The Skisni metameantone sequence

Merry Christmas everybody,

About metameantones, if I can borrow this term from Erv Wilson,
here is a specially useful one, I think, I found after my relatively small
previous 1,4933585565602 fifth, of 694,27 c. :

1,4959535062432 or 697,278405 c., that I will call a "skisni" 5th.

This ratio allows the differential coherence of every 8/5 in a meantone
chain :

8 - x^4 = 2x

and the equal-beating of the fourth, and the major third in all G - C - E
type major triads :

4 - 2x = x^4 - 4

Here is a Skisni sequence of frequencies starting from 2^n but there are
plenty - (transposed by octaves below here and there) :

C = 256
G = 383
D = 573
A = 857
E = 641
B = 959
F# = 1435
C# = 1073
G# = 1605
D# = 2401 (7^4)
A# = 1797
E# = 2687
C-- = 4019
G-- = 3005

(1282 = 2048 - 766 and so on - it should only come back near 2^n with x^74)

What's nice is that E (four 5ths) are only sligthly higher than 5 (641/640
here), and B (five 5ths) is just about a skisma lower than 15/2 (959/960) -
for that reason if you follow B by a chain of seven pure 5ths after five of
those skisni 5ths you fall back on C, only 0,077 c. higher :
x^5 . 3^7 = 16384,73 (2^14 = 16384...)

And that's already some sort of well-temperament - though I don't know if
it's much in use - (you could also have it exact with more or less same
properties, then the meantone fifth would be of 1,495940194).

Another recurrent sequence I call Melkis (= 1,33415744713, x^4 - 2x = 1/2)
combines quite well with 3 or 4 skisni 5ths, and 4 or zero pure 5ths to
produce more common types of well-temperaments, without harmonic deviation,
that made some adepts around here after a workshop I gave in 2006 about
harmonic temperaments.

Any feedback is welcome.

- - - - - - - -
Jacques Dudon

Hello Jacques!

While I was searching for the applicability of the meantone fifths to
Turkish Maqam music, I noticed that at least 5 Ahenks (diapasons)
could be met admirably with a bike-chain of 3 modified meantone
temperaments containing 12 tones each and made up of 1/6 comma
tempered fifths which are seperated by 35/32 and 81/65 cents apart
respectively.

This is an interesting development, since I had proposed a 36-tone
system to explain maqamat as far back as 7 years ago in my master's
thesis.

The modified meantones in question are cyclic with 3 super-pyth fifths
705 cents in size in each.

Here is the tuning with perde names (SCALA scale file attached):

0: 1/1 Kaba Rast / RAST / Gerdaniye / Tiz Gerdaniye
1: 47 cents Dik (Rast)...
2: 76 cents Nerm (Zengule)...
3: 95 cents Kaba Nim Zengule / Nim Zengule, Nim Şehnaz
4: 155 cents (35/32) Kaba Zengule / Zengule / Şehnaz
5: 184 cents Kaba Dik Zengule / Dik Zengule / Dik Şehnaz
6: 197 cents Kaba Dügah / DÜGAH / Muhayyer
7: 250 cents Dik...
8: 273 cents Nerm...
9: 305 cents Kaba Kürdi / Kürdi / Sünbüle
10: 352 cents Kaba Nerm Segah / Nerm Segah / Tiz Nerm Segah
(Ushshaq)
11: 381 cents (81/65) Kaba Segah / SEGAH / Tiz Segah
12: 393 cents Kaba Buselik / BUSELIK / Tiz Buselik
13: 460 cents Dik...
14: 476 cents Nerm...
15: 502 cents Kaba Tschargah / TSCHARGAH / Tiz Tschargah
16: 549 cents Dik...
17: 578 cents Nerm...
18: 590 cents Kaba Nim Hijaz / Nim Hijaz / Tiz Nim Hijaz
19: 657 cents Kaba Hijaz / Hijaz / Tiz Hijaz (Saba)
20: 686 cents Kaba Dik Hijaz / Dik Hijaz / Dik Nim Hijaz
21: 698 cents Yegah / NEVA / Tiz Neva
22: 745 cents Dik...
23: 774 cents Nerm...
24: 800 cents Kaba Nim Hisar / Nim Hisar / Tiz Nim Hisar
25: 854 cents Kaba Hisar / Hisar / Tiz Hisar
26: 883 cents Kaba Dik Hisar / Dik Hisar / Tiz Dik Hisar
27: 895 cents Ashiran / HÜSEYNI / Tiz Hüseyni
28: 955 cents Dik...
29: 971 cents Nerm...
30: 1003 cents Ajem Ashiran / Ajem / Tiz Ajem Ashiran
31: 1050 cents Dik Ajem Ashiran / Dik Ajem / Tiz Dik Ajem Ashiran
32: 1079 cents Nerm Evdj / EVDJ / Tiz Evdj
33: 1092 cents Gevasht / Mahur / Tiz Mahur
34: 1158 cents Dik...
35: 1181 cents Nerm...
36: 2/1 Rast / GERDANIYE / Tiz Gerdaniye

Süpürde Ahenk starts perde rast on C, the 0th step.
In Bolahenk, perde rast is D, the 6th step.
In Mansur, perde rast is G, the 21st step.
In Kiz, perde rast is A, the 27th step.
In Mustahsen, perde rast is B, the 33rd step.

