Racapitulation on Yarman-36 variants (new 441/440 tweaked)

Hello again,

Instead of my previous announcement regarding "Yarman-48", which I
scratched out entirely for reasons which will be disclosed below, I
shall recapitulate on the Yarman-36 variants, namely, Yarman-36a, b
and c.

It will be recalled that Yarman-36a (previously "Yarman-36") was
attained by fixing the diapason for A4 at 438.410457150843 Hz and
tuning fifths by listening to the integer beats therefrom.

Layer I was tuned thus:

A= 438.41 Hz

A'-E = -1 (narrow fifth beating once per second)
E-B = -2
B'-F# = -1
F#-c# = +1 (wide fifth beating once per second)
C#-G# = +1
G#'-Eb = 0 (pure fifth)

A-D = -2
D-G' = -1
G-C = -1
c-F = -2
F-Bb' = -1
Bb-Eb = -2

Layer II was tuned thus:

G of Layer I to E of Layer II = +3 (wide 6:5 beating thrice per second)

E-B = -2
B'-F# = -1
F#-c# = -1
C#-G# = -1

E-A' = -1
A-D = -1
D-G' = -1
G-C = -2
c-F = 0
F-Bb' = 0
Bb-Eb = -1

Layer III was tuned thus:

C of Layer I to E of Layer III = 11:9 (exact)

E-B = -2
B'-F# = -1
F#-c# = -2
C#-G# = -1

E-A' = -1
A-D = -1
D-G' = -1
G-C = -1
c-F = 0
F-Bb' = 0
Bb-Eb = -1

The details of this tuning have already been described on my webpage,
alas in Turkish, but fortunately with video demonstrations on the
SCALA Chromatic Keyboard:

http://www.ozanyarman.com/yarman36.html

The diapason for A can easily be set at 440 Hz using the same tuning
method. The results will be almost the same.

Recall that I have postulated the correct Rast scale to be:

0: 1/1 C unison, perfect prime
1: 55/49 D quasi-equal major second
2: 550/441 Ed
3: 147/110 F
4: 3/2 G perfect fifth
5: 165/98 A
6: 275/147 Bd
7: 441/220 C Werckmeister's undecimal
septenarian schisma +1 octave

As described in my previous message:

/tuning/topicId_84942.html#84964

Let me, at this point, evaluate the features of the WerkmeisterianRast scale. 3/2 is preserved, but all other intervals are tweaked by
441/440. The reason for this was the desire to acquire a segah lower
than 5/4. 441/440 is a curious schisma to work with. This little
interval is 11-limit in nature, which, of course, vibes well with the
general high prime limit characteristics of Maqam music.

Once segah was lowered to the desired position, preserving the
symmetry of intervals required the widening of the octave, which
produced a pleasing result. The consequent intervals in the
Werckmeisterian Rast are:

0-1: 55:49 C-D quasi-equal major second
1-2: 10:9 D-Ed minor whole tone
2-3: 64827:60500 Ed-F
3-4: 55:49 F-G quasi-equal major second
4-5: 55:49 G-A quasi-equal major second
5-6: 10:9 A-Bd minor whole tone
6-7: 64827:60500 Bd-c

In cents:

200 + 182 + 120 + 200 + 200 + 182 + 120.

Note, that 64827:60500 is practically the same as 15:14.

The Werckmeisterian Rast yields splendid proportional beat rates
between given chords. For example, the beating of 6:5 to 5:4 in 4:5:6
is -1.5 per second on degrees 0 and 4. Beating of 5:4 to 6:5 in
10:12:15 is -1 per second on degrees 1, 2, and 3.

From this Rast scale, I derived two kinds of Major and two kinds of
Minor chords:

0 2 4 | 2 2 | 382 320 | 882:1100:1323 Highly consonant Major
1 3 5 | 2 2 | 302 400 | 6050:7203:9075 Very consonant Minor
2 4 6 | 2 2 | 320 382 | 1100:1323:1650 Fine consonant Minor
3 5 7 | 2 2 | 400 302 | 4802:6050:7203 Mediocre consonant Major

The consonance level of these chords is purely subjective. I prefer
the first two chords above the others. I still don't know exactly why
so. Nevertheless, I will base my following arguments on this table of
consonances.

