about ozan system

Hi all

Some days ago I sent a mail about TOP . as I have mentioned there we can
see symmetrical arrangement in such a system with a symmetry center.

Now for ozan system :

1-as all know , The basis of the system is 33-ED(4/3) , that is 33 equal
divisions of 4/3.

2- in ozan's system , we have 33 equal interval of 15.092 cent (498.45 /
33) from 1/1 to 4/3 and another 33 equal interval from 2/1 to 3/2 with
inverse trend so we can place center of symmetry between gap of 4/3 and
3/2

3-the gap which is equal to 9/8 consists of 12 equal interval and one
interval of 22.803 cent.

4- so we have 78 equal interval of 15.092 cent and one interval of
22.803 cent , that is :

a=15.092

b=22.803

na=78

nb=1

the ozan system is arranged as : 45a b 33a which has 2 unequal
intervallic package with connecting interval of 22.803 cent.

in this system we havn't any center of symmetry in intervals
arrangement. But , we could have 39a b 39a to have center of symmetry
and then gap of 4/3 and 3/2 is filled as below :

498.045

513.137

528.23

543.322

558.414

573.506

588.599

611.402

626.494

641.586

656.678

671.77

686.862

701.954

This new system is based on 2 equal intervallic package and 1 interval
as connecting point.

Now another thing :

We could consider 2 non-octave equal systems based on 33-ED(4/3) having
79 or 80 degrees:

1- in 79-degree system ozan added the difference of (1200-1192.287) to
all degrees from 46 to 79 to make a similar package like 1 to 33 but in
different direction.

2- in 80-degree system we can delete degree of 46 and subtract
difference of (1207.379-1200 from end of system to 47 which is now 46
degree.

here you can see an example resulted from 10-ED(4/3) which is very
similar to 24-edo and have assymetric structure of 13a b 10a which
can't be changed to a symmetric arrangement :

0

49.8045 ............... 49.8045

99.609 ............... 49.8045

149.4135............... 49.8045

199.218 ............... 49.8045

249.0225............... 49.8045

298.827 ............... 49.8045

348.6315............... 49.8045

398.436 ............... 49.8045

448.2405............... 49.8045

498.045 ............... 49.8045

547.8495............... 49.8045

597.654 ............... 49.8045

647.4585............... 49.8045

701.955 ............... 54.4965

751.7595............... 49.8045

801.564 ............... 49.8045

851.3685............... 49.8045

901.173 ............... 49.8045

950.9775............... 49.8045

1000.782............... 49.8045

1050.5865............... 49.8045

1100.391............... 49.8045

1150.1955............... 49.8045

1200 ............... 49.8045

________________________________

From: Mohajeri Shahin
Sent: Sunday, February 26, 2006 5:31 PM
To: 'tuning@yahoogroups.com'
Subject: to wallyesterpaulrus :about TOP

Dear paul

hi

After reading your sent article , I saw order in setting of intervals in
scale. For example in NAUTLIUS TOP , the order is :

If 41 cent =a

If 42 cent = b

And b=a+1 and na=nb+1

We have na=15 from (a) and nb=14 from (b) and the setting in this top is
:

ababababababab(a)bababababababa

Now we can have a circle by connecting start and end of the line so, 2
(a) are next to each other) and octave is placed in point of connection

Another example :

In magic top we have structure with a=27.52 and b=31.36 and na=22 and
nb=19 :

We have 2 number of this setting :(bababababab) or 6b+5a as package A
and 1 number of this setting : ( babababababab ) or 7b+6a as package B .
the n we have final structure as aAaa(B)aaAa , (B) is center of symmetry
.now if we connect start and end of the line we can have a circle which
octave is point of connection.

In meantone top we have :

a=41.06

b=35.13

A=(bab)

structure :aAaabaaAaa(A)aaAaabaaAa. (A) is center of symmetry.octave is
point of connection of start and end of the line.

Now I have an example : consider

a=35 with na=20

b=33 with nb=15

structure is : aabaabababababababababababababaabaa

35 70 103 148 183 216 ........... 1195

what is your view?

