Since we are all getting worked up in Bach's intended 12-tone keyboard tuning again...

It seems we are hopelessly beleagured trying to decide on the ultimate tuning which works for Bach, or that which he would have preferred among all the infinite temperament choices in the late Baroque era. Joining the contention with my 250 pounds of flesh, here is a revised 24-tone maqam tuning containing an embedded 12-tone circular temperament ordinaire inspired by Rameau's 1/4 PC modified meantone, which may unite the maqam tradition with the Bachist keyboard tradition:

1/1
84.360
38/35
192.180
9/8
292.180
17/14
16/13
5/4
19/15
4/3
584.079
36/25
696.090
3/2
788.270
18/11
888.270
27/16
16/9
20/11
13/7
15/8
21/11
2/1

With cents and perde names:

0: 1/1 0.000 RAST
1: 84.360 cents Nim Zengule (Shuri)
2: 38/35 142.373 Zengule
3: 192.180 cents Nerm Dugah
4: 9/8 203.910 DUGAH
5: 292.180 cents Kurdi
6: 17/14 336.130 Dik Kurdi (Ushshaq)
7: 16/13 359.472 Nerm Segah
8: 5/4 386.314 SEGAH
9: 19/15 409.244 Buselik
10: 4/3 498.045 TSCHARGAH
11: 584.079 cents Hijaz
12: 36/25 631.283 Dik Hijaz (Saba)
13: 696.090 cents Nerm Neva
14: 3/2 701.955 NEVA
15: 788.270 cents Nim Hisar (Beyati)
16: 18/11 852.592 Hisar
17: 888.270 cents Dik Hisar
18: 27/16 905.865 HUSEYNI
19: 16/9 996.090 Ajem
20: 20/11 1034.996 Dik Ajem
21: 13/7 1071.702 Nerm EVDJ
22: 15/8 1088.269 EVDJ
23: 21/11 1119.463 Mahur
24: 2/1 1200.000 GERDANIYE

Here is the 12-tone circular tuning (mode 1 3 1 3 2 1 2 2 2 2 3 2) out of the above:
|
0: 1/1 C Dbb unison, perfect prime
1: 84.360 cents C# Db
2: 192.180 cents D Ebb
3: 292.180 cents D# Eb
4: 5/4 E Fb major third
5: 4/3 F Gbb perfect fourth
6: 584.079 cents F# Gb
7: 696.090 cents G Abb
8: 788.270 cents G# Ab
9: 888.270 cents A Bbb
10: 16/9 A# Bb Pythagorean minor seventh
11: 15/8 B Cb classic major seventh
12: 2/1 C Dbb octave

Here go the cycle of fifths. I think it's pretty neat. Wouldn't Bach love it?
|
0: 0.000 cents 0.000 0 0 commas C
7: 696.090 cents -5.865 -180 -1/4 Pyth. commas G
2: 696.090 cents -11.730 -360 -1/2 Pyth. commas D
9: 696.090 cents -17.595 -540 -3/4 Pyth. commas A
4: 698.044 cents -21.506 -660 -syntonic comma, Didymus comma E
11: 701.955 cents -21.506 -660 -syntonic comma, Didymus comma B
6: 695.810 cents -27.651 -849 -9/7 synt. commas F#
1: 700.281 cents -29.325 -900 -5/4 Pyth. commas C#
8: 703.910 cents -27.370 -840 -7/6 Pyth. commas G#
3: 703.910 cents -25.415 -780 -13/12 Pyth. commas Eb
10: 703.910 cents -23.460 -720 -Pythagorean comma, ditonic co Bb
5: 701.955 cents -23.460 -720 -Pythagorean comma, ditonic co F
12: 701.955 cents -23.460 -720 -Pythagorean comma, ditonic co C
Average absolute difference: 21.5286 cents
Root mean square difference: 23.5049 cents
Maximum absolute difference: 29.3250 cents
Maximum formal fifth difference: 6.1447 cents
|