Notation is as follows:

First bike-chain starting from degree 0 gets the naturals, sharps and
flats.
Second bike-chain 35/32 higher gets the ƀ flats and ‡ sharps.
Third bike-chain 81/65 higher gets the d flats and H sharps (# sharps
whose lower line is omitted).

Some maqams are given below in this notation:

RAST
C D Ed F G A Bd c, Bb A G F Ed D C

MAHUR
C D E F G A B c

PENCHGAH
C D Ed FH G A B c

NIHAVEND
C D Eb F G Ab B c, Bb Ab G F Eb D c

BUSELIK
D E F G A Bb cH d, c Bb A G F E D

HIJAZ
D Eb FH G A Bƀ c d, C Bb A G FH Eb D

KURDI
D Eb F G A Bb c d, c Bb Ab G F Eb D

HUSEYNI
D Eƀ F G A Bƀ c d, c Bƀ A G F Eƀ D

USHSHAQ
D Eƀ F G A Bb c, Bb A G F Eƀ D

SABA
D Ed F Gƀ A Bb c dƀ e f, ed___eƀ d c Bb A Gƀ F Eƀ D

KARJIGHAR
D Eƀ F G Aƀ Bd c d

SEGAH
Ed F G Ad Bd c dH e, eb d c Bb Ad G F Ed

HUZZAM
Ed F G Aƀ Bd c dH e, eb d c Bb Ad G F Ed

MUSTEAR
Ed FH G Ad Bd c dH e

Cordially,
Oz.

On Dec 24, 2008, at 9:16 PM, Jacques Dudon wrote:

> Merry Christmas everybody,
>
> About metameantones, if I can borrow this term from Erv Wilson,
> here is a specially useful one, I think, I found after my relatively
> small
> previous 1,4933585565602 fifth, of 694,27 c. :
>
> 1,4959535062432 or 697,278405 c., that I will call a "skisni" 5th.
>
> This ratio allows the differential coherence of every 8/5 in a
> meantone
> chain :
>
> 8 - x^4 = 2x
>
> and the equal-beating of the fourth, and the major third in all G -
> C - E
> type major triads :
>
> 4 - 2x = x^4 - 4
>
> Here is a Skisni sequence of frequencies starting from 2^n but there
> are
> plenty - (transposed by octaves below here and there) :
>
> C = 256
> G = 383
> D = 573
> A = 857
> E = 641
> B = 959
> F# = 1435
> C# = 1073
> G# = 1605
> D# = 2401 (7^4)
> A# = 1797
> E# = 2687
> C-- = 4019
> G-- = 3005
>
> (1282 = 2048 - 766 and so on - it should only come back near 2^n
> with x^74)
>
> What's nice is that E (four 5ths) are only sligthly higher than 5
> (641/640
> here), and B (five 5ths) is just about a skisma lower than 15/2
> (959/960) -
> for that reason if you follow B by a chain of seven pure 5ths after
> five of
> those skisni 5ths you fall back on C, only 0,077 c. higher :
> x^5 . 3^7 = 16384,73 (2^14 = 16384...)
>
> And that's already some sort of well-temperament - though I don't
> know if
> it's much in use - (you could also have it exact with more or less
> same
> properties, then the meantone fifth would be of 1,495940194).
>
> Another recurrent sequence I call Melkis (= 1,33415744713, x^4 - 2x
> = 1/2)
> combines quite well with 3 or 4 skisni 5ths, and 4 or zero pure 5ths
> to
> produce more common types of well-temperaments, without harmonic
> deviation,
> that made some adepts around here after a workshop I gave in 2006
> about
> harmonic temperaments.
>
> Any feedback is welcome.
>
> - - - - - - - -
> Jacques Dudon
>
>
>
>

2008/12/25 Ozan Yarman <ozanyarman@...>:
> While I was searching for the applicability of the meantone fifths to
> Turkish Maqam music, I noticed that at least 5 Ahenks (diapasons) could be
> met admirably with a bike-chain of 3 modified meantone temperaments
> containing 12 tones each and made up of 1/6 comma tempered fifths which are
> seperated by 35/32 and 81/65 cents apart respectively.

I used the cents values you gave below to test my equal temperament
finding script. The best fit is to 79 notes to the octave. I know
how much you like 79!

Other matches are for 31 and 48 steps. 31 is a meantone and 48 gives
you the bike-chains. Then 31+48=79.

You can also think about this by starting with 55-equal, which is a
good approximation to 1/6 comma meantone. 35/32 comes out as about
7.11 steps. That's a reasonable match. But 81/65 is about 17.46
steps. Dividing the steps in two will give you a better fit, to give
110-equal. 79+31=110.

> This is an interesting development, since I had proposed a 36-tone system to
> explain maqamat as far back as 7 years ago in my master's thesis.
> The modified meantones in question are cyclic with 3 super-pyth fifths 705
> cents in size in each.
> Here is the tuning with perde names (SCALA scale file attached):
<snip>

It looks like something around 1/6-comma meantone would make it
simpler. The system naturally rounds up to 48 rather than 36 notes,
though, so there must be a phantom chain you aren't using.