Shaahin Mohaajeri

Tombak Player & Researcher , Composer

www.geocities.com/acousticsoftombak

My tombak musics : www.rhythmweb.com/gdg

My articles in ''Harmonytalk'':

www.harmonytalk.com/archives/000296.html

www.harmonytalk.com/archives/000288.html

My article in DrumDojo:

www.drumdojo.com/world/persia/tonbak_acoustics.htm

________________________________

From: Mohajeri Shahin
Sent: Sunday, February 26, 2006 5:31 PM
To: 'tuning@yahoogroups.com'
Subject: to wallyesterpaulrus :about TOP

Dear paul

hi

After reading your sent article , I saw order in setting of intervals in
scale. For example in NAUTLIUS TOP , the order is :

If 41 cent =a

If 42 cent = b

And b=a+1 and na=nb+1

We have na=15 from (a) and nb=14 from (b) and the setting in this top is
:

ababababababab(a)bababababababa

Now we can have a circle by connecting start and end of the line so, 2
(a) are next to each other) and octave is placed in point of connection

Another example :

In magic top we have structure with a=27.52 and b=31.36 and na=22 and
nb=19 :

We have 2 number of this setting :(bababababab) or 6b+5a as package A
and 1 number of this setting : ( babababababab ) or 7b+6a as package B .
the n we have final structure as aAaa(B)aaAa , (B) is center of symmetry
.now if we connect start and end of the line we can have a circle which
octave is point of connection.

In meantone top we have :

a=41.06

b=35.13

A=(bab)

structure :aAaabaaAaa(A)aaAaabaaAa. (A) is center of symmetry.octave is
point of connection of start and end of the line.

Now I have an example : consider

a=35 with na=20

b=33 with nb=15

structure is : aabaabababababababababababababaabaa

35 70 103 148 183 216 ........... 1195

what is your view?

Shaahin Mohaajeri

Tombak Player & Researcher , Composer

www.geocities.com/acousticsoftombak

My tombak musics : www.rhythmweb.com/gdg

My articles in ''Harmonytalk'':

www.harmonytalk.com/archives/000296.html

www.harmonytalk.com/archives/000288.html

My article in DrumDojo:

www.drumdojo.com/world/persia/tonbak_acoustics.htm

When you carry the larger comma to the 40th step, you disrupt the system so that I cannot anymore jump from B to Gb. I require both 39th and the 40th steps to be reachable by perfect fifths, so that the `accidental half-tones` can be flexibly arranged, where:

Gb equals F#

while also:

Gb does not equal F#
----- Original Message -----
From: Mohajeri Shahin
To: tuning@yahoogroups.com
Sent: 05 Mart 2006 Pazar 12:08
Subject: [tuning] about ozan system

Hi all

Some days ago I sent a mail about TOP . as I have mentioned there we can see symmetrical arrangement in such a system with a symmetry center.

Now for ozan system :

1-as all know , The basis of the system is 33-ED(4/3) , that is 33 equal divisions of 4/3.

2- in ozan's system , we have 33 equal interval of 15.092 cent (498.45 / 33) from 1/1 to 4/3 and another 33 equal interval from 2/1 to 3/2 with inverse trend so we can place center of symmetry between gap of 4/3 and 3/2

3-the gap which is equal to 9/8 consists of 12 equal interval and one interval of 22.803 cent.

4- so we have 78 equal interval of 15.092 cent and one interval of 22.803 cent , that is :

a=15.092

b=22.803

na=78

nb=1

the ozan system is arranged as : 45a b 33a which has 2 unequal intervallic package with connecting interval of 22.803 cent.

in this system we havn't any center of symmetry in intervals arrangement. But , we could have 39a b 39a to have center of symmetry and then gap of 4/3 and 3/2 is filled as below :

498.045

513.137

528.23

543.322

558.414

573.506

588.599

611.402

626.494

641.586

656.678

671.77

686.862

701.954

This new system is based on 2 equal intervallic package and 1 interval as connecting point.

Now another thing :

We could consider 2 non-octave equal systems based on 33-ED(4/3) having 79 or 80 degrees:

1- in 79-degree system ozan added the difference of (1200-1192.287) to all degrees from 46 to 79 to make a similar package like 1 to 33 but in different direction.