And the harmonic statistics:

Temperings of 3/2 5/4 6/5
0: 0.000: -5.8650 0.0000 -23.4613
1: 84.360: 1.9550 27.3713 -13.6876
2: 192.180: -5.8650 5.5853 -9.7763
3: 292.180: 1.9550 17.5963 -23.7422
4: 386.314: 0.0000 15.6426 -5.8650
5: 498.045: 0.0000 3.9113 -25.4163
6: 584.079: -1.6741 25.6972 -11.4503
7: 696.090: -5.8650 5.8650 -15.6413
8: 788.270: 1.9550 25.4163 -15.6426
9: 888.270: -3.9113 9.7763 -3.9113
10: 996.090: 0.0000 9.7763 -27.3713
11: 1088.269: -6.1447 17.5976 -11.7300
12: 1200.000: -5.8650 0.0000 -23.4613
Total abs. diff. : 35.1900 164.2354 187.6954
Average abs. diff.: 2.9325 13.6863 15.6413
Highest abs. diff.: 6.1447 27.3713 27.3713
|

And here is a 17-tone circular tuning (mode 1 1 2 1 2 2 1 1 1 2 1 1 2 1 2 2 1) extracted from the same:

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

The cycle of fifths follow:

|
0: 0.000 cents 0.000 0 0 commas
10: 701.955 cents 0.000 0 0 commas
3: 701.955 cents 0.000 0 0 commas
13: 701.955 cents 0.000 0 0 commas
6: 703.379 cents 1.424 44 Eratosthenes' comma
16: 710.219 cents 9.688 297 undecimal semicomma
9: 711.820 cents 19.553 600 diaschisma
2: 711.091 cents 28.688 880
12: 710.219 cents 36.952 1134 undecimal minor diesis
5: 706.880 cents 41.877 1285
15: 712.229 cents 52.152 1601
8: 712.377 cents 62.574 1920
1: 700.281 cents 60.900 1869
11: 703.910 cents 62.855 1929
4: 703.910 cents 64.810 1989
14: 703.910 cents 66.765 2049 Pythagorean double diminished third
7: 701.955 cents 66.765 2049 Pythagorean double diminished third
17: 701.955 cents 66.765 2049 Pythagorean double diminished third
Average absolute difference: 37.7511 cents
Root mean square difference: 47.6237 cents
Maximum absolute difference: 66.7650 cents
Maximum formal fifth difference: 10.4223 cents

I encourage everyone with the means to try it.

Cordially,
Oz.

do they measure in pounds in turkey not kilos?