Maybe it's possible to have 4 shorter chains and get a 31 note MOS.

Graham

Hi Ozan,

Sorry for the delay ! I was not able to answer you, having a big problem
with my internet connexion. And now it's back I have thousands mails to deal
with.

Only using just-intonation myself, I am nevertheless convinced of the
utility of temperaments, that I would not be surprised to find unconsciously
in use in middle-East music practices as well.
On the opposite, the usage of 12-ET by qanun makers in Turkey and other
places, to base their tunings on, even if by chance the musicians make good
music out of it, is quite a pity - because even if 12-ET is convenient with
an electronic tuner, it is a poor temperament.

What you do is totally different because it combines a useful irregular
temperament, with coherent JI.
If I get it right, you must have already 45/32 (590c.), 35/32 and 1575/1024
(745c.), 81/65 and 729/416 (971c.) among your 36 tones.
I am really fond of 35/32 myself, but can you explain 81/65 ? anything
specially turkish ?

Do you get near to any other JI ratios in your system and could you send a
list of it, if any ?
(measures in cents alone are not what I understand better - thanks !)

About the Skisni meantone I was explaining previously, it leads to a 1/5
pythagorean comma and has no interest for your system.
But other reccurent sequences would get close to 1/6 syntonic comma, such as
Evelyne :

x^5 = 15 - 5x

where x = 1,4968965147934 = 698,37 c.

that suggests a sequence such as 320 479 717 1073 1606 2405 3600
(=45/32) 5390 8065 12065
from which it is easy to get back to 1/1 (320) in three 705c. steps as you
said :

320 479 717 1073 1606 2405 3600 5390 8065 12065 > 567 852 1280

it could also be divided in 5 and rounded to :

256 383 1147 1717 1285 481 720 539 1613 2415 > 3629 1363 1024

(modulo octaves...)

- - - - - -
Jacques

le 24/12/08 21:57, Ozan Yarman à ozanyarman@... a écrit :

Hello Jacques!

While I was searching for the applicability of the meantone fifths to
Turkish Maqam music, I noticed that at least 5 Ahenks (diapasons) could be
met admirably with a bike-chain of 3 modified meantone temperaments
containing 12 tones each and made up of 1/6 comma tempered fifths which are
seperated by 35/32 and 81/65 cents apart respectively.

This is an interesting development, since I had proposed a 36-tone system to
explain maqamat as far back as 7 years ago in my master's thesis.

The modified meantones in question are cyclic with 3 super-pyth fifths 705
cents in size in each.

Here is the tuning with perde names (SCALA scale file attached):

0: 1/1 Kaba Rast / RAST / Gerdaniye / Tiz Gerdaniye
1: 47 cents Dik (Rast)...
2: 76 cents Nerm (Zengule)...
3: 95 cents Kaba Nim Zengule / Nim Zengule, Nim S¸ehnaz
4: 155 cents (35/32) Kaba Zengule / Zengule / S¸ehnaz
5: 184 cents Kaba Dik Zengule / Dik Zengule / Dik S¸ehnaz
6: 197 cents Kaba Dügah / DÜGAH / Muhayyer
7: 250 cents Dik...
8: 273 cents Nerm...
9: 305 cents Kaba Kürdi / Kürdi / Sünbüle
10: 352 cents Kaba Nerm Segah / Nerm Segah / Tiz Nerm Segah (Ushshaq)
11: 381 cents (81/65) Kaba Segah / SEGAH / Tiz Segah
12: 393 cents Kaba Buselik / BUSELIK / Tiz Buselik
13: 460 cents Dik...
14: 476 cents Nerm...
15: 502 cents Kaba Tschargah / TSCHARGAH / Tiz Tschargah
16: 549 cents Dik...
17: 578 cents Nerm...
18: 590 cents Kaba Nim Hijaz / Nim Hijaz / Tiz Nim Hijaz
19: 657 cents Kaba Hijaz / Hijaz / Tiz Hijaz (Saba)
20: 686 cents Kaba Dik Hijaz / Dik Hijaz / Dik Nim Hijaz
21: 698 cents Yegah / NEVA / Tiz Neva
22: 745 cents Dik...
23: 774 cents Nerm...
24: 800 cents Kaba Nim Hisar / Nim Hisar / Tiz Nim Hisar
25: 854 cents Kaba Hisar / Hisar / Tiz Hisar
26: 883 cents Kaba Dik Hisar / Dik Hisar / Tiz Dik Hisar
27: 895 cents Ashiran / HÜSEYNI / Tiz Hüseyni
28: 955 cents Dik...
29: 971 cents Nerm...
30: 1003 cents Ajem Ashiran / Ajem / Tiz Ajem Ashiran
31: 1050 cents Dik Ajem Ashiran / Dik Ajem / Tiz Dik Ajem Ashiran
32: 1079 cents Nerm Evdj / EVDJ / Tiz Evdj
33: 1092 cents Gevasht / Mahur / Tiz Mahur
34: 1158 cents Dik...
35: 1181 cents Nerm...
36: 2/1 Rast / GERDANIYE / Tiz Gerdaniye

Süpürde Ahenk starts perde rast on C, the 0th step.
In Bolahenk, perde rast is D, the 6th step.
In Mansur, perde rast is G, the 21st step.
In Kiz, perde rast is A, the 27th step.
In Mustahsen, perde rast is B, the 33rd step.

