9ed9/8: Difference between revisions

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Adopt template: ED intro. -reference to Turkish music (this interpretation instead of plain 53edo is absurd). Misc. cleanup
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{{Infobox ET}}
{{Infobox ET}}
'''9ED9/8''' is the [[Equal-step tuning|equal division]] of the [[9/8|Pythagorean whole tone]] into nine parts of 22.6567 [[cent|cents]] each, corresponding to 52.9645 [[edo]]. This tuning is used in Ottoman classical music theory, in which ninth root of the 9/8 whole tone is treated as the minimum interval.
{{ED intro}} It corresponds to 52.9645 [[edo]], which is closely related to [[53edo]] but with the whole tone instead of the octave tuned pure.  


==Intervals==
== Intervals ==
{| class="wikitable right-2"
{| class="wikitable right-2"
|-
|-
! degree
! #
! cents value
! Cents Value
! ratio
! Ratio
|-
|-
| 0
| 0
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|}
|}


== Just approximation ==
== Approximation to JI ==
=== 15-odd-limit mappings ===
=== 15-odd-limit mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 9ed9/8 (ordered by absolute error).
The following table shows how [[15-odd-limit intervals]] are represented in 9ed9/8 (ordered by absolute error).
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{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
|+ Direct mapping (even if inconsistent)
|+ Direct approximation (even if inconsistent)
|-
|-
! Interval(s)
! Interval(s)
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| 10.491
| 10.491
|}
|}
==See also==
* [http://en.wikipedia.org/wiki/Ottoman_classical_music Ottoman classical music - Wikipedia]
* [http://en.wikipedia.org/wiki/Makam Makam - Wikipedia]


[[Category:Equal-step tuning]]
[[Category:Equal-step tuning]]
[[Category:Edonoi]]
[[Category:Edonoi]]
[[Category:Whole tone]]
[[Category:Whole tone]]

Revision as of 09:06, 24 May 2024

← 8ed9/8 9ed9/8 10ed9/8 →
Prime factorization 32
Step size 22.6567 ¢ 
Octave 53\9ed9/8 (1200.8 ¢)
(convergent)
Twelfth 84\9ed9/8 (1903.16 ¢) (→ 28\3ed9/8)
Consistency limit 10
Distinct consistency limit 10

9 equal divisions of 9/8 (abbreviated 9ed9/8) is a nonoctave tuning system that divides the interval of 9/8 into 9 equal parts of about 22.7 ¢ each. Each step represents a frequency ratio of (9/8)1/9, or the 9th root of 9/8. It corresponds to 52.9645 edo, which is closely related to 53edo but with the whole tone instead of the octave tuned pure.

Intervals

# Cents Value Ratio
0 0.0000 1/1
1 22.6567 (9/8)1/9
2 45.3133 (9/8)2/9
3 67.9700 (9/8)1/3
4 90.6267 (9/8)4/9
5 113.2833 (9/8)5/9
6 135.9400 (9/8)2/3
7 158.5967 (9/8)7/9
8 181.2533 (9/8)8/9
9 203.9100 9/8
10 226.5667 (9/8)10/9
11 249.2233 (9/8)11/9
12 271.8800 (9/8)4/3
13 294.5367 (9/8)13/9
14 317.1933 (9/8)14/9
15 339.8500 (9/8)5/3
16 362.5067 (9/8)16/9
17 385.1633 (9/8)17/9
18 407.8200 (9/8)2 = 81/64
19 430.4767 (9/8)19/9
20 453.1333 (9/8)20/9
21 475.7900 (9/8)7/3
22 498.4467 (9/8)22/9
23 521.1033 (9/8)23/9
24 543.7600 (9/8)8/3
25 566.4167 (9/8)25/9
26 589.0733 (9/8)26/9
27 611.7300 (9/8)3 = 729/512
28 634.3867 (9/8)28/9
29 657.0433 (9/8)29/9
30 679.7000 (9/8)10/3
31 702.3567 (9/8)31/9
32 725.0133 (9/8)32/9
33 747.6700 (9/8)11/3
34 770.3267 (9/8)34/9
35 792.9833 (9/8)35/9
36 815.6400 (9/8)4 = 6561/4096
37 838.2967 (9/8)37/9
38 860.9533 (9/8)38/9
39 883.6100 (9/8)13/3
40 906.2667 (9/8)40/9
41 928.9233 (9/8)41/9
42 951.5800 (9/8)14/3
43 974.2367 (9/8)43/9
44 996.8933 (9/8)44/9
45 1019.5500 (9/8)5 = 59049/32768
46 1042.2067 (9/8)46/9
47 1064.8633 (9/8)47/9
48 1087.5200 (9/8)16/3
49 1110.1767 (9/8)49/9
50 1132.8333 (9/8)50/9
51 1155.4900 (9/8)17/3
52 1178.1467 (9/8)52/9
53 1200.8033 (9/8)53/9
54 1223.4600 (9/8)6 = 531441/262144

Approximation to JI

15-odd-limit mappings

The following table shows how 15-odd-limit intervals are represented in 9ed9/8 (ordered by absolute error).

Direct approximation (even if inconsistent)
Interval(s) Error (abs, ¢)
9/8 0.000
3/2, 4/3 0.402
26/15 0.679
15/8, 5/3 0.749
16/9 0.803
13/10 1.081
5/4, 10/9 1.150
15/13 1.482
6/5, 16/15 1.552
20/13 1.884
9/5, 8/5 1.954
13/8, 13/9 2.231
13/12 2.633
16/13, 18/13 3.034
24/13 3.436
12/7 4.206
22/13 4.524
9/7, 8/7 4.607
7/6 5.009
13/11 5.327
7/4, 14/9 5.411
10/7 5.758
22/15 6.006
15/14 6.159
11/10 6.408
7/5 6.561
15/11 6.809
13/7 6.838
28/15 6.963
11/6 7.156
20/11 7.211
11/9, 11/8 7.558
14/13 7.642
12/11 7.960
18/11, 16/11 8.361
14/11 9.688
11/7 10.491
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