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

ozanyarman@... wrote:
>
> It seems we are hopelessly beleagured trying to decide on the ultimate
> tuning which works for Bach, or that which he would have preferred
> among all the infinite temperament choices in the late Baroque era.
> Joining the contention with my 250 pounds of flesh, here is a revised
> 24-tone maqam tuning containing an embedded 12-tone circular
> temperament ordinaire inspired by Rameau's 1/4 PC modified meantone,
> which may unite the maqam tradition with the Bachist keyboard tradition:
>
> 1/1
> 84.360
> 38/35
> 192.180
> 9/8
> 292.180
> 17/14
> 16/13
> 5/4
> 19/15
> 4/3
> 584.079
> 36/25
> 696.090
> 3/2
> 788.270
> 18/11
> 888.270
> 27/16
> 16/9
> 20/11
> 13/7
> 15/8
> 21/11
> 2/1
>
> With cents and perde names:
>
> 0: 1/1 0.000 RAST
> 1: 84.360 cents Nim Zengule (Shuri)
> 2: 38/35 142.373 Zengule
> 3: 192.180 cents Nerm Dugah
> 4: 9/8 203.910 DUGAH
> 5: 292.180 cents Kurdi
> 6: 17/14 336.130 Dik Kurdi (Ushshaq)
> 7: 16/13 359.472 Nerm Segah
> 8: 5/4 386.314 SEGAH
> 9: 19/15 409.244 Buselik
> 10: 4/3 498.045 TSCHARGAH
> 11: 584.079 cents Hijaz
> 12: 36/25 631.283 Dik Hijaz (Saba)
> 13: 696.090 cents Nerm Neva
> 14: 3/2 701.955 NEVA
> 15: 788.270 cents Nim Hisar (Beyati)
> 16: 18/11 852.592 Hisar
> 17: 888.270 cents Dik Hisar
> 18: 27/16 905.865 HUSEYNI
> 19: 16/9 996.090 Ajem
> 20: 20/11 1034.996 Dik Ajem
> 21: 13/7 1071.702 Nerm EVDJ
> 22: 15/8 1088.269 EVDJ
> 23: 21/11 1119.463 Mahur
> 24: 2/1 1200.000 GERDANIYE
>
> Here is the 12-tone circular tuning (mode 1 3 1 3 2 1 2 2 2 2 3 2) out
> of the above:
> |
> 0: 1/1 C Dbb unison, perfect prime
> 1: 84.360 cents C# Db
> 2: 192.180 cents D Ebb
> 3: 292.180 cents D# Eb
> 4: 5/4 E Fb major third
> 5: 4/3 F Gbb perfect fourth
> 6: 584.079 cents F# Gb
> 7: 696.090 cents G Abb
> 8: 788.270 cents G# Ab
> 9: 888.270 cents A Bbb
> 10: 16/9 A# Bb Pythagorean minor
> seventh
> 11: 15/8 B Cb classic major seventh
> 12: 2/1 C Dbb octave
>
> Here go the cycle of fifths. I think it's pretty neat. Wouldn't Bach
> love it?
> |
> 0: 0.000 cents 0.000 0 0
> commas C
> 7: 696.090 cents -5.865 -180 -1/4 Pyth.
> commas G
> 2: 696.090 cents -11.730 -360 -1/2 Pyth.
> commas D
> 9: 696.090 cents -17.595 -540 -3/4 Pyth.
> commas A
> 4: 698.044 cents -21.506 -660 -syntonic comma, Didymus
> comma E
> 11: 701.955 cents -21.506 -660 -syntonic comma, Didymus
> comma B
> 6: 695.810 cents -27.651 -849 -9/7 synt.
> commas F#
> 1: 700.281 cents -29.325 -900 -5/4 Pyth.
> commas C#
> 8: 703.910 cents -27.370 -840 -7/6 Pyth.
> commas G#
> 3: 703.910 cents -25.415 -780 -13/12 Pyth.
> commas Eb
> 10: 703.910 cents -23.460 -720 -Pythagorean comma, ditonic
> co Bb
> 5: 701.955 cents -23.460 -720 -Pythagorean comma, ditonic
