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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}}


==Intervals==
== 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 [[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.
 
=== Harmonics ===
{{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)}}
 
=== Subsets and supersets ===
9ed9/8 is the first odd composite ed9/8, containing [[3ed9/8]] as a subset.
 
== Intervals ==
{| class="wikitable right-2"
{| class="wikitable right-2"
|-
|-
! degree
! #
! cents value
! Cents
! ratio
! Ratio
|-
|-
| 0
| 0
| 0.0000
| 0.0
| '''[[1/1]]'''
| '''[[1/1]]'''
|-
|-
| 1
| 1
| 22.6567
| 22.7
| (9/8)<sup>1/9</sup>
| (9/8)<sup>1/9</sup>
|-
|-
| 2
| 2
| 45.3133
| 45.3
| (9/8)<sup>2/9</sup>
| (9/8)<sup>2/9</sup>
|-
|-
| 3
| 3
| 67.9700
| 68.0
| (9/8)<sup>1/3</sup>
| (9/8)<sup>1/3</sup>
|-
|-
| 4
| 4
| 90.6267
| 90.6
| (9/8)<sup>4/9</sup>
| (9/8)<sup>4/9</sup>
|-
|-
| 5
| 5
| 113.2833
| 113.3
| (9/8)<sup>5/9</sup>
| (9/8)<sup>5/9</sup>
|-
|-
| 6
| 6
| 135.9400
| 135.9
| (9/8)<sup>2/3</sup>
| (9/8)<sup>2/3</sup>
|-
|-
| 7
| 7
| 158.5967
| 158.6
| (9/8)<sup>7/9</sup>
| (9/8)<sup>7/9</sup>
|-
|-
| 8
| 8
| 181.2533
| 181.3
| (9/8)<sup>8/9</sup>
| (9/8)<sup>8/9</sup>
|-
|-
| 9
| 9
| 203.9100
| 203.9
| '''[[9/8]]'''
| '''[[9/8]]'''
|-
|-
| 10
| 10
| 226.5667
| 226.6
| (9/8)<sup>10/9</sup>
| (9/8)<sup>10/9</sup>
|-
|-
| 11
| 11
| 249.2233
| 249.2
| (9/8)<sup>11/9</sup>
| (9/8)<sup>11/9</sup>
|-
|-
| 12
| 12
| 271.8800
| 271.9
| (9/8)<sup>4/3</sup>
| (9/8)<sup>4/3</sup>
|-
|-
| 13
| 13
| 294.5367
| 294.5
| (9/8)<sup>13/9</sup>
| (9/8)<sup>13/9</sup>
|-
|-
| 14
| 14
| 317.1933
| 317.2
| (9/8)<sup>14/9</sup>
| (9/8)<sup>14/9</sup>
|-
|-
| 15
| 15
| 339.8500
| 339.9
| (9/8)<sup>5/3</sup>
| (9/8)<sup>5/3</sup>
|-
|-
| 16
| 16
| 362.5067
| 362.5
| (9/8)<sup>16/9</sup>
| (9/8)<sup>16/9</sup>
|-
|-
| 17
| 17
| 385.1633
| 385.2
| (9/8)<sup>17/9</sup>
| (9/8)<sup>17/9</sup>
|-
|-
| 18
| 18
| 407.8200
| 407.8
| (9/8)<sup>2</sup> = [[81/64]]
| (9/8)<sup>2</sup> = [[81/64]]
|-
|-
| 19
| 19
| 430.4767
| 430.5
| (9/8)<sup>19/9</sup>
| (9/8)<sup>19/9</sup>
|-
|-
| 20
| 20
| 453.1333
| 453.1
| (9/8)<sup>20/9</sup>
| (9/8)<sup>20/9</sup>
|-
|-
| 21
| 21
| 475.7900
| 475.8
| (9/8)<sup>7/3</sup>
| (9/8)<sup>7/3</sup>
|-
|-
| 22
| 22
| 498.4467
| 498.4
| (9/8)<sup>22/9</sup>
| (9/8)<sup>22/9</sup>
|-
|-
| 23
| 23
| 521.1033
| 521.1
| (9/8)<sup>23/9</sup>
| (9/8)<sup>23/9</sup>
|-
|-
| 24
| 24
| 543.7600
| 543.8
| (9/8)<sup>8/3</sup>
| (9/8)<sup>8/3</sup>
|-
|-
| 25
| 25
| 566.4167
| 566.4
| (9/8)<sup>25/9</sup>
| (9/8)<sup>25/9</sup>
|-
|-
| 26
| 26
| 589.0733
| 589.1
| (9/8)<sup>26/9</sup>
| (9/8)<sup>26/9</sup>
|-
|-
| 27
| 27
| 611.7300
| 611.7
| (9/8)<sup>3</sup> = [[729/512]]
| (9/8)<sup>3</sup> = [[729/512]]
|-
|-
| 28
| 28
| 634.3867
