9ed9/8: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Lupus (talk | contribs)
11 edits
Added just approximation
→Theory: expand on its tuning characteristics
 
(11 intermediate revisions by 5 users not shown)
Line 1: Line 1:
'''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.
{{Infobox ET}}
{{ED intro}}


==Intervals==
== Theory ==
{| class="wikitable"
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]], but it has a sharp tendency, with all the [[harmonic]]s within 1 to 16 but [[11/1|11]] tuned sharp.  
|-
 
! | degree
=== Harmonics ===
! | cents value
{{Harmonics in equal|9|9|8}}
! | ratio
{{Harmonics in equal|9|9|8|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 9ed9/8 (continued)}}
|-
 
| | 0
=== Subsets and supersets ===
| | 0.0000
9ed9/8 is the first odd composite ed9/8, containing [[3ed9/8]] as a subset.
| | '''[[1/1]]'''
 
|-
== Intervals ==
| | 1
{| class="wikitable right-2"
| | 22.6567
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">1/9</font>
|-
| | 2
| | 45.3133
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">2/9</font>
|-
| | 3
| | 67.9700
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">1/3</font>
|-
| | 4
| | 90.6267
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">4/9</font>
|-
| | 5
| | 113.2833
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">5/9</font>
|-
| | 6
| | 135.9400
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">2/3</font>
|-
| | 7
| | 158.5967
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">7/9</font>
|-
| | 8
| | 181.2533
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">8/9</font>
|-
| | 9
| | 203.9100
| | '''[[9/8]]'''
|-
| | 10
| | 226.5667
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">10/9</font>
|-
| | 11
| | 249.2233
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">11/9</font>
|-
| | 12
| | 271.8800
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">4/3</font>
|-
| | 13
| | 294.5367
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">13/9</font>
|-
| | 14
| | 317.1933
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">14/9</font>
|-
| | 15
| | 339.8500
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">5/3</font>
|-
| | 16
| | 362.5067
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">16/9</font>
|-
| | 17
| | 385.1633
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">17/9</font>
|-
| | 18
| | 407.8200
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">2</font> = [[81/64]]
|-
| | 19
| | 430.4767
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">19/9</font>
|-
| | 20
| | 453.1333
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">20/9</font>
|-
| | 21
| | 475.7900
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">7/3</font>
|-
| | 22
| | 498.4467
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">22/9</font>
|-
| | 23
| | 521.1033
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">23/9</font>
|-
| | 24
| | 543.7600
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">8/3</font>
|-
| | 25
| | 566.4167
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">25/9</font>
|-
| | 26
| | 589.0733
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">26/9</font>
|-
| | 27
| | 611.7300
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">3</font> = [[729/512]]
|-
| | 28
| | 634.3867
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">28/9</font>
|-
| | 29
| | 657.0433
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">29/9</font>
|-
| | 30
| | 679.7000
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">10/3</font>
|-
| | 31
| | 702.3567
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">31/9</font>
|-
|-
| | 32
! #
| | 725.0133
! Cents
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">32/9</font>
! Ratio
|-
|-
| | 33
| 0
| | 747.6700
| 0.0
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">11/3</font>
| '''[[1/1]]'''
|-
|-
| | 34
| 1
| | 770.3267
| 22.7
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">34/9</font>
| (9/8)<sup>1/9</sup>
|-
|-
| | 35
| 2
| | 792.9833
| 45.3
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">35/9</font>
| (9/8)<sup>2/9</sup>
|-
|-
| | 36
| 3
| | 815.6400
| 68.0
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">4</font> = 6561/4096
| (9/8)<sup>1/3</sup>
|-
|-
| | 37
| 4
| | 838.2967
| 90.6
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">37/9</font>
| (9/8)<sup>4/9</sup>
|-
|-
| | 38
| 5
| | 860.9533
| 113.3
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">38/9</font>
| (9/8)<sup>5/9</sup>
|-
|-
| | 39
| 6
| | 883.6100
| 135.9
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">13/3</font>
| (9/8)<sup>2/3</sup>
|-
|-
| | 40
| 7
| | 906.2667
| 158.6
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">40/9</font>
| (9/8)<sup>7/9</sup>
|-
|-
| | 41
| 8
| | 928.9233
| 181.3
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">41/9</font>
| (9/8)<sup>8/9</sup>
|-
|-
| | 42
| 9
| | 951.5800
| 203.9
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">14/3</font>
| '''[[9/8]]'''
|-
|-
| | 43
| 10
| | 974.2367
| 226.6
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">43/9</font>
| (9/8)<sup>10/9</sup>
|-
|-
| | 44
| 11
| | 996.8933
| 249.2
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">44/9</font>
| (9/8)<sup>11/9</sup>
|-
|-
| | 45
| 12
| | 1019.5500
| 271.9
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">5</font> = 59049/32768
| (9/8)<sup>4/3</sup>
|-
|-
| | 46
| 13
| | 1042.2067
| 294.5
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">46/9</font>
| (9/8)<sup>13/9</sup>
|-
|-
| | 47
| 14
| | 1064.8633
| 317.2
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">47/9</font>
| (9/8)<sup>14/9</sup>
|-
|-
| | 48
| 15
| | 1087.5200
| 339.9
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">16/3</font>
| (9/8)<sup>5/3</sup>
|-
|-
| | 49
| 16
| | 1110.1767
| 362.5
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">49/9</font>
| (9/8)<sup>16/9</sup>
|-
|-
| | 50
| 17
| | 1132.8333
| 385.2
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">50/9</font>
| (9/8)<sup>17/9</sup>
|-
|-
| | 51
| 18
| | 1155.4900
| 407.8
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">17/3</font>
| (9/8)<sup>2</sup> = [[81/64]]
|-
|-
| | 52
| 19
| | 1178.1467
| 430.5
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">52/9</font>
| (9/8)<sup>19/9</sup>
|-
|-
| | 53
| 20
| | 1200.8033
| 453.1
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">53/9</font>
| (9/8)<sup>20/9</sup>
|-
|-
| | 54
| 21
| | 1223.4600
| 475.8
| | (9/8)<font style="vertical-align:super;font-size:0.8em;">6</font> = 531441/262144
| (9/8)<sup>7/3</sup>
|}
 