> co F
> 12: 701.955 cents -23.460 -720 -Pythagorean comma, ditonic
> co C
> Average absolute difference: 21.5286 cents
> Root mean square difference: 23.5049 cents
> Maximum absolute difference: 29.3250 cents
> Maximum formal fifth difference: 6.1447 cents
> |
>
> And the harmonic statistics:
>
> Temperings of 3/2 5/4 6/5
> 0: 0.000: -5.8650 0.0000 -23.4613
> 1: 84.360: 1.9550 27.3713 -13.6876
> 2: 192.180: -5.8650 5.5853 -9.7763
> 3: 292.180: 1.9550 17.5963 -23.7422
> 4: 386.314: 0.0000 15.6426 -5.8650
> 5: 498.045: 0.0000 3.9113 -25.4163
> 6: 584.079: -1.6741 25.6972 -11.4503
> 7: 696.090: -5.8650 5.8650 -15.6413
> 8: 788.270: 1.9550 25.4163 -15.6426
> 9: 888.270: -3.9113 9.7763 -3.9113
> 10: 996.090: 0.0000 9.7763 -27.3713
> 11: 1088.269: -6.1447 17.5976 -11.7300
> 12: 1200.000: -5.8650 0.0000 -23.4613
> Total abs. diff. : 35.1900 164.2354 187.6954
> Average abs. diff.: 2.9325 13.6863 15.6413
> Highest abs. diff.: 6.1447 27.3713 27.3713
> |
>
> And here is a 17-tone circular tuning (mode 1 1 2 1 2 2 1 1 1 2 1 1 2
> 1 2 2 1) extracted from the same:
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 84.360 cents
> 2: 38/35 142.373
> 3: 9/8 203.910 major whole tone
> 4: 292.180 cents
> 5: 16/13 359.472 tridecimal neutral third
> 6: 19/15 409.244 undevicesimal ditone
> 7: 4/3 498.045 perfect fourth
> 8: 584.079 cents
> 9: 36/25 631.283 classic diminished fifth
> 10: 3/2 701.955 perfect fifth
> 11: 788.270 cents
> 12: 18/11 852.592 undecimal neutral sixth
> 13: 27/16 905.865 Pythagorean major sixth
> 14: 16/9 996.090 Pythagorean minor seventh
> 15: 13/7 1071.702 16/3-tone
> 16: 21/11 1119.463 undecimal major seventh
> 17: 2/1 1200.000 octave
>
> The cycle of fifths follow:
>
> |
> 0: 0.000 cents 0.000 0 0 commas
> 10: 701.955 cents 0.000 0 0 commas
> 3: 701.955 cents 0.000 0 0 commas
> 13: 701.955 cents 0.000 0 0 commas
> 6: 703.379 cents 1.424 44 Eratosthenes' comma
> 16: 710.219 cents 9.688 297 undecimal semicomma
> 9: 711.820 cents 19.553 600 diaschisma
> 2: 711.091 cents 28.688 880
> 12: 710.219 cents 36.952 1134 undecimal minor diesis
> 5: 706.880 cents 41.877 1285
> 15: 712.229 cents 52.152 1601
> 8: 712.377 cents 62.574 1920
> 1: 700.281 cents 60.900 1869
> 11: 703.910 cents 62.855 1929
> 4: 703.910 cents 64.810 1989
> 14: 703.910 cents 66.765 2049 Pythagorean double
> diminished third
> 7: 701.955 cents 66.765 2049 Pythagorean double
> diminished third
> 17: 701.955 cents 66.765 2049 Pythagorean double
> diminished third
> Average absolute difference: 37.7511 cents
> Root mean square difference: 47.6237 cents
> Maximum absolute difference: 66.7650 cents
> Maximum formal fifth difference: 10.4223 cents
>
> I encourage everyone with the means to try it.
>
> Cordially,
> Oz.
>
>

