A yet another revised 21-tone minimal tuning for maqam music

This revised 21-tone tuning has many of the charms of its predecessor and new features. It embodies a beautiful 12-tone cyclic quasi-temperament for Western Classical harmony as well as a 17-tone cyclic scale for the xenharmony-oriented through fairly simple ratios:

1/1
20/19
38/35
9/8
19/16
11/9
37/30
59/47
19/15
4/3
7/5
36/25
3/2
19/12
18/11
42/25
16/9
11/6
13/7
15/8
21/11
2/1

Notice, that mode 1 2 1 3 2 1 2 1 2 1 3 2 extracts the 12-tone cyclic quasi-temperament:

0.000 0.000 0 0 commas
701.955 0.000 0 0 commas
701.955 0.000 0 0 commas
694.243 -7.712 -237 -septimal kleisma
695.512 -14.155 -434
694.604 -21.506 -660 -syntonic comma
694.243 -29.218 -897
706.289 -24.884 -764
706.757 -20.082 -616
701.955 -20.082 -616
698.577 -23.460 -720 -Pythagorean comma
701.955 -23.460 -720 -Pythagorean comma
701.955 -23.460 -720 -Pythagorean comma
Average absolute difference: 17.3349 cents
Root mean square difference: 20.5669 cents
Maximum absolute difference: 29.2178 cents
Maximum formal fifth difference: 7.7115 cents

And mode 1 1 1 1 2 2 1 1 1 1 1 1 1 1 2 2 1 extracts the 17-tone cyclic scale:

0.000 0.000 0 0 commas
701.955 0.000 0 0 commas
701.955 0.000 0 0 commas
694.243 -7.712 -237 -septimal kleisma
711.091 1.424 44 Eratosthenes' comma
710.219 9.688 297 undecimal semicomma
711.820 19.553 600 diaschisma
711.091 28.688 880
710.219 36.952 1134 undecimal minor diesis
710.483 45.480 1396
708.626 52.152 1601
710.810 61.007 1872
706.289 65.341 2005
706.757 70.143 2153
701.955 70.143 2153
698.577 66.765 2049 Pyth. double dim. third
701.955 66.765 2049 Pyth. double dim. third
701.955 66.765 2049 Pyth. double dim. third
Average absolute difference: 39.3281 cents
Root mean square difference: 49.1925 cents
Maximum absolute difference: 70.1430 cents
Maximum formal fifth difference: 9.8646 cents

Here is a list of perdes in cents:

0 RAST
89 Nim Zengule (Shuri)
142 Zengule
204 DUGAH
298 Kurdi
347 Ushshaq *
363 Nerm Segah (Segahche)
394 SEGAH
409 Buselik
498 TCHARGAH
583 Hijaz
631 Dik Hijaz (Saba)
702 NEVA
796 Nim Hisar (Beyati)
853 Hisar
898 HUSEYNI (tempered Hisarek)
996 Ajem
1049 Dik Ajem *
1072 Nerm Evdj
1088 EVDJ
1119 Mahur
1200 GERDANIYE

(* non-cyclic tones)

While possibilities of transposition of maqamat are somewhat limited compared to, say, 41 or 53-tET, this is by no means detrimental to serious music-making.

For optimal keyboard mapping In SCALA, choose Equal 45. You have to imagine nerm segah, buselik, nerm evdj and mahur as raised white keys instead of black. I suppose such a keyboard design is feasible for an acoustic piano.

Cordially,
Oz.

A final touch to this tuning for a better 12-tone cycle, is the replacement of 9/8 with 37/33 at 198 cents.

Oz.
----- Original Message -----
From: Ozan Yarman
To: Tuning List
Sent: 20 Nisan 2008 Pazar 6:08
Subject: A yet another revised 21-tone minimal tuning for maqam music

This revised 21-tone tuning has many of the charms of its predecessor and new features. It embodies a beautiful 12-tone cyclic quasi-temperament for Western Classical harmony as well as a 17-tone cyclic scale for the xenharmony-oriented through fairly simple ratios:

1/1
20/19
38/35
9/8
19/16
11/9
37/30
59/47
19/15
4/3
7/5
36/25
3/2
19/12
18/11
42/25
16/9
11/6
13/7
15/8
21/11
2/1

Notice, that mode 1 2 1 3 2 1 2 1 2 1 3 2 extracts the 12-tone cyclic quasi-temperament:

0.000 0.000 0 0 commas
701.955 0.000 0 0 commas
701.955 0.000 0 0 commas
694.243 -7.712 -237 -septimal kleisma
695.512 -14.155 -434
694.604 -21.506 -660 -syntonic comma
694.243 -29.218 -897
706.289 -24.884 -764
706.757 -20.082 -616
701.955 -20.082 -616
698.577 -23.460 -720 -Pythagorean comma
701.955 -23.460 -720 -Pythagorean comma
701.955 -23.460 -720 -Pythagorean comma
Average absolute difference: 17.3349 cents
Root mean square difference: 20.5669 cents
Maximum absolute difference: 29.2178 cents
Maximum formal fifth difference: 7.7115 cents

And mode 1 1 1 1 2 2 1 1 1 1 1 1 1 1 2 2 1 extracts the 17-tone cyclic scale:

0.000 0.000 0 0 commas
701.955 0.000 0 0 commas
701.955 0.000 0 0 commas
694.243 -7.712 -237 -septimal kleisma
711.091 1.424 44 Eratosthenes' comma
710.219 9.688 297 undecimal semicomma
711.820 19.553 600 diaschisma
711.091 28.688 880
710.219 36.952 1134 undecimal minor diesis
710.483 45.480 1396
708.626 52.152 1601
710.810 61.007 1872
706.289 65.341 2005
706.757 70.143 2153
701.955 70.143 2153
698.577 66.765 2049 Pyth. double dim. third
701.955 66.765 2049 Pyth. double dim. third
701.955 66.765 2049 Pyth. double dim. third
Average absolute difference: 39.3281 cents
Root mean square difference: 49.1925 cents
Maximum absolute difference: 70.1430 cents
Maximum formal fifth difference: 9.8646 cents

Here is a list of perdes in cents:

0 RAST
89 Nim Zengule (Shuri)
142 Zengule
204 DUGAH
298 Kurdi
347 Ushshaq *
363 Nerm Segah (Segahche)
394 SEGAH
409 Buselik
498 TCHARGAH
583 Hijaz
631 Dik Hijaz (Saba)
702 NEVA
796 Nim Hisar (Beyati)
853 Hisar
898 HUSEYNI (tempered Hisarek)
996 Ajem
1049 Dik Ajem *
1072 Nerm Evdj
1088 EVDJ
1119 Mahur
1200 GERDANIYE

(* non-cyclic tones)

While possibilities of transposition of maqamat are somewhat limited compared to, say, 41 or 53-tET, this is by no means detrimental to serious music-making.

For optimal keyboard mapping In SCALA, choose Equal 45. You have to imagine nerm segah, buselik, nerm evdj and mahur as raised white keys instead of black. I suppose such a keyboard design is feasible for an acoustic piano.

Cordially,
Oz.