;
>>> what I want to share here is mainly the sheer excitement of
>>> exploring
>>> these different flavors.
>
>> I'm pleased you find the mujannab-tanini-mujannab (JTJ) tetrachord
>> useful.
>
> Yes, I have seen that notation in some notes to a CD offering a
> performance of some musical examples from Safi al-Din al-Urmawi. Here
> a point is that T or tanini, while it often means 9:8 or 9 commas (a
> usual whole tone), may also mean 8:7 or 10 commas.
>

Exactly.

> Generally, I understand that JTJ may be called `Iraq.
>

Also Hijazi by Ladikli Mehmet Chelebi during the reign of Mehmet the
Conqueror.

> A tetrachord of 132-231-132 or roughly 6-10-6 commas, is what I would
> call Buzurg, after the lower tetrachord of a tuning by that name given
> by Qutb al-Din al-Shirazi as 1/1-14/13-16/13-4/3-56/39-3/2 or
> 14/13x8/7x13/12x14/13x117/112 (128-231-139-128-76). Of course, I know
> that this name means many different things to many different people.
>

I like that tetrachord! In fact, you can hear that tetrachord chanted
by our best muezzins in Istanbul during the evening call to prayer.

> Here the difference between 14:13 and 13:12, or 169:168, is tempered
> out, so that 132 cents may stand for either.
>

I too cannot differentiate between the two 13-limit ratios in
79MOS2deg159-tET.

>>> Please let me hasten to agree that for the musical styles and
>>> situations you address, these requirements are indeed vital: the >
>>> 79-MOS addresses them superbly, and Ozan24 shows how they can be >
>>> addressed quite ingeniously with much more modest resources.
>
>> I think it best to call it Yarman24. Sorting out Scala scales that I
>> came up with during my presence on the tuning list is still in the
>> back of my mind.
>
> Noted! I will remember: Yarman24.
>
>> Thank you for your acknowledgment of my points above!
>
> Reading some of your discussions about recent Turkish theory and the
> 79-MOS has helped me to get a better perspective.
>

Once again, a big thank you!

>>> Indeed I might regard the 79-MOS as a palatial estate filled with
>>> many fragrant gardens, each offering a different musical flavor
>>> from some region or style of the Maqam/Dastgah world.
>
>> Oh that is a very charming description. However I tend to think it
>> is more like a jungle!
>
> At times, I now realize through a bit of consideration, the need to
> get as many intervals and practical chains of fifths as possible into
> even a 79-note tuning may require some compromises, as I found when
> trying to locate some medieval European progressions you mention
> below.
>
> From one perspective, a full 159-EDO might be ideal; from another, the
> artistry of a 79-tone qanun such as your wonderful innovation shows
> that the art of the intricate but possible is not to be disdained!
>

> With the regular 704.607-cent tuning, also, there are some real
> compromises even in covering a narrower ground: for example, the use
> of a single 132-cent size for both 14/13 and 13/12. Curiously, both
> this tuning and the 79-MOS involve some compromises here, although at
> some locations you have a virtually just 14:13 at 127-129 cents
> (varying minutely in size among these locations).

Surely, the happenstance actualization of 14:13 here and there cannot
be construed as a positive aspect of 79MOS2deg159tET.

>
>
>>> In contrast, the 704.607-cent or "e-based" tuning is like a small
>>> and quiet garden with curious plants and flowers that may offer
>>> some special colors and flavors. It is meant to be a charming
>>> sample from the vast world of Maqam/Dastgah music -- but by no
>>> means a complete or even representative one! >
>
>> Despite all this modesty, I am inclined to admit that the 705 cent
>> tuning produces much agreeable results from the perspective of maqam
>> theory.
>
> This is most welcome feedback. What I would love to do is to explore
> this system with some guidance on your part, for I relish our
> dialogues, and wonder if the process may possibly lead to any new
> insights or refinements as to maqam music in a kind of 24-note system
> which may not be of the most familiar type either for European music
> or for al-Musiqa al-Sharqiyya ("Eastern Music") as it is often called
> in Arabic, I now know.

I don't see why not!

>
>
>>> A regular temperament with fifths tempered by about 2.65 cents
>>> might be at about the maximum degree of impurity for 3:2 and 4:3
>>> that I would be comfortable using when seeking something not too
>>> far from a medieval European sound.
>
>> That is indeed about as much impurity one can tolerate in the
>> building of maqam tetrachords and pentachords too.
>
> Happily, it seems to be just sufficient, when tempering in the wide
> direction, to obtain some reasonable septimal flavors from chains of
> 13, 14, or 15 generators for 9:7, 7:6, and 7:4.
>

That is swell.

>>> As someone whose musical formation and training focus largely on
>>> medieval European music, I feel right at home. However, here it
>>> might not be out of place to quote the proverb of a 16th-century
>>> English musician named Thomas Morley -- "Such lips, such lettuce,"
>>> said while watching a donkey eating thistles!
>
>> LOL! That is a cute proverb, but I feel it does grave injustice to
>> your attire. Your tuning does indeed seem to reflect the musical
>> spirit of Medieval Europe as well as house many maqam/dastgah
>> flavours.
>
> Now the challenge is for me to learn some of those flavors, with your
> fine examples of qanun music for inspiration.
>

You are most kind. I am, however, quite the amateur in maqam music.
What clarity I possess in tuning and theory comes from much cunning
and shrewdness, yet little skill!

>>> In a setting where it is important either to represent the full
>>> range
>>> of Maqam/Dastgah colors, or to support music on a "common practice"
>>> footing in the usual sense of major/minor tonality, it is very easy
>>> to show why this tuning will fail, Ozan24 will succeed with the
>>> same number of notes per octave, and the 79-MOS on the qanun or
>>> another instrument will, of course, succeed brilliantly.
>
>> Very nicely stated. Conversely, Yarman24 and 79 MOS 2deg159-tET will
>> surely fail in representing Medieval cadences that you have
>> accentuated many times in our correspondences.
>
> Here I found, upon considering the 79-MOS closely, that indeed one
> will often need to use tempered rather than pure fourths and fifths in
> such cadences. This is one necessary compromise involved in making
> available the chains of tempered fifths required for the smooth
> management of maqam flavors and modulations.
>

Very keenly described.

>>> Instead of almost nothing being acknowledged as a middle or neutral
>>> interval, we might slightly exaggerate by saying that almost
>>> everything might seem to be a middle interval or to produce one,
>>> including the apotome of 132 cents, almost 6 commas, or not quite 9
>>> yarmans (each equal to about 2/3 of a Holdrian comma, although the
>>> size of a yarman may minutely vary, like the secor, depending on
>>> the implementation of the 79-MOS).
>
>> How nice of you to dub the 2/3 Holderian commas as a yarman unit,
>> and how pleasing to see that you noticed the variation of its size
>> depending on which approach one uses to derive 79 MOS 2deg159-tET.
>
> Yes, that is a great subtlety of the tuning, as when I noticed at one
> location an approximation of 14:13 at not quite 127 cents, and then
> used Scala to find that a range of sizes from 127-129 cents or so are
> present from different perdeler.
>
> [...]
>
>> And your tuning does a splendid job too.
>
> The challenge is for me to learn to use it well.
>

Hopefully you will succeed without further ado!

>>> Cordially,
>>> Oz.
>
> With many thanks,
>
> Margo
>
>

Appreciatively,
Oz.