They measure in kilos, but I am converting to pounds.

Oz.

On May 18, 2008, at 2:30 AM, Kraig Grady wrote:

> do they measure in pounds in turkey not kilos?
>
>
> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
> _'''''''_ ^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria <http://> anaphoriasouth.blogspot.com/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
>
>
>
> ozanyarman@... wrote:
>>
>> It seems we are hopelessly beleagured trying to decide on the >> ultimate
>> tuning which works for Bach, or that which he would have preferred
>> among all the infinite temperament choices in the late Baroque era.
>> Joining the contention with my 250 pounds of flesh, here is a revised
>> 24-tone maqam tuning containing an embedded 12-tone circular
>> temperament ordinaire inspired by Rameau's 1/4 PC modified meantone,
>> which may unite the maqam tradition with the Bachist keyboard >> tradition:
>>
>> 1/1
>> 84.360
>> 38/35
>> 192.180
>> 9/8
>> 292.180
>> 17/14
>> 16/13
>> 5/4
>> 19/15
>> 4/3
>> 584.079
>> 36/25
>> 696.090
>> 3/2
>> 788.270
>> 18/11
>> 888.270
>> 27/16
>> 16/9
>> 20/11
>> 13/7
>> 15/8
>> 21/11
>> 2/1
>>
>> With cents and perde names:
>>
>> 0: 1/1 0.000 RAST
>> 1: 84.360 cents Nim Zengule (Shuri)
>> 2: 38/35 142.373 Zengule
>> 3: 192.180 cents Nerm Dugah
>> 4: 9/8 203.910 DUGAH
>> 5: 292.180 cents Kurdi
>> 6: 17/14 336.130 Dik Kurdi (Ushshaq)
>> 7: 16/13 359.472 Nerm Segah
>> 8: 5/4 386.314 SEGAH
>> 9: 19/15 409.244 Buselik
>> 10: 4/3 498.045 TSCHARGAH
>> 11: 584.079 cents Hijaz
>> 12: 36/25 631.283 Dik Hijaz (Saba)
>> 13: 696.090 cents Nerm Neva
>> 14: 3/2 701.955 NEVA
>> 15: 788.270 cents Nim Hisar (Beyati)
>> 16: 18/11 852.592 Hisar
>> 17: 888.270 cents Dik Hisar
>> 18: 27/16 905.865 HUSEYNI
>> 19: 16/9 996.090 Ajem
>> 20: 20/11 1034.996 Dik Ajem
>> 21: 13/7 1071.702 Nerm EVDJ
>> 22: 15/8 1088.269 EVDJ
>> 23: 21/11 1119.463 Mahur
>> 24: 2/1 1200.000 GERDANIYE
>>
>> Here is the 12-tone circular tuning (mode 1 3 1 3 2 1 2 2 2 2 3 2) >> out
>> of the above:
>> |
>> 0: 1/1 C Dbb unison, perfect prime
>> 1: 84.360 cents C# Db
>> 2: 192.180 cents D Ebb
>> 3: 292.180 cents D# Eb
>> 4: 5/4 E Fb major third
>> 5: 4/3 F Gbb perfect fourth
>> 6: 584.079 cents F# Gb
>> 7: 696.090 cents G Abb
>> 8: 788.270 cents G# Ab
>> 9: 888.270 cents A Bbb
>> 10: 16/9 A# Bb Pythagorean minor
>> seventh
>> 11: 15/8 B Cb classic major seventh
>> 12: 2/1 C Dbb octave
>>
>> Here go the cycle of fifths. I think it's pretty neat. Wouldn't Bach
>> love it?
>> |
>> 0: 0.000 cents 0.000 0 0
>> commas C
>> 7: 696.090 cents -5.865 -180 -1/4 Pyth.
>> commas G
>> 2: 696.090 cents -11.730 -360 -1/2 Pyth.
>> commas D
>> 9: 696.090 cents -17.595 -540 -3/4 Pyth.
>> commas A
>> 4: 698.044 cents -21.506 -660 -syntonic comma, Didymus
>> comma E
>> 11: 701.955 cents -21.506 -660 -syntonic comma, Didymus
>> comma B
>> 6: 695.810 cents -27.651 -849 -9/7 synt.
>> commas F#
>> 1: 700.281 cents -29.325 -900 -5/4 Pyth.
>> commas C#
>> 8: 703.910 cents -27.370 -840 -7/6 Pyth.