Notation is as follows:

First bike-chain starting from degree 0 gets the naturals, sharps and flats.
Second bike-chain 35/32 higher gets the ? flats and ý sharps.
Third bike-chain 81/65 higher gets the d flats and H sharps (# sharps whose
lower line is omitted).

Some maqams are given below in this notation:

RAST
C D Ed F G A Bd c, Bb A G F Ed D C

MAHUR
C D E F G A B c

PENCHGAH
C D Ed FH G A B c

NIHAVEND
C D Eb F G Ab B c, Bb Ab G F Eb D c

BUSELIK
D E F G A Bb cH d, c Bb A G F E D

HIJAZ
D Eb FH G A B? c d, C Bb A G FH Eb D

KURDI
D Eb F G A Bb c d, c Bb Ab G F Eb D

HUSEYNI
D E? F G A B? c d, c B? A G F E? D

USHSHAQ
D E? F G A Bb c, Bb A G F E? D

SABA
D Ed F G? A Bb c d? e f, ed___e? d c Bb A G? F E? D

KARJIGHAR
D E? F G A? Bd c d

SEGAH
Ed F G Ad Bd c dH e, eb d c Bb Ad G F Ed

HUZZAM
Ed F G A? Bd c dH e, eb d c Bb Ad G F Ed

MUSTEAR
Ed FH G Ad Bd c dH e

Cordially,
Oz.

On Dec 24, 2008, at 9:16 PM, Jacques Dudon wrote:

Merry Christmas everybody,

About metameantones, if I can borrow this term from Erv Wilson,
here is a specially useful one, I think, I found after my relatively small
previous 1,4933585565602 fifth, of 694,27 c. :

1,4959535062432 or 697,278405 c., that I will call a "skisni" 5th.

This ratio allows the differential coherence of every 8/5 in a meantone
chain :

8 - x^4 = 2x

and the equal-beating of the fourth, and the major third in all G - C - E
type major triads :

4 - 2x = x^4 - 4

Here is a Skisni sequence of frequencies starting from 2^n but there are
plenty - (transposed by octaves below here and there) :

C = 256
G = 383
D = 573
A = 857
E = 641
B = 959
F# = 1435
C# = 1073
G# = 1605
D# = 2401 (7^4)
A# = 1797
E# = 2687
C-- = 4019
G-- = 3005

(1282 = 2048 - 766 and so on - it should only come back near 2^n with x^74)

What's nice is that E (four 5ths) are only sligthly higher than 5 (641/640
here), and B (five 5ths) is just about a skisma lower than 15/2 (959/960) -
for that reason if you follow B by a chain of seven pure 5ths after five of
those skisni 5ths you fall back on C, only 0,077 c. higher :
x^5 . 3^7 = 16384,73 (2^14 = 16384...)

And that's already some sort of well-temperament - though I don't know if
it's much in use - (you could also have it exact with more or less same
properties, then the meantone fifth would be of 1,495940194).

Another recurrent sequence I call Melkis (= 1,33415744713, x^4 - 2x = 1/2)
combines quite well with 3 or 4 skisni 5ths, and 4 or zero pure 5ths to
produce more common types of well-temperaments, without harmonic deviation,
that made some adepts around here after a workshop I gave in 2006 about
harmonic temperaments.

Any feedback is welcome.

- - - - - - - -
Jacques Dudon

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Jacques Dudon
Atelier d'Exploration Harmonique
83340 LE THORONET
09 60 04 92 63 (internet, fixe)
04 94 73 87 78 (autre fixe)
fotosonix@...
http://aeh.free.fr

Dear Graham, I do not consider the chain of meantone fifths to close up in a cycle as a necessity, except to attain a 12-tone chain using super-pythagorean fifths. EQ temps are not as attractive as they used to be. It's time for intelligent designs!

Oz.

On Dec 29, 2008, at 11:03 AM, Graham Breed wrote:

> 2008/12/25 Ozan Yarman <ozanyarman@...>:
>> While I was searching for the applicability of the meantone fifths to
>> Turkish Maqam music, I noticed that at least 5 Ahenks (diapasons) >> could be
>> met admirably with a bike-chain of 3 modified meantone temperaments
>> containing 12 tones each and made up of 1/6 comma tempered fifths >> which are
>> seperated by 35/32 and 81/65 cents apart respectively.
>
> I used the cents values you gave below to test my equal temperament
> finding script. The best fit is to 79 notes to the octave. I know
> how much you like 79!
>
> Other matches are for 31 and 48 steps. 31 is a meantone and 48 gives
> you the bike-chains. Then 31+48=79.
>
> You can also think about this by starting with 55-equal, which is a
> good approximation to 1/6 comma meantone. 35/32 comes out as about
> 7.11 steps. That's a reasonable match. But 81/65 is about 17.46
> steps. Dividing the steps in two will give you a better fit, to give
> 110-equal. 79+31=110.
>
>> This is an interesting development, since I had proposed a 36-tone >> system to
>> explain maqamat as far back as 7 years ago in my master's thesis.
>> The modified meantones in question are cyclic with 3 super-pyth >> fifths 705
>> cents in size in each.
>> Here is the tuning with perde names (SCALA scale file attached):
> <snip>
>
> It looks like something around 1/6-comma meantone would make it
> simpler. The system naturally rounds up to 48 rather than 36 notes,
> though, so there must be a phantom chain you aren't using.
>
> Maybe it's possible to have 4 shorter chains and get a 31 note MOS.
>
>
> Graham