== Just approximation ==
=== 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 mapping (even if inconsistent)
|-
|-
! Interval(s)
| 22
! Error (abs, [[cent|¢]])
| 498.4
| (9/8)<sup>22/9</sup>
|-
|-
| [[9/8]]
| 23
|0.0
| 521.1
| (9/8)<sup>23/9</sup>
|-
|-
| [[3/2]], [[4/3]]
| 24
|0.402
| 543.8
| (9/8)<sup>8/3</sup>
|-
|-
| [[26/15]]
| 25
|0.679
| 566.4
| (9/8)<sup>25/9</sup>
|-
|-
| [[15/8]], [[5/3]]
| 26
|0.749
| 589.1
| (9/8)<sup>26/9</sup>
|-
|-
| [[16/9]]
| 27
|0.803
| 611.7
| (9/8)<sup>3</sup> = [[729/512]]
|-
|-
| [[13/10]]
| 28
|1.081
| 634.4
| (9/8)<sup>28/9</sup>
|-
|-
| [[5/4]], [[10/9]]
| 29
|1.15
| 657.0
| (9/8)<sup>29/9</sup>
|-
|-
| [[15/13]]
| 30
|1.482
| 679.7
| (9/8)<sup>10/3</sup>
|-
|-
| [[6/5]], [[16/15]]
| 31
|1.552
| 702.4
| (9/8)<sup>31/9</sup>
|-
|-
| [[20/13]]
| 32
|1.884
| 725.0
| (9/8)<sup>32/9</sup>
|-
|-
| [[9/5]], [[8/5]]
| 33
|1.954
| 747.7
| (9/8)<sup>11/3</sup>
|-
|-
| [[13/8]], [[13/9]]
| 34
|2.231
| 770.3
| (9/8)<sup>34/9</sup>
|-
|-
| [[13/12]]
| 35
|2.633
| 792.0
| (9/8)<sup>35/9</sup>
|-
|-
| [[16/13]], [[18/13]]
| 36
|3.034
| 815.6
| (9/8)<sup>4</sup> = [[6561/4096]]
|-
|-
| [[24/13]]
| 37
|3.436
| 838.3
| (9/8)<sup>37/9</sup>
|-
|-
| [[12/7]]
| 38
|4.206
| 861.0
| (9/8)<sup>38/9</sup>
|-
|-
| [[22/13]]
| 39
|4.524
| 883.6
| (9/8)<sup>13/3</sup>
|-
|-
| [[9/7]], [[8/7]]
| 40
|4.607
| 906.3
| (9/8)<sup>40/9</sup>
|-
|-
| [[7/6]]
| 41
|5.009
| 928.9
| (9/8)<sup>41/9</sup>
|-
|-
| [[13/11]]
| 42
|5.327
| 951.6
| (9/8)<sup>14/3</sup>
|-
|-
| [[7/4]], [[14/9]]
| 43
|5.411
| 974.2
| (9/8)<sup>43/9</sup>
|-
|-
| [[10/7]]
| 44
|5.758
| 996.9
| (9/8)<sup>44/9</sup>
|-
|-
| [[22/15]]
| 45
|6.006
| 1019.6
| (9/8)<sup>5</sup> = 59049/32768
|-
|-
| [[15/14]]
| 46
|6.159
| 1042.2
| (9/8)<sup>46/9</sup>
|-
|-
| [[11/10]]
| 47
|6.408
| 1064.9
| (9/8)<sup>47/9</sup>
|-
|-
| [[7/5]]
| 48
|6.561
| 1087.5
| (9/8)<sup>16/3</sup>
|-
|-
| [[15/11]]
| 49
|6.809
| 1110.2
| (9/8)<sup>49/9</sup>
|-
|-
| [[13/7]]
| 50
|6.838
| 1132.8
| (9/8)<sup>50/9</sup>
|-
|-
| [[28/15]]
| 51
|6.963
| 1155.5
| (9/8)<sup>17/3</sup>
|-
|-
| [[11/6]]
| 52
|7.156
| 1178.1
| (9/8)<sup>52/9</sup>
|-
|-
| [[20/11]]
| 53
|7.211
| 1200.8
| (9/8)<sup>53/9</sup>
|-
|-
| [[11/9]], [[11/8]]
| 54
|7.558
| 1223.5
|-
| (9/8)<sup>6</sup> = 531441/262144
| [[14/13]]
|7.642
|-
| [[12/11]]
|7.96
|-
| [[18/11]], [[16/11]]
|8.361
|-
| [[14/11]]
|9.688
|-
| [[11/7]]
|10.491
|}
|}