Since 320 + 382 cent Minor is not as desirable (at least to me) as the
302 + 400 cent Minor, only three layers of 12-tone cycles reminiscent
of Yarman-36a will be sufficient for 441/220 octave variants. That is
to say, the ear no longer seeks 10:12:15 or its beating analogs. As
strange as it might seem, 6050:7203:9075 is most welcome to my ears,
and is equally or more consonant than 1100:1323:1650.

In fact, this quasi-equal Minor is often observed in Nihavend, Buselik
and Kurdilihijazkar nowadays.

Members will notice the proximity of this Minor chord to equal-
tempered Minor of common-practice Western music.

Therefore, there will be no need for a 48-tone sibling next to the
Yarman-36 variants.

400 + 302 cent Major is an inescapable fact. It is a component of
layer I. But it is not sufficient. Perde segah must be incorporated.
Thus layer II must be placed at 550/441.

Hence, we arrive at Yarman-36b. This variant is a combination of 24-
tone equal division of 441/220 and a 12-tone equal division of 441/220
placed at 550:441 from 1/1. It is a regular temperament, suitable for
delineating theory on paper:

1/1
50.164
81.401
100.328
150.491
181.729
200.655
250.819
282.056
300.983
351.146
550/441
401.310
451.474
482.711
501.638
551.801
583.039
601.965
652.129
683.366
702.293
752.456
783.694
802.620
852.784
884.021
902.948
953.111
984.349
1003.275
1053.439
1084.676
1103.603
1153.766
1185.004
441/220

Now comes the true gem. The last variant is Yarman-36c, where the
chain of fifths starting from C are tuned by pure fifths 6 upwards,
and 3 downwards. The octave is, once more, taken as 441/220. The beat
rates of the Major chords (6:5 to 5:4) have thus been -1.5 per second.
However, the cycle must be completed. We require slightly wider-than-
pure fifths for this. Hence the F#-C# and C#-G# fifths are to be
widened as much as the F#-Bb-C# and C#-F-G# chords produce beat rates
between 6:5 and 5:4 equal to -7/5 (-1.4) per second. The cycle is thus:

0: 0.000 cents 0.000 0 0 commas
7: 701.955 cents 0.000 0 0 commas
2: 701.955 cents 0.000 0 0 commas
9: 701.955 cents 0.000 0 0 commas
4: 701.955 cents 0.000 0 0 commas
11: 701.955 cents 0.000 0 0 commas
6: 701.955 cents 0.000 0 0 commas
1: 703.140 cents 1.185 36
8: 703.061 cents 2.291 70
3: 703.715 cents 4.051 124
10: 701.955 cents 4.051 124
5: 701.955 cents 4.051 124
12: 701.955 cents 4.051 124
Average absolute difference: 1.6401 cents
Root mean square difference: 2.5637 cents
Maximum absolute difference: 4.0511 cents
Maximum formal fifth difference: 1.7600 cents

And the pitches are:

1/1
99.150
55/49
7203/6050
3025/2401
147/110
166375/117649
3/2
802.211
165/98
21609/12100
9075/4802
441/220

We arrive at a what I would like to call a special "Pythagorean
Ordinaire" (PO) tuning.

This concludes Layer I. Layer II is created by shifting the same 12-
tone PO scale by 550/441 and layer III is created by shifting PO by
60:49. The resultant tuning is given below:

0: 1/1 0.000 unison, perfect prime
1: 121000/117649 48.622
2: 3327500/3176523 80.388
3: 99.150 cents 99.150
4: 12/11 150.637 3/4-tone, undecimal neutral
second
5: 10/9 182.404 minor whole tone
6: 55/49 199.980 quasi-equal major second
7: 6655000/5764801 248.601
8: 183012500/155649627 280.368
9: 7203/6050 301.995
10: 60/49 350.617 smaller approximation to
neutral third
11: 550/441 382.384
12: 3025/2401 399.960 two (quasi-equal major second)
13: 449.767 cents 449.767
14: 481.533 cents 481.533
15: 147/110 501.975
16: 3300/2401 550.597
17: 30250/21609 582.363
18: 166375/117649 599.940
19: 882/605 652.612
20: 49/33 684.379
21: 3/2 701.955 perfect fifth
22: 181500/117649 750.577
23: 1663750/1058841 782.343
24: 802.211 cents 802.211
25: 18/11 852.592 undecimal neutral sixth
26: 5/3 884.359 major sixth, BP sixth
27: 165/98 901.935
28: 9982500/5764801 950.556
29: 91506250/51883209 982.323
30: 21609/12100 1003.950
31: 90/49 1052.572
32: 275/147 1084.339
33: 9075/4802 1101.915
34: 1152.827 cents 1152.827
35: 1184.594 cents 1184.594
36: 441/220 1203.930 Werckmeister's undecimal
septenarian schisma +1 octave