Dear Jacques,

I apologize also for the delay... I just returned from Pakistan a few
days ago and found out that the ceiling of the apartment where I work
has crashed down. Fortunately I wasn't there when it happened! It'll
take some time till I get settled back.

In the meanwhile, I shall answer to your post to the best of my ability.

I agree with you almost entirely that 12-ET is a poor choice given the
wealth of even proportional-beating modified meantones. More on that
below.

Yes, I wish to attain the coherent resonance of JI intervals using
tolerable meantone fifths, and to stack the 12-tone temperaments
thereby achieved into a bike-chain to arrive at a tuning comprising
all the tones needed for all maqams in at least 5 ahenks.

This optimization requires some sacrifices, particularly in the 7-
limit. The remedy is to stack another layer, increasing the number oftones to 48. But I do not want to do that, as more tones complicate
the music-making process.

81/65 is an interval that represents the region between 5:4 and 4:3
less 15:14. I find that a 382 cent major third works wonders for
maqams Rast, Segah and Huzzam. 5:4 is just too steep as Rauf Yekta
points out. He would rather the 3-limit 8192:6561, but I do not feel it to be right. Proportional beat rates might be a clue in desiring a
narrower major third. More on that below.

Before I departed for Pakistan, I tuned my piano for the first time
listening to the beats of the fifths. The resultant scale was:

1st layer:

TONE HZ RF5th Cents5th BRAT CentsPitch

Eb 312.749 303.592
0.668 -698.261 2
Bb=> Bb 468.123 1001.852
0.668 -699.488 2
F 350.593 501.340
0.668 -698.660 2
c=> C 262.444 0
0.668 -697.552 2
G=> G 392.667 697.552
0.668 -699.013 2
D 294.000 196.565
0.668 -698.025 2
A=> A 440.000 894.590
1.498 699.330 -1
E 329.500 393.920
1.498 700.203 -1
B=> B 493.750 1094.122
1.496 697.274 -2
F#=> F# 369.313 591.396
1.503 705.077 1
C# 277.484 96.473
1.502 704.033 1
G#=> G# 416.727 800.507
1.501 703.085 0.408
Eb 312.749

Re-organized
pitches: LAYER1 Hz Cents Intervals 5th beat freq.

C 262.4444444 0 -2
C# 277.484375 96.4733738 96.4733738 1
D 294 196.5649163 100.0915425 -2
Eb 312.7489712 303.5916965 107.0267802 -2
E 329.5 393.9196885 90.327992 -1
F 350.5925926 501.3401479 107.4204595 -2
F# 369.3125 591.3960427 90.0558948 2
G 392.6666667 697.5516902 106.1556474 -2
G# 416.7265625 800.5068041 102.9551139 0.816197274
A 440 894.5897587 94.08295463 -2
Bb 468.1234568 1001.852399 107.2626406 -2
B 493.75 1094.122428 92.27002868 -4
c 524.8888889 1200.000000 105.877572 -4

Hearing how pleasing the chords were, I decided to keep this
temperament and add more layers. The second layer's E was at (6G+C)/8
higher, at an excellent 382 cents compared to the C of the 1st layer.
The BRATS are -3 or the multiples thereof:

LAYER2 Hz Cents Intervals 5th beat freq.

C 260.6071386 -12.16254651 -1.44898767
C# 274.625 78.54108776 90.70363427 -0.875
D 292.2021605 185.9457445 107.4046567 -1.625
Eb 307.84375 276.2234644 90.27771996 1.25
E 327.3055556 382.3512386 106.1277742 -1.75
F 347.2304688 484.6577864 102.3065478 0.737147955
F# 366.75 579.3418893 94.68410289 -1.75
G 390.186214 686.5808971 107.2390078 -1.75
G# 411.5 778.656447 92.07554989 -3.125
A 437.4907407 884.6885103 106.0320633 -3.25
Bb 462.390625 980.5201071 95.83159681 1.75
B 490.0833333 1081.218032 100.6979247 -3.25
c 521.2142771 1187.837453 106.6194217 -2.897975341

4:5:6 resonates wonderfully in this setup. What remains is to add a
third layer 35:32 away from C:

LAYER3 Hz Cents Intervals

C 270.0195313 49.26204833
C# 287.0486111 155.1396203 105.877572
D 303.4985352 251.6129941 96.4733738
Eb 321.5625 351.7045366 100.0915425
E 342.0691872 458.7313168 107.0267802
F 360.390625 549.0593088 90.327992
F# 383.4606481 656.4797683 107.4204595
G 403.9355469 746.5356631 90.0558948
G# 429.4791667 852.6913105 106.1556474
A 455.7946777 955.6464244 102.9551139
Bb 481.25 1049.729379 94.08295463
B 512.0100309 1156.99202 107.2626406