>> commas G#
>> 3: 703.910 cents -25.415 -780 -13/12 Pyth.
>> commas Eb
>> 10: 703.910 cents -23.460 -720 -Pythagorean comma, ditonic
>> co Bb
>> 5: 701.955 cents -23.460 -720 -Pythagorean comma, ditonic
>> co F
>> 12: 701.955 cents -23.460 -720 -Pythagorean comma, ditonic
>> co C
>> Average absolute difference: 21.5286 cents
>> Root mean square difference: 23.5049 cents
>> Maximum absolute difference: 29.3250 cents
>> Maximum formal fifth difference: 6.1447 cents
>> |
>>
>> And the harmonic statistics:
>>
>> Temperings of 3/2 5/4 6/5
>> 0: 0.000: -5.8650 0.0000 -23.4613
>> 1: 84.360: 1.9550 27.3713 -13.6876
>> 2: 192.180: -5.8650 5.5853 -9.7763
>> 3: 292.180: 1.9550 17.5963 -23.7422
>> 4: 386.314: 0.0000 15.6426 -5.8650
>> 5: 498.045: 0.0000 3.9113 -25.4163
>> 6: 584.079: -1.6741 25.6972 -11.4503
>> 7: 696.090: -5.8650 5.8650 -15.6413
>> 8: 788.270: 1.9550 25.4163 -15.6426
>> 9: 888.270: -3.9113 9.7763 -3.9113
>> 10: 996.090: 0.0000 9.7763 -27.3713
>> 11: 1088.269: -6.1447 17.5976 -11.7300
>> 12: 1200.000: -5.8650 0.0000 -23.4613
>> Total abs. diff. : 35.1900 164.2354 187.6954
>> Average abs. diff.: 2.9325 13.6863 15.6413
>> Highest abs. diff.: 6.1447 27.3713 27.3713
>> |
>>
>> And here is a 17-tone circular tuning (mode 1 1 2 1 2 2 1 1 1 2 1 1 2
>> 1 2 2 1) extracted from the same:
>>
>> 0: 1/1 0.000 unison, perfect prime
>> 1: 84.360 cents
>> 2: 38/35 142.373
>> 3: 9/8 203.910 major whole tone
>> 4: 292.180 cents
>> 5: 16/13 359.472 tridecimal neutral third
>> 6: 19/15 409.244 undevicesimal ditone
>> 7: 4/3 498.045 perfect fourth
>> 8: 584.079 cents
>> 9: 36/25 631.283 classic diminished fifth
>> 10: 3/2 701.955 perfect fifth
>> 11: 788.270 cents
>> 12: 18/11 852.592 undecimal neutral sixth
>> 13: 27/16 905.865 Pythagorean major sixth
>> 14: 16/9 996.090 Pythagorean minor seventh
>> 15: 13/7 1071.702 16/3-tone
>> 16: 21/11 1119.463 undecimal major seventh
>> 17: 2/1 1200.000 octave
>>
>> The cycle of fifths follow:
>>
>> |
>> 0: 0.000 cents 0.000 0 0 commas
>> 10: 701.955 cents 0.000 0 0 commas
>> 3: 701.955 cents 0.000 0 0 commas
>> 13: 701.955 cents 0.000 0 0 commas
>> 6: 703.379 cents 1.424 44 Eratosthenes' comma
>> 16: 710.219 cents 9.688 297 undecimal semicomma
>> 9: 711.820 cents 19.553 600 diaschisma
>> 2: 711.091 cents 28.688 880
>> 12: 710.219 cents 36.952 1134 undecimal minor diesis
>> 5: 706.880 cents 41.877 1285
>> 15: 712.229 cents 52.152 1601
>> 8: 712.377 cents 62.574 1920
>> 1: 700.281 cents 60.900 1869
>> 11: 703.910 cents 62.855 1929
>> 4: 703.910 cents 64.810 1989
>> 14: 703.910 cents 66.765 2049 Pythagorean double
>> diminished third
>> 7: 701.955 cents 66.765 2049 Pythagorean double
>> diminished third
>> 17: 701.955 cents 66.765 2049 Pythagorean double
>> diminished third
>> Average absolute difference: 37.7511 cents
>> Root mean square difference: 47.6237 cents
>> Maximum absolute difference: 66.7650 cents
>> Maximum formal fifth difference: 10.4223 cents
>>
>> I encourage everyone with the means to try it.
>>
>> Cordially,
>> Oz.
>>
>>