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'''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]]
Retrieved from "https://en.xen.wiki/w/9ed9/8"