The product is a three-layered bikechain of cleverly spaced modified
meantones:

CENTS JI PERDE

0.000 1/1 RAST
49.262 36/35
78.541 23/22
96.473 55/52 Nim Zengule
155.140 35/32 Zengule
185.946 10/9 Dik Zengule
196.565 9/8 DUGAH
251.613 52/45
276.223 75/64
303.592 6/5 Kurdi
351.705 27/22 Nerm Segah
382.351 56/45 SEGAH
393.920 5/4 Buselik
458.731 125/96
484.658 33/25
501.340 4/3 TCHARGAH
549.059 11/8
579.342 7/5
591.396 45/32 Nim Hijaz
656.480 35/24 Hijaz
686.581 125/84 Dik Hijaz
697.552 3/2 NEVA
746.536 192/125
778.656 25/16
800.507 35/22 Nim Hisar
852.691 18/11 Hisar
884.689 5/3 Dik Hisar
894.590 42/25 HUSEYNI (440 hz)
955.646 125/72
980.520 44/25
1001.852 16/9 Ajem
1049.729 11/6 Nerm Evdj
1081.218 28/15 EVDJ
1094.122 15/8 Mahur
1156.992 125/64
1187.837 125/63
1200.000 2/1 GERDANIYE

Cordially,
Oz.

On Jan 8, 2009, at 6:50 PM, Jacques Dudon wrote:

> Hi Ozan,
>
> Sorry for the delay ! I was not able to answer you, having a big
> problem with my internet connexion. And now it's back I have
> thousands mails to deal with.
>
> Only using just-intonation myself, I am nevertheless convinced of
> the utility of temperaments, that I would not be surprised to find
> unconsciously in use in middle-East music practices as well.
> On the opposite, the usage of 12-ET by qanun makers in Turkey and
> other places, to base their tunings on, even if by chance the
> musicians make good music out of it, is quite a pity - because even
> if 12-ET is convenient with an electronic tuner, it is a poor
> temperament.
>
> What you do is totally different because it combines a useful
> irregular temperament, with coherent JI.
> If I get it right, you must have already 45/32 (590c.), 35/32 and
> 1575/1024 (745c.), 81/65 and 729/416 (971c.) among your 36 tones.
> I am really fond of 35/32 myself, but can you explain 81/65 ?
> anything specially turkish ?
>
> Do you get near to any other JI ratios in your system and could you
> send a list of it, if any ?
> (measures in cents alone are not what I understand better - thanks !)
>
> About the Skisni meantone I was explaining previously, it leads to a
> 1/5 pythagorean comma and has no interest for your system.
> But other reccurent sequences would get close to 1/6 syntonic comma, > such as Evelyne :
>
> x^5 = 15 - 5x
>
> where x = 1,4968965147934 = 698,37 c.
>
> that suggests a sequence such as 320 479 717 1073 1606 2405
> 3600 (=45/32) 5390 8065 12065
> from which it is easy to get back to 1/1 (320) in three 705c. steps
> as you said :
>
> 320 479 717 1073 1606 2405 3600 5390 8065 12065 > 567
> 852 1280
>
> it could also be divided in 5 and rounded to :
>
> 256 383 1147 1717 1285 481 720 539 1613 2415 > 3629 1363
> 1024
>
> (modulo octaves...)
>
> - - - - - -
> Jacques
>
>
>
> le 24/12/08 21:57, Ozan Yarman à ozanyarman@... a écrit :
>
>> Hello Jacques!
>>
>> While I was searching for the applicability of the meantone fifths
>> to Turkish Maqam music, I noticed that at least 5 Ahenks
>> (diapasons) could be met admirably with a bike-chain of 3 modified
>> meantone temperaments containing 12 tones each and made up of 1/6
>> comma tempered fifths which are seperated by 35/32 and 81/65 cents
>> apart respectively.
>>
>> This is an interesting development, since I had proposed a 36-tone
>> system to explain maqamat as far back as 7 years ago in my master's
>> thesis.
>>
>> The modified meantones in question are cyclic with 3 super-pyth
>> fifths 705 cents in size in each.
>>
>> Here is the tuning with perde names (SCALA scale file attached):
>>
>> 0: 1/1 Kaba Rast / RAST / Gerdaniye / Tiz Gerdaniye
>> 1: 47 cents Dik (Rast)...
>> 2: 76 cents Nerm (Zengule)...
>> 3: 95 cents Kaba Nim Zengule / Nim Zengule, Nim S¸ehnaz
>> 4: 155 cents (35/32) Kaba Zengule / Zengule / S¸ehnaz
>> 5: 184 cents Kaba Dik Zengule / Dik Zengule / Dik S¸ehnaz
>> 6: 197 cents Kaba Dügah / DÜGAH / Muhayyer
>> 7: 250 cents Dik...
>> 8: 273 cents Nerm...
>> 9: 305 cents Kaba Kürdi / Kürdi / Sünbüle
>> 10: 352 cents Kaba Nerm Segah / Nerm Segah / Tiz Nerm Segah
>> (Ushshaq)
>> 11: 381 cents (81/65) Kaba Segah / SEGAH / Tiz Segah
>> 12: 393 cents Kaba Buselik / BUSELIK / Tiz Buselik
>> 13: 460 cents Dik...
>> 14: 476 cents Nerm...
>> 15: 502 cents Kaba Tschargah / TSCHARGAH / Tiz Tschargah
>> 16: 549 cents Dik...
>> 17: 578 cents Nerm...
>> 18: 590 cents Kaba Nim Hijaz / Nim Hijaz / Tiz Nim Hijaz
>> 19: 657 cents Kaba Hijaz / Hijaz / Tiz Hijaz (Saba)
>> 20: 686 cents Kaba Dik Hijaz / Dik Hijaz / Dik Nim Hijaz
>> 21: 698 cents Yegah / NEVA / Tiz Neva
>> 22: 745 cents Dik...
>> 23: 774 cents Nerm...
>> 24: 800 cents Kaba Nim Hisar / Nim Hisar / Tiz Nim Hisar
>> 25: 854 cents Kaba Hisar / Hisar / Tiz Hisar
>> 26: 883 cents Kaba Dik Hisar / Dik Hisar / Tiz Dik Hisar
>> 27: 895 cents Ashiran / HÜSEYNI / Tiz Hüseyni
>> 28: 955 cents Dik...
>> 29: 971 cents Nerm...
>> 30: 1003 cents Ajem Ashiran / Ajem / Tiz Ajem Ashiran
>> 31: 1050 cents Dik Ajem Ashiran / Dik Ajem / Tiz Dik Ajem Ashiran
>> 32: 1079 cents Nerm Evdj / EVDJ / Tiz Evdj
>> 33: 1092 cents Gevasht / Mahur / Tiz Mahur
>> 34: 1158 cents Dik...
>> 35: 1181 cents Nerm...
>> 36: 2/1 Rast / GERDANIYE / Tiz Gerdaniye
>>
>> Süpürde Ahenk starts perde rast on C, the 0th step.
>> In Bolahenk, perde rast is D, the 6th step.
>> In Mansur, perde rast is G, the 21st step.
>> In Kiz, perde rast is A, the 27th step.
>> In Mustahsen, perde rast is B, the 33rd step.
>>
>> Notation is as follows:
>>
>> First bike-chain starting from degree 0 gets the naturals, sharps
>> and flats.
>> Second bike-chain 35/32 higher gets the ? flats and ‡ sharps.
>> Third bike-chain 81/65 higher gets the d flats and H sharps (#
>> sharps whose lower line is omitted).
>>
>> Some maqams are given below in this notation:
>>
>> RAST
>> C D Ed F G A Bd c, Bb A G F Ed D C
>>
>> MAHUR
>> C D E F G A B c
>>
>> PENCHGAH
>> C D Ed FH G A B c
>>
>> NIHAVEND
>> C D Eb F G Ab B c, Bb Ab G F Eb D c
>>
>> BUSELIK
>> D E F G A Bb cH d, c Bb A G F E D
>>
>> HIJAZ
>> D Eb FH G A B? c d, C Bb A G FH Eb D
>>
>> KURDI
>> D Eb F G A Bb c d, c Bb Ab G F Eb D
>>
>> HUSEYNI
>> D E? F G A B? c d, c B? A G F E? D
>>
>> USHSHAQ
>> D E? F G A Bb c, Bb A G F E? D
>>
>> SABA
>> D Ed F G? A Bb c d? e f, ed___e? d c Bb A G? F E? D
>>
>> KARJIGHAR
>> D E? F G A? Bd c d
>>
>> SEGAH
>> Ed F G Ad Bd c dH e, eb d c Bb Ad G F Ed
>>
>> HUZZAM
>> Ed F G A? Bd c dH e, eb d c Bb Ad G F Ed
>>
>> MUSTEAR
>> Ed FH G Ad Bd c dH e
>>
>>
>> Cordially,
>> Oz.
>>
>>
>> On Dec 24, 2008, at 9:16 PM, Jacques Dudon wrote:
>>
>>> Merry Christmas everybody,
>>>
>>> About metameantones, if I can borrow this term from Erv Wilson,
>>> here is a specially useful one, I think, I found after my
>>> relatively small
>>> previous 1,4933585565602 fifth, of 694,27 c. :
>>>
>>> 1,4959535062432 or 697,278405 c., that I will call a "skisni" 5th.
>>>
>>> This ratio allows the differential coherence of every 8/5 in a >>> meantone
>>> chain :
>>>
>>> 8 - x^4 = 2x
>>>
>>> and the equal-beating of the fourth, and the major third in all G
>>> - C - E
>>> type major triads :
>>>
>>> 4 - 2x = x^4 - 4
>>>
>>> Here is a Skisni sequence of frequencies starting from 2^n but
>>> there are
>>> plenty - (transposed by octaves below here and there) :
>>>
>>> C = 256
>>> G = 383
>>> D = 573
>>> A = 857
>>> E = 641
>>> B = 959
>>> F# = 1435
>>> C# = 1073
>>> G# = 1605
>>> D# = 2401 (7^4)
>>> A# = 1797
>>> E# = 2687
>>> C-- = 4019
>>> G-- = 3005
>>>
>>> (1282 = 2048 - 766 and so on - it should only come back near 2^n
>>> with x^74)
>>>
>>> What's nice is that E (four 5ths) are only sligthly higher than 5
>>> (641/640
>>> here), and B (five 5ths) is just about a skisma lower than 15/2
>>> (959/960) -
>>> for that reason if you follow B by a chain of seven pure 5ths
>>> after five of
>>> those skisni 5ths you fall back on C, only 0,077 c. higher :
>>> x^5 . 3^7 = 16384,73 (2^14 = 16384...)
>>>
>>> And that's already some sort of well-temperament - though I don't
>>> know if
>>> it's much in use - (you could also have it exact with more or less
>>> same
>>> properties, then the meantone fifth would be of 1,495940194).
>>>
>>> Another recurrent sequence I call Melkis (= 1,33415744713, x^4 -
>>> 2x = 1/2)
>>> combines quite well with 3 or 4 skisni 5ths, and 4 or zero pure
>>> 5ths to
>>> produce more common types of well-temperaments, without harmonic
>>> deviation,
>>> that made some adepts around here after a workshop I gave in 2006
>>> about
>>> harmonic temperaments.
>>>
>>> Any feedback is welcome.
>>>
>>> - - - - - - - -
>>> Jacques Dudon
>>>
>>>
>>>
>>>
>>
>>
>
>
> - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
> Jacques Dudon
> Atelier d'Exploration Harmonique
> 83340 LE THORONET
> 09 60 04 92 63 (internet, fixe)
> 04 94 73 87 78 (autre fixe)
> fotosonix@wanadoo.fr
> http://aeh.free.fr
>
>