==See also==
== See also ==
*[http://en.wikipedia.org/wiki/Ottoman_classical_music Ottoman classical music - Wikipedia]
* [[31edf]] – relative edf
*[http://en.wikipedia.org/wiki/Makam Makam - Wikipedia]
* [[53edo]] – relative edo
 
* [[84edt]] – relative edt
[[Category:Equal-step tuning]]
* [[137ed6]] – relative ed6
[[Category:Edonoi]]
[[Category:Whole tone]]

Latest revision as of 14:24, 18 April 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, but it has a sharp tendency, with all the harmonics within 1 to 16 but 11 tuned sharp.

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)

Subsets and supersets

9ed9/8 is the first odd composite ed9/8, containing 3ed9/8 as a subset.

Intervals

# Cents Ratio
0 0.0 1/1
1 22.7 (9/8)1/9
2 45.3 (9/8)2/9
3 68.0 (9/8)1/3
4 90.6 (9/8)4/9
5 113.3 (9/8)5/9
6 135.9 (9/8)2/3
7 158.6 (9/8)7/9
8 181.3 (9/8)8/9
9 203.9 9/8
10 226.6 (9/8)10/9
11 249.2 (9/8)11/9
12 271.9 (9/8)4/3
13 294.5 (9/8)13/9
14 317.2 (9/8)14/9
15 339.9 (9/8)5/3
16 362.5 (9/8)16/9
17 385.2 (9/8)17/9
18 407.8 (9/8)2 = 81/64
19 430.5 (9/8)19/9
20 453.1 (9/8)20/9
21 475.8 (9/8)7/3
22 498.4 (9/8)22/9
23 521.1 (9/8)23/9
24 543.8 (9/8)8/3
25 566.4 (9/8)25/9
26 589.1 (9/8)26/9
27 611.7 (9/8)3 = 729/512
28 634.4 (9/8)28/9
29 657.0 (9/8)29/9
30 679.7 (9/8)10/3
31 702.4 (9/8)31/9
32 725.0 (9/8)32/9
33 747.7 (9/8)11/3
34 770.3 (9/8)34/9
35 792.0 (9/8)35/9
36 815.6 (9/8)4 = 6561/4096
37 838.3 (9/8)37/9
38 861.0 (9/8)38/9
39 883.6 (9/8)13/3
40 906.3 (9/8)40/9
41 928.9 (9/8)41/9
42 951.6 (9/8)14/3
43 974.2 (9/8)43/9
44 996.9 (9/8)44/9
45 1019.6 (9/8)5 = 59049/32768
46 1042.2 (9/8)46/9
47 1064.9 (9/8)47/9
48 1087.5 (9/8)16/3
49 1110.2 (9/8)49/9
50 1132.8 (9/8)50/9
51 1155.5 (9/8)17/3
52 1178.1 (9/8)52/9
53 1200.8 (9/8)53/9
54 1223.5 (9/8)6 = 531441/262144

See also

Retrieved from "https://en.xen.wiki/index.php?title=9ed9/8&oldid=192846"