| 634.4
| (9/8)<sup>28/9</sup>
| (9/8)<sup>28/9</sup>
|-
|-
| 29
| 29
| 657.0433
| 657.0
| (9/8)<sup>29/9</sup>
| (9/8)<sup>29/9</sup>
|-
|-
| 30
| 30
| 679.7000
| 679.7
| (9/8)<sup>10/3</sup>
| (9/8)<sup>10/3</sup>
|-
|-
| 31
| 31
| 702.3567
| 702.4
| (9/8)<sup>31/9</sup>
| (9/8)<sup>31/9</sup>
|-
|-
| 32
| 32
| 725.0133
| 725.0
| (9/8)<sup>32/9</sup>
| (9/8)<sup>32/9</sup>
|-
|-
| 33
| 33
| 747.6700
| 747.7
| (9/8)<sup>11/3</sup>
| (9/8)<sup>11/3</sup>
|-
|-
| 34
| 34
| 770.3267
| 770.3
| (9/8)<sup>34/9</sup>
| (9/8)<sup>34/9</sup>
|-
|-
| 35
| 35
| 792.9833
| 792.0
| (9/8)<sup>35/9</sup>
| (9/8)<sup>35/9</sup>
|-
|-
| 36
| 36
| 815.6400
| 815.6
| (9/8)<sup>4</sup> = [[6561/4096]]
| (9/8)<sup>4</sup> = [[6561/4096]]
|-
|-
| 37
| 37
| 838.2967
| 838.3
| (9/8)<sup>37/9</sup>
| (9/8)<sup>37/9</sup>
|-
|-
| 38
| 38
| 860.9533
| 861.0
| (9/8)<sup>38/9</sup>
| (9/8)<sup>38/9</sup>
|-
|-
| 39
| 39
| 883.6100
| 883.6
| (9/8)<sup>13/3</sup>
| (9/8)<sup>13/3</sup>
|-
|-
| 40
| 40
| 906.2667
| 906.3
| (9/8)<sup>40/9</sup>
| (9/8)<sup>40/9</sup>
|-
|-
| 41
| 41
| 928.9233
| 928.9
| (9/8)<sup>41/9</sup>
| (9/8)<sup>41/9</sup>
|-
|-
| 42
| 42
| 951.5800
| 951.6
| (9/8)<sup>14/3</sup>
| (9/8)<sup>14/3</sup>
|-
|-
| 43
| 43
| 974.2367
| 974.2
| (9/8)<sup>43/9</sup>
| (9/8)<sup>43/9</sup>
|-
|-
| 44
| 44
| 996.8933
| 996.9
| (9/8)<sup>44/9</sup>
| (9/8)<sup>44/9</sup>
|-
|-
| 45
| 45
| 1019.5500
| 1019.6
| (9/8)<sup>5</sup> = 59049/32768
| (9/8)<sup>5</sup> = 59049/32768
|-
|-
| 46
| 46
| 1042.2067
| 1042.2
| (9/8)<sup>46/9</sup>
| (9/8)<sup>46/9</sup>
|-
|-
| 47
| 47
| 1064.8633
| 1064.9
| (9/8)<sup>47/9</sup>
| (9/8)<sup>47/9</sup>
|-
|-
| 48
| 48
| 1087.5200
| 1087.5
| (9/8)<sup>16/3</sup>
| (9/8)<sup>16/3</sup>
|-
|-
| 49
| 49
| 1110.1767
| 1110.2
| (9/8)<sup>49/9</sup>
| (9/8)<sup>49/9</sup>
|-
|-
| 50
| 50
| 1132.8333
| 1132.8
| (9/8)<sup>50/9</sup>
| (9/8)<sup>50/9</sup>
|-
|-
| 51
| 51
| 1155.4900
| 1155.5
| (9/8)<sup>17/3</sup>
| (9/8)<sup>17/3</sup>
|-
|-
| 52
| 52
| 1178.1467
| 1178.1
| (9/8)<sup>52/9</sup>
| (9/8)<sup>52/9</sup>
|-
|-
| 53
| 53
| 1200.8033
| 1200.8
| (9/8)<sup>53/9</sup>
| (9/8)<sup>53/9</sup>
|-
|-
| 54
| 54
| 1223.4600
| 1223.5
| (9/8)<sup>6</sup> = 531441/262144
| (9/8)<sup>6</sup> = 531441/262144
|}
|}


== Just approximation ==
== See also ==
=== 15-odd-limit mappings ===
* [[31edf]] – relative edf
The following table shows how [[15-odd-limit intervals]] are represented in 9ed9/8 (ordered by absolute error).
* [[53edo]] – relative edo
 
* [[84edt]] – relative edt
{| class="wikitable center-all"
* [[137ed6]] – relative ed6
|-
|+ Direct mapping (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
|}
 
==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:Edonoi]]
[[Category:Whole tone]]
Retrieved from "https://en.xen.wiki/w/9ed9/8"