Ozan Yarman wrote:
> Dear Graham, I do not consider the chain of meantone fifths to close > up in a cycle as a necessity, except to attain a 12-tone chain using > super-pythagorean fifths. EQ temps are not as attractive as they used > to be. It's time for intelligent designs!

Nowhere in my message did I suggest closing the chain of fifths. What I noted is that the scale you gave suggests a system with a 79 note scale, which is interesting because you came up with a different 79 note MOS for maqam music. Also, that system might also have a 31 note MOS that would be suitable for your purposes.

There's no hurry, but these are things you might want to look at. I can't speculate much because I don't have a list of the pitches you want to approximate and the tuning tolerances.

Graham

Ah, then it is quite interesting that we reach 79 notes once more! Can
you give the 79-tone scale (emphasizing the 31-note subset) in cents
or as a SCL file?

Cordially,
Oz.

On Jan 24, 2009, at 10:58 AM, Graham Breed wrote:

> Ozan Yarman wrote:
>> Dear Graham, I do not consider the chain of meantone fifths to close
>> up in a cycle as a necessity, except to attain a 12-tone chain using
>> super-pythagorean fifths. EQ temps are not as attractive as they used
>> to be. It's time for intelligent designs!
>
> Nowhere in my message did I suggest closing the chain of
> fifths. What I noted is that the scale you gave suggests a
> system with a 79 note scale, which is interesting because
> you came up with a different 79 note MOS for maqam music.
> Also, that system might also have a 31 note MOS that would
> be suitable for your purposes.
>
> There's no hurry, but these are things you might want to
> look at. I can't speculate much because I don't have a list
> of the pitches you want to approximate and the tuning
> tolerances.
>
>
> Graham
>
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>

✩ ✩ ✩
www.ozanyarman.com

Ozan Yarman wrote:
> Ah, then it is quite interesting that we reach 79 notes once more! Can > you give the 79-tone scale (emphasizing the 31-note subset) in cents > or as a SCL file?

Here's a 31 note scale:

0.0
32.6
76.3
109.0
152.7
185.3
229.0
261.7
305.4
338.0
381.7
425.4
458.1
501.8
534.4
578.1
610.8
654.5
687.1
730.8
763.5
807.2
850.9
883.5
927.2
959.9
1003.6
1036.2
1079.9
1112.6
1156.3
1200.0

It's tuned to an optimum for 13-limit harmony. There are different ways of mapping the 13-limit so don't take it too seriously. You'll have to find the best tonic. Likewise for this 79 note scale:

0.0
21.5
32.6
54.2
65.2
76.3
97.9
109.0
130.5
141.6
152.7
174.2
185.3
206.8
217.9
229.0
250.6
261.7
283.2
294.3
305.4
326.9
338.0
359.5
370.6
381.7
403.3
414.4
425.4
447.0
458.1
479.6
490.7
501.8
523.3
534.4
555.9
567.0
578.1
599.7
610.8
632.3
643.4
654.5
676.0
687.1
708.6
719.7
730.8
752.4
763.5
785.0
796.1
807.2
828.7
839.8
850.9
872.4
883.5
905.0
916.1
927.2
948.8
959.9
981.4
992.5
1003.6
1025.1
1036.2
1057.7
1068.8
1079.9
1101.5
1112.6
1134.1
1145.2
1156.3
1177.8
1188.9
1200.0

Graham