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== Theory ==
== Theory ==
18edf corresponds to [[31edo]] with an [[octave stretching]] of about 9 [[cent]]s. Consequently, it does not provide 31edo's good approximations of most low harmonics, but it provides good approximations to many simple ratios in the thirds region: subminor [[7/6]] (+6{{cent}}), minor [[6/5]] (-3{{cent}}), neutral [[11/9]] (+4{{cent}}), major [[5/4]] (+4{{cent}}), and supermajor [[9/7]] (-6{{cent}}). These intervals may be used to form a variety of [[triad]]s and [[tetrad]]s in close harmony along with the tuning's pure fifth.
In comparison, [[20edf]] (and [[Carlos Gamma]]) offers more accurate pental (minor and major) and undecimal (neutral) thirds, but less accurate septimal (subminor and supermajor) thirds.
=== Regular temperaments ===
18edf is related to the [[regular temperament]] which [[tempering out|tempers out]] 2401/2400 and 8589934592/8544921875 in the [[7-limit]]; with 5632/5625, 46656/46585, and 166698/166375 in the [[11-limit]], which is supported by [[31edo]], [[369edo]], [[400edo]], [[431edo]], and [[462edo]].
18edf is related to the [[regular temperament]] which [[tempering out|tempers out]] 2401/2400 and 8589934592/8544921875 in the [[7-limit]]; with 5632/5625, 46656/46585, and 166698/166375 in the [[11-limit]], which is supported by [[31edo]], [[369edo]], [[400edo]], [[431edo]], and [[462edo]].


Lookalikes: [[31edo]], [[39cET]], [[49edt]], [[72ed5]], [[80ed6]]
=== Harmonics ===
{{Harmonics in equal|18|3|2|intervals=integer|columns=11}}
{{Harmonics in equal|18|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18edf (continued)}}


=== Harmonics ===
=== Subsets and supersets ===
{{Harmonics in equal|18|3|2}}
Since 18 factors into primes as {{nowrap| 2 × 3<sup>2</sup> }}, 18edf has subset edfs {{EDs|equave=f| 2, 3, 6, and 9 }}.
{{Harmonics in equal|18|3|2|start=12|collapsed=1}}