For copy/paste purposes:

1/1
121000/117649
3327500/3176523
99.150
12/11
10/9
55/49
6655000/5764801
183012500/155649627
7203/6050
60/49
550/441
3025/2401
449.767
481.533
147/110
3300/2401
30250/21609
166375/117649
882/605
49/33
3/2
181500/117649
1663750/1058841
802.211
18/11
5/3
165/98
9982500/5764801
91506250/51883209
21609/12100
90/49
275/147
9075/4802
1152.827
1184.594
441/220

I am currently scribbling down the groundwork for a new makam theory
based on the Yarman-36 variants. More on this later...

Please try out Yarman-36c. Look especially for 7 and 11-limit chords.
I find them most pleasing.

Cordially,
Oz.

✩ ✩ ✩
www.ozanyarman.com

Ok,

I will be putting this into Scala and see what it sounds like.

I will say this - a beat around 1 Hz sounds pleasant and akin to a chorus
effect.

Chris

On Sun, Oct 11, 2009 at 7:54 PM, Ozan Yarman <ozanyarman@ozanyarman.com>wrote:

>
>
> Hello again,
>
> Instead of my previous announcement regarding "Yarman-48", which I
> scratched out entirely for reasons which will be disclosed below, I
> shall recapitulate on the Yarman-36 variants, namely, Yarman-36a, b
> and c.
>
> It will be recalled that Yarman-36a (previously "Yarman-36") was
> attained by fixing the diapason for A4 at 438.410457150843 Hz and
> tuning fifths by listening to the integer beats therefrom.
>
> Layer I was tuned thus:
>
> A= 438.41 Hz
>
> A'-E = -1 (narrow fifth beating once per second)
> E-B = -2
> B'-F# = -1
> F#-c# = +1 (wide fifth beating once per second)
> C#-G# = +1
> G#'-Eb = 0 (pure fifth)
>
> A-D = -2
> D-G' = -1
> G-C = -1
> c-F = -2
> F-Bb' = -1
> Bb-Eb = -2
>
> Layer II was tuned thus:
>
> G of Layer I to E of Layer II = +3 (wide 6:5 beating thrice per second)
>
> E-B = -2
> B'-F# = -1
> F#-c# = -1
> C#-G# = -1
>
> E-A' = -1
> A-D = -1
> D-G' = -1
> G-C = -2
> c-F = 0
> F-Bb' = 0
> Bb-Eb = -1
>
> Layer III was tuned thus:
>
> C of Layer I to E of Layer III = 11:9 (exact)
>
> E-B = -2
> B'-F# = -1
> F#-c# = -2
> C#-G# = -1
>
> E-A' = -1
> A-D = -1
> D-G' = -1
> G-C = -1
> c-F = 0
> F-Bb' = 0
> Bb-Eb = -1
>
> The details of this tuning have already been described on my webpage,
> alas in Turkish, but fortunately with video demonstrations on the
> SCALA Chromatic Keyboard:
>
> http://www.ozanyarman.com/yarman36.html
>
> The diapason for A can easily be set at 440 Hz using the same tuning
> method. The results will be almost the same.
>
> Recall that I have postulated the correct Rast scale to be:
>
> 0: 1/1 C unison, perfect prime
> 1: 55/49 D quasi-equal major second
> 2: 550/441 Ed
> 3: 147/110 F
> 4: 3/2 G perfect fifth
> 5: 165/98 A
> 6: 275/147 Bd
> 7: 441/220 C Werckmeister's undecimal
> septenarian schisma +1 octave
>
> As described in my previous message:
>
> /tuning/topicId_84942.html#84964
>
> Let me, at this point, evaluate the features of the Werkmeisterian
> Rast scale. 3/2 is preserved, but all other intervals are tweaked by
> 441/440. The reason for this was the desire to acquire a segah lower
> than 5/4. 441/440 is a curious schisma to work with. This little
> interval is 11-limit in nature, which, of course, vibes well with the
> general high prime limit characteristics of Maqam music.
>
> Once segah was lowered to the desired position, preserving the
