9ed9/8: Difference between revisions

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Cleanup; -virtually duplicate stuff from 53edo that fails to undo octave reduction
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{{Infobox ET}}
{{Infobox ET}}
{{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.
{{ED intro}}  


== Harmonics ==
== Theory ==
{{Harmonics in equal
9ed9/8 corresponds to 52.9645…[[edo]], which is closely related to [[53edo]] but with the whole tone instead of the octave tuned pure. Like [[53edo]], 9ed9/8 is [[consistent]] to the [[integer limit|10-integer-limit]].
| steps = 9
 
| num = 9
=== Harmonics ===
| denom = 8
{{Harmonics in equal|9|9|8}}
}}
{{Harmonics in equal|9|9|8|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 9ed9/8 (continued)}}
{{Harmonics in equal
| steps = 9
| num = 9
| denom = 8
| start = 12
| collapsed = 1
}}


== Intervals ==
== Intervals ==
Line 20: Line 13:
|-
|-
! #
! #
! Cents Value
! Cents
! Ratio
! Ratio
|-
|-
Line 243: Line 236:
| (9/8)<sup>6</sup> = 531441/262144
| (9/8)<sup>6</sup> = 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).
{| class="wikitable center-all"
|-
|+ Direct approximation (even if inconsistent)
|-
! Interval(s)
! Error (abs, [[cent|¢]])
|-
| [[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
|}
[[Category:Equal-step tuning]]
[[Category:Edonoi]]
[[Category:Whole tone]]

Revision as of 10:53, 24 March 2025

← 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.

Theory

9ed9/8 corresponds to 52.9645…edo, which is closely related to 53edo but with the whole tone instead of the octave tuned pure. Like 53edo, 9ed9/8 is consistent to the 10-integer-limit.

Harmonics

Approximation of harmonics in 9ed9/8
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.80 +1.21 +1.61 +0.46 +2.01 +7.02 +2.41 +2.41 +1.26 -5.15 +2.81
Relative (%) +3.5 +5.3 +7.1 +2.0 +8.9 +31.0 +10.6 +10.6 +5.6 -22.7 +12.4
Steps
(reduced) 		53
(8) 		84
(3) 		106
(7) 		123
(6) 		137
(2) 		149
(5) 		159
(6) 		168
(6) 		176
(5) 		183
(3) 		190
(1)
Approximation of harmonics in 9ed9/8 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +0.18 +7.82 +1.66 +3.21 -11.12 +3.21 +0.24 +2.06 +8.22 -4.34 +9.33 +3.62
Relative (%) +0.8 +34.5 +7.3 +14.2 -49.1 +14.2 +1.0 +9.1 +36.3 -19.2 +41.2 +16.0
Steps
(reduced) 		196
(7) 		202
(4) 		207
(0) 		212
(5) 		216
(0) 		221
(5) 		225
(0) 		229
(4) 		233
(8) 		236
(2) 		240
(6) 		243
(0)

Intervals

# Cents 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
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