== Intervals ==
== Intervals ==
{| class="wikitable mw-collapsible"
{| class="wikitable center-1 right-2 mw-collapsible"
|+ style="font-size: 105%;" | Intervals of 18edf
|-
|-
! Degree
! #
! Cents value
! Cents
! Corresponding<br />JI intervals
! Approximate ratios
! Comments
|-
|-
! colspan="2" | 0
| 0
| '''exact [[1/1]]'''
| 0.0
|
| [[1/1]]
|-
|-
| 1
| 1
| 38.9975
| 39.0
| 45/44
| [[33/32]], [[36/35]], [[49/48]], [[50/49]], [[64/63]]
|
|-
|-
| 2
| 2
| 77.995
| 78.0
|
| [[21/20]], [[22/21]], [[25/24]], [[28/27]]
|  
|-
|-
| 3
| 3
| 116.9925
| 117.0
| 16/15
| [[15/14]], [[16/15]]
|
|-
|-
| 4
| 4
| 155.99
| 156.0
| 128/117
| [[11/10]], [[12/11]]
|
|-
|-
| 5
| 5
| 194.9875
| 195.0
| 28/25
| [[9/8]], [[10/9]]
|
|-
|-
| 6
| 6
| 233.985
| 234.0
| 8/7
| [[8/7]]
|
|-
|-
| 7
| 7
| 272.9825
| 273.0
| 7/6
| [[7/6]]
|
|-
|-
| 8
| 8
| 311.98
| 312.0
| 6/5
| [[6/5]]
|
|-
|-
| 9
| 9
| 350.9775
| 351.0
| 60/49, 49/40
| [[11/9]], [[16/13]]
|
|-
|-
| 10
| 10
| 389.975
| 390.0
| 5/4
| [[5/4]]
|
|-
|-
| 11
| 11
| 428.9725
| 429.0
| 9/7
| [[9/7]], [[14/11]]
|
|-
|-
| 12
| 12
| 467.97
| 468.0
|
| [[13/10]], [[21/16]]
|  
|-
|-
| 13
| 13
| 506.9675
| 507.0
| 75/56
| [[4/3]]
|
|-
|-
| 14
| 14
| 545.965
| 546.0
|
| [[11/8]], [[15/11]]
|  
|-
|-
| 15
| 15
| 584.9625
| 585.0
|
| [[7/5]]
|  
|-
|-
| 16
| 16
| 623.96
| 624.0
|
| [[10/7]]
|  
|-
|-
| 17
| 17
| 662.9575
| 663.0
| [[22/15]]
| [[16/11]], [[22/15]]
|
|-
|-
| 18
| 18
| 701.955
| 702.0
| '''exact [[3/2]]'''
| [[3/2]]
| just perfect fifth
|-
|-
| 19
| 19
| 740.9525
| 741.0
| 135/88
| [[20/13]], [[32/21]]
|
|-
|-
| 20
| 20
| 779.95
| 780.0
|
| [[11/7]], [[14/9]]
|  
|-
|-
| 21
| 21
| 818.9475
| 818.9
| 8/5
| [[8/5]]
|
|-
|-
| 22
| 22
| 857.945
| 857.9
| 64/39
| [[18/11]]
|
|-
|-
| 23
| 23
| 896.9425
| 896.9
| 42/25
| [[5/3]]
|
|-
|-
| 24
| 24
| 935.94
| 935.9
| 12/7
| [[12/7]]
|
|-
|-
| 25
| 25
| 974.9375
| 974.9
| 7/4
| [[7/4]]
|
|-
|-
| 26
| 26
| 1013.935
| 1013.9
| 9/5
| [[9/5]]
|
|-
|-
| 27
| 27
| 1052.9325
| 1052.9
| 90/49, 147/80
| [[11/6]]
|
|-
|-
| 28
| 28
| 1091.93
| 1091.9
| 15/8
| [[15/8]]
|
|-
|-
| 29
| 29
| 1130.9275
| 1130.9
| 27/14
| [[27/14]]
|
|-
|-
| 30
| 30
| 1169.925
| 1169.9
|
| [[35/18]], [[49/25]], [[63/32]]
|  
|-
|-
| 31
| 31
| 1208.9225
| 1208.9
| 225/112
| [[2/1]]
|
|-
|-
| 32
| 32
| 1247.92
| 1247.9
|
| [[33/16]], [[45/22]], [[49/24]], [[55/27]]
|  
|-
|-
| 33
| 33
| 1286.9175
| 1286.9
|
| [[21/10]], [[25/12]]
|  
|-
|-
| 34
| 34
| 1325.915
| 1325.9
|
| [[15/7]]
|  
|-
|-
| 35
| 35
| 1364.9125
| 1364.9
|
| [[11/5]]
|  
|-
|-
| 36
| 36
| 1403.91
| 1403.9
| '''exact''' 9/4
| [[9/4]]
|
|}
|}


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{{Todo|cleanup|expand|inline=1|comment=say what the temperaments are like and why one would want to use them, and for what}}
{{Todo|cleanup|expand|inline=1|comment=say what the temperaments are like and why one would want to use them, and for what}}
== See also ==
* [[31edo]] – relative edo
* [[49edt]] – relative edt
* [[72ed5]] – relative ed5
* [[80ed6]] – relative ed6
* [[87ed7]] – relative ed7
* [[107ed11]] – relative ed11
* [[111ed12]] – relative ed12
* [[138ed22]] – relative ed22
* [[204ed96]] – close to the zeta-optimized tuning for 31edo
* [[39cET]]
[[Category:31edo]]
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