> symmetry of intervals required the widening of the octave, which
> produced a pleasing result. The consequent intervals in the
> Werckmeisterian Rast are:
>
> 0-1: 55:49 C-D quasi-equal major second
> 1-2: 10:9 D-Ed minor whole tone
> 2-3: 64827:60500 Ed-F
> 3-4: 55:49 F-G quasi-equal major second
> 4-5: 55:49 G-A quasi-equal major second
> 5-6: 10:9 A-Bd minor whole tone
> 6-7: 64827:60500 Bd-c
>
> In cents:
>
> 200 + 182 + 120 + 200 + 200 + 182 + 120.
>
> Note, that 64827:60500 is practically the same as 15:14.
>
> The Werckmeisterian Rast yields splendid proportional beat rates
> between given chords. For example, the beating of 6:5 to 5:4 in 4:5:6
> is -1.5 per second on degrees 0 and 4. Beating of 5:4 to 6:5 in
> 10:12:15 is -1 per second on degrees 1, 2, and 3.
>
> From this Rast scale, I derived two kinds of Major and two kinds of
> Minor chords:
>
> 0 2 4 | 2 2 | 382 320 | 882:1100:1323 Highly consonant Major
> 1 3 5 | 2 2 | 302 400 | 6050:7203:9075 Very consonant Minor
> 2 4 6 | 2 2 | 320 382 | 1100:1323:1650 Fine consonant Minor
> 3 5 7 | 2 2 | 400 302 | 4802:6050:7203 Mediocre consonant Major
>
> The consonance level of these chords is purely subjective. I prefer
> the first two chords above the others. I still don't know exactly why
> so. Nevertheless, I will base my following arguments on this table of
> consonances.
>
> Since 320 + 382 cent Minor is not as desirable (at least to me) as the
> 302 + 400 cent Minor, only three layers of 12-tone cycles reminiscent
> of Yarman-36a will be sufficient for 441/220 octave variants. That is
> to say, the ear no longer seeks 10:12:15 or its beating analogs. As
> strange as it might seem, 6050:7203:9075 is most welcome to my ears,
> and is equally or more consonant than 1100:1323:1650.
>
> In fact, this quasi-equal Minor is often observed in Nihavend, Buselik
> and Kurdilihijazkar nowadays.
>
> Members will notice the proximity of this Minor chord to equal-
> tempered Minor of common-practice Western music.
>
> Therefore, there will be no need for a 48-tone sibling next to the
> Yarman-36 variants.
>
> 400 + 302 cent Major is an inescapable fact. It is a component of
> layer I. But it is not sufficient. Perde segah must be incorporated.
> Thus layer II must be placed at 550/441.
>
> Hence, we arrive at Yarman-36b. This variant is a combination of 24-
> tone equal division of 441/220 and a 12-tone equal division of 441/220
> placed at 550:441 from 1/1. It is a regular temperament, suitable for
> delineating theory on paper:
>
> 1/1
> 50.164
> 81.401
> 100.328
> 150.491
> 181.729
> 200.655
> 250.819
> 282.056
> 300.983
> 351.146
> 550/441
> 401.310
> 451.474
> 482.711
> 501.638
> 551.801
> 583.039
> 601.965
> 652.129
> 683.366
> 702.293
> 752.456
> 783.694
> 802.620
> 852.784
> 884.021
> 902.948
> 953.111
> 984.349
> 1003.275
> 1053.439
> 1084.676
> 1103.603
> 1153.766
> 1185.004
> 441/220
>
> Now comes the true gem. The last variant is Yarman-36c, where the
> chain of fifths starting from C are tuned by pure fifths 6 upwards,
> and 3 downwards. The octave is, once more, taken as 441/220. The beat
> rates of the Major chords (6:5 to 5:4) have thus been -1.5 per second.
> However, the cycle must be completed. We require slightly wider-than-