2- in 80-degree system we can delete degree of 46 and subtract difference of (1207.379-1200 from end of system to 47 which is now 46 degree.

here you can see an example resulted from 10-ED(4/3) which is very similar to 24-edo and have assymetric structure of 13a b 10a which can't be changed to a symmetric arrangement :

0

49.8045 …………… 49.8045

99.609 …………… 49.8045

149.4135…………… 49.8045

199.218 …………… 49.8045

249.0225…………… 49.8045

298.827 …………… 49.8045

348.6315…………… 49.8045

398.436 …………… 49.8045

448.2405…………… 49.8045

498.045 …………… 49.8045

547.8495…………… 49.8045

597.654 …………… 49.8045

647.4585…………… 49.8045

701.955 …………… 54.4965

751.7595…………… 49.8045

801.564 …………… 49.8045

851.3685…………… 49.8045

901.173 …………… 49.8045

950.9775…………… 49.8045

1000.782…………… 49.8045

1050.5865…………… 49.8045

1100.391…………… 49.8045

1150.1955…………… 49.8045

1200 …………… 49.8045

Dear brother ozan

Doroud bar to ba'ad ( that is as-salam-o alaik in arabic word)

Very sorry to hear such events. Be sure my heart is with you and I wish
I were there to embrace you to feel my sense about .....

I want god the fast recovery of your mother and bless of your grand
father soul.

Shaahin Mohaajeri

Tombak Player & Researcher , Composer

www.geocities.com/acousticsoftombak

My tombak musics : www.rhythmweb.com/gdg

My articles in ''Harmonytalk'':

www.harmonytalk.com/archives/000296.html

www.harmonytalk.com/archives/000288.html

My article in DrumDojo:

www.drumdojo.com/world/persia/tonbak_acoustics.htm

________________________________

Hi all,

On Sun, 5 Mar 2006, Mohajeri Shahin wrote:
>
> Some days ago I sent a mail about TOP . as I have mentioned there we can
> see symmetrical arrangement in such a system with a symmetry center.
>
> Now for ozan system :
>
> 1-as all know , The basis of the system is 33-ED(4/3) , that is 33 equal
> divisions of 4/3.
>
> 2- in ozan's system , we have 33 equal interval of 15.092 cent (498.45 /
> 33) from 1/1 to 4/3 and another 33 equal interval from 2/1 to 3/2 with
> inverse trend so we can place center of symmetry between gap of 4/3 and
> 3/2
>
> 3-the gap which is equal to 9/8 consists of 12 equal interval and one
> interval of 22.803 cent.
>
> 4- so we have 78 equal interval of 15.092 cent and one interval of
> 22.803 cent , that is :
>
> a=15.092
> b=22.803
> na=78
> nb=1
>
> the ozan system is arranged as : 45a b 33a which has 2 unequal
> intervallic package with connecting interval of 22.803 cent.
>
> in this system we havn't any center of symmetry in intervals
> arrangement. But , we could have 39a b 39a to have center of symmetry
> and then gap of 4/3 and 3/2 is filled as below :
>
> 498.045
> 513.137
> 528.23
> 543.322
> 558.414
> 573.506
> 588.599
> 611.402
> 626.494
> 641.586
> 656.678
> 671.77
> 686.862
> 701.954
>
> This new system is based on 2 equal intervallic package and 1 interval
> as connecting point.
[snip]

Depending on what kinds of F# and Gb you wanted,
(assuming you are using this for maqam music, you
probably do want these), there are several other
ways to fill the 9/8 gap between F and G. The most
obvious of these still gives only two step sizes, and
divides the 9/8 gap into 13 equal steps each of
(701.954 - 498.045)/13 = 203.909/13 = 15.685 cents.

a=15.092
b=15.685
na=66
nb=13

The step sizes a and b are much closer to each other
than in Ozan's tuning.

This is symmetrical about 600 cents, which is NOT a
scale degree, but falls exactly between degrees 39
and 40 of the scale, which are at 592.155 & 607.840
cents.

The important questions here are: what notes between
F and G do you want to approximate well, and how
closely?

Regards,
Yahya

--
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Splendid suggestion! Although the jump from B to Gb is a little wide for my
tastes, one can overlook that particular glitch in this alternative version
you suggest:

Run SCALA

File>New>Equal Temperament>33 equal divisions of 4/3
command prompt>copy (scale number) 0 to (scale number) 1
File>New>Equal Temperament>13 equal divisions of 9/8
Modify>Move>raise all pitches after degree 0 by 4/3
Combine>Merge>With (scale number) 1
command prompt>copy (scale number) 0 to (scale number) 1
File>New>Equal Temperament>33 equal divisions of 4/3
Modify>Move>raise all pitches after degree 0 by 3/2
Combine>Merge>With (scale number) 1
command prompt>set nota e79

Voila!