> pure fifths for this. Hence the F#-C# and C#-G# fifths are to be
> widened as much as the F#-Bb-C# and C#-F-G# chords produce beat rates
> between 6:5 and 5:4 equal to -7/5 (-1.4) per second. The cycle is thus:
>
> 0: 0.000 cents 0.000 0 0 commas
> 7: 701.955 cents 0.000 0 0 commas
> 2: 701.955 cents 0.000 0 0 commas
> 9: 701.955 cents 0.000 0 0 commas
> 4: 701.955 cents 0.000 0 0 commas
> 11: 701.955 cents 0.000 0 0 commas
> 6: 701.955 cents 0.000 0 0 commas
> 1: 703.140 cents 1.185 36
> 8: 703.061 cents 2.291 70
> 3: 703.715 cents 4.051 124
> 10: 701.955 cents 4.051 124
> 5: 701.955 cents 4.051 124
> 12: 701.955 cents 4.051 124
> Average absolute difference: 1.6401 cents
> Root mean square difference: 2.5637 cents
> Maximum absolute difference: 4.0511 cents
> Maximum formal fifth difference: 1.7600 cents
>
> And the pitches are:
>
> 1/1
> 99.150
> 55/49
> 7203/6050
> 3025/2401
> 147/110
> 166375/117649
> 3/2
> 802.211
> 165/98
> 21609/12100
> 9075/4802
> 441/220
>
> We arrive at a what I would like to call a special "Pythagorean
> Ordinaire" (PO) tuning.
>
> This concludes Layer I. Layer II is created by shifting the same 12-
> tone PO scale by 550/441 and layer III is created by shifting PO by
> 60:49. The resultant tuning is given below:
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 121000/117649 48.622
> 2: 3327500/3176523 80.388
> 3: 99.150 cents 99.150
> 4: 12/11 150.637 3/4-tone, undecimal neutral
> second
> 5: 10/9 182.404 minor whole tone
> 6: 55/49 199.980 quasi-equal major second
> 7: 6655000/5764801 248.601
> 8: 183012500/155649627 280.368
> 9: 7203/6050 301.995
> 10: 60/49 350.617 smaller approximation to
> neutral third
> 11: 550/441 382.384
> 12: 3025/2401 399.960 two (quasi-equal major second)
> 13: 449.767 cents 449.767
> 14: 481.533 cents 481.533
> 15: 147/110 501.975
> 16: 3300/2401 550.597
> 17: 30250/21609 582.363
> 18: 166375/117649 599.940
> 19: 882/605 652.612
> 20: 49/33 684.379
> 21: 3/2 701.955 perfect fifth
> 22: 181500/117649 750.577
> 23: 1663750/1058841 782.343
> 24: 802.211 cents 802.211
> 25: 18/11 852.592 undecimal neutral sixth
> 26: 5/3 884.359 major sixth, BP sixth
> 27: 165/98 901.935
> 28: 9982500/5764801 950.556
> 29: 91506250/51883209 982.323
> 30: 21609/12100 1003.950
> 31: 90/49 1052.572
> 32: 275/147 1084.339
> 33: 9075/4802 1101.915
> 34: 1152.827 cents 1152.827
> 35: 1184.594 cents 1184.594
> 36: 441/220 1203.930 Werckmeister's undecimal
> septenarian schisma +1 octave
>
> For copy/paste purposes:
>
> 1/1
> 121000/117649
> 3327500/3176523
> 99.150
> 12/11
> 10/9
> 55/49
> 6655000/5764801
> 183012500/155649627
> 7203/6050
> 60/49
> 550/441
> 3025/2401
> 449.767
> 481.533
> 147/110
> 3300/2401
> 30250/21609
> 166375/117649
> 882/605
> 49/33
> 3/2
> 181500/117649
> 1663750/1058841
> 802.211
> 18/11
> 5/3
> 165/98
> 9982500/5764801
> 91506250/51883209
> 21609/12100
> 90/49
> 275/147
> 9075/4802
> 1152.827
> 1184.594
> 441/220
>
> I am currently scribbling down the groundwork for a new makam theory
> based on the Yarman-36 variants. More on this later...
>
> Please try out Yarman-36c. Look especially for 7 and 11-limit chords.
> I find them most pleasing.
>
> Cordially,
> Oz.
>
> ✩ ✩ ✩
> www.ozanyarman.com
>
>
>