0: 1/1 C unison, perfect prime
1: 15.092 cents C/
2: 30.185 cents C//
3: 45.277 cents C^ Db(
4: 60.369 cents C) Dbv
5: 75.461 cents C#\ Db\\
6: 90.554 cents C# Db\
7: 105.646 cents C#/ Db
8: 120.738 cents C#// Db/
9: 135.830 cents C#^ D(
10: 150.923 cents C#) Dv
11: 166.015 cents D\\
12: 181.107 cents D\
13: 196.200 cents D
14: 211.292 cents D/
15: 226.384 cents D//
16: 241.476 cents D^ Eb(
17: 256.569 cents D) Ebv
18: 271.661 cents D#\ Eb\\
19: 286.753 cents D# Eb\
20: 301.845 cents D#/ Eb
21: 316.938 cents D#// Eb/
22: 332.030 cents D#^ E(
23: 347.122 cents D#) Ev
24: 362.215 cents E\\
25: 377.307 cents E\
26: 392.399 cents E
27: 407.491 cents E/ Fb
28: 422.584 cents E// Fb/
29: 437.676 cents E^ F(
30: 452.768 cents E) Fv
31: 467.860 cents E#\ F\\
32: 482.953 cents E# F\
33: 4/3 F perfect fourth
34: 513.730 cents F/
35: 529.416 cents F//
36: 545.101 cents F^ Gb(
37: 560.787 cents F) Gbv
38: 576.472 cents F#\ Gb\\
39: 592.157 cents F# Gb\
40: 607.843 cents F#/ Gb
41: 623.528 cents F#// Gb/
42: 639.213 cents F#^ G(
43: 654.899 cents F#) Gv
44: 670.584 cents G\\
45: 686.270 cents G\
46: 3/2 G perfect fifth
47: 717.047 cents G/
48: 732.140 cents G//
49: 747.232 cents G^ Ab(
50: 762.324 cents G) Abv
51: 777.416 cents G#\ Ab\\
52: 792.509 cents G# Ab\
53: 807.601 cents G#/ Ab
54: 822.693 cents G#// Ab/
55: 837.785 cents G#^ A(
56: 852.878 cents G#) Av
57: 867.970 cents A\\
58: 883.062 cents A\
59: 898.155 cents A
60: 913.247 cents A/
61: 928.339 cents A//
62: 943.431 cents A^ Bb(
63: 958.524 cents A) Bbv
64: 973.616 cents A#\ Bb\\
65: 988.708 cents A# Bb\
66: 1003.800 cents A#/ Bb
67: 1018.893 cents A#// Bb/
68: 1033.985 cents A#^ B(
69: 1049.077 cents A#) Bv
70: 1064.170 cents B\\
71: 1079.262 cents B\
72: 1094.354 cents B
73: 1109.446 cents B/ Cb
74: 1124.539 cents B// Cb/
75: 1139.631 cents B^ C(
76: 1154.723 cents B) Cv
77: 1169.815 cents B#\ C\\
78: 1184.908 cents B# C\
79: 2/1 C octave

One requires 636-tET to approximate all intervals with an error of less than
1 cents.

Here is the cycle:

|
0: 0.000 cents 0.000 0 0 commas C
46: 701.955 cents 0.000 0 0 commas G
13: 694.245 cents -7.710 -237 D
59: 701.955 cents -7.710 -237 A
26: 694.245 cents -15.421 -473 E
72: 701.955 cents -15.421 -473 B
39: 697.803 cents -19.573 -601 F#
6: 698.396 cents -23.131 -710 C#
52: 701.955 cents -23.131 -710 G#
19: 694.245 cents -30.842 -947 D#
65: 701.955 cents -30.842 -947 A#
32: 694.245 cents -38.552 -1183 E#
78: 701.955 cents -38.552 -1183 B#
45: 701.362 cents -39.145 -1201 G\
12: 694.838 cents -46.263 -1420 D\
58: 701.955 cents -46.263 -1420 A\
25: 694.245 cents -53.973 -1656 E\
71: 701.955 cents -53.973 -1656 B\
38: 697.210 cents -58.718 -1802 F#\
5: 698.989 cents -61.684 -1893 C#\
51: 701.955 cents -61.684 -1893 G#\
18: 694.245 cents -69.394 -2130 D#\
64: 701.955 cents -69.394 -2130 A#\
31: 694.245 cents -77.105 -2366 F\\
77: 701.955 cents -77.105 -2366 C\\
44: 700.769 cents -78.291 -2403 G\\
11: 695.431 cents -84.815 -2603 D\\
57: 701.955 cents -84.815 -2603 A\\
24: 694.245 cents -92.525 -2840 E\\
70: 701.955 cents -92.525 -2840 B\\
37: 696.617 cents -97.863 -3003 F)
4: 699.583 cents -100.236 -3076 C)
50: 701.955 cents -100.236 -3076 G)
17: 694.245 cents -107.946 -3313 D)
63: 701.955 cents -107.946 -3313 A)
30: 694.245 cents -115.657 -3550 Fv
76: 701.955 cents -115.657 -3550 Cv
43: 700.176 cents -117.436 -3604 Gv
10: 696.024 cents -123.367 -3786 Dv
56: 701.955 cents -123.367 -3786 Av
23: 694.245 cents -131.078 -4023 Ev
69: 701.955 cents -131.078 -4023 Bv
36: 696.024 cents -137.009 -4205 F^
3: 700.176 cents -138.788 -4259 C^
49: 701.955 cents -138.788 -4259 G^
16: 694.245 cents -146.499 -4496 D^
62: 701.955 cents -146.499 -4496 A^
29: 694.245 cents -154.209 -4733 F(
75: 701.955 cents -154.209 -4733 C(
42: 699.583 cents -156.582 -4806 G(
9: 696.617 cents -161.920 -4969 D(
55: 701.955 cents -161.920 -4969 A(
22: 694.245 cents -169.630 -5206 E(
68: 701.955 cents -169.630 -5206 B(
35: 695.431 cents -176.154 -5406 F//
2: 700.769 cents -177.341 -5443 C//
48: 701.955 cents -177.341 -5443 G//
15: 694.245 cents -185.051 -5679 D//
61: 701.955 cents -185.051 -5679 A//
28: 694.245 cents -192.761 -5916 E//
74: 701.955 cents -192.761 -5916 B//
41: 698.989 cents -195.727 -6007 Gb/
8: 697.210 cents -200.472 -6153 Db/
54: 701.955 cents -200.472 -6153 Ab/
21: 694.245 cents -208.182 -6389 Eb/
67: 701.955 cents -208.182 -6389 Bb/
34: 694.838 cents -215.300 -6608 F/
1: 701.362 cents -215.893 -6626 C/
47: 701.955 cents -215.893 -6626 G/
14: 694.245 cents -223.603 -6863 D/
60: 701.955 cents -223.603 -6863 A/
27: 694.245 cents -231.314 -7099 Fb
73: 701.955 cents -231.314 -7099 Cb
40: 698.396 cents -234.872 -7208 Gb
7: 697.803 cents -239.024 -7336 Db
53: 701.955 cents -239.024 -7336 Ab
20: 694.245 cents -246.735 -7572 Eb
66: 701.955 cents -246.735 -7572 Bb
33: 694.245 cents -254.445 -7809 F
79: 701.955 cents -254.445 -7809 C
Average absolute difference: 128.8329 cents
Root mean square difference: 149.3033 cents
Maximum absolute difference: 254.4451 cents
Maximum formal fifth difference: 7.7105 cents

Oz.

SNIP

>
> Depending on what kinds of F# and Gb you wanted,
> (assuming you are using this for maqam music, you
> probably do want these), there are several other
> ways to fill the 9/8 gap between F and G. The most
> obvious of these still gives only two step sizes, and
> divides the 9/8 gap into 13 equal steps each of
> (701.954 - 498.045)/13 = 203.909/13 = 15.685 cents.
>
> a=15.092
> b=15.685
> na=66
> nb=13
>
> The step sizes a and b are much closer to each other
> than in Ozan's tuning.
>
> This is symmetrical about 600 cents, which is NOT a
> scale degree, but falls exactly between degrees 39
> and 40 of the scale, which are at 592.155 & 607.840
> cents.
>
> The important questions here are: what notes between
> F and G do you want to approximate well, and how
> closely?
>
> Regards,
> Yahya
>