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'''18EDF''' is the [[EDF|equal division of the just perfect fifth]] into 18 parts of 38.9975 [[cent|cents]] each, corresponding to 30.7712 [[edo]]. It is related to the regular temperament which tempers out 2401/2400 and 8589934592/8544921875 in the 7-limit; 5632/5625, 46656/46585, and 166698/166375 in the 11-limit, which is supported by [[31edo]], [[369edo]], [[400edo]], 431edo, and 462edo.
{{Infobox ET}}
{{ED intro}}


Lookalikes: [[31edo]], [[49edt]]
== 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.


==Intervals==
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.
{| class="wikitable"
 
=== 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]].
 
=== 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)}}
 
=== Subsets and supersets ===
Since 18 factors into primes as {{nowrap| 2 × 3<sup>2</sup> }}, 18edf has subset edfs {{EDs|equave=f| 2, 3, 6, and 9 }}.
 
== Intervals ==
{| class="wikitable center-1 right-2 mw-collapsible"
|-
|-
! | degree
! #
! | cents value
! Cents
! | corresponding <br>JI intervals
! Approximate ratios
! | comments
|-
|-
| | 0
| 0
| | 0.0000
| 0.0
| | '''exact [[1/1]]'''
| [[1/1]]
| |
|-
|-
| | 1
| 1
| | 38.9975
| 39.0
| | 45/44
| [[33/32]], [[36/35]], [[49/48]], [[50/49]], [[64/63]]
| |
|-
|-
| | 2
| 2
| | 77.9950
| 78.0
| |
| [[21/20]], [[22/21]], [[25/24]], [[28/27]]
| |
|-
|-
| | 3
| 3
| | 116.9925
| 117.0
| |
| [[15/14]], [[16/15]]
| |
|-
|-
| | 4
| 4
| | 155.9900
| 156.0
| | 128/117
| [[11/10]], [[12/11]]
| |
|-
|-
| | 5
| 5
| | 194.9875
| 195.0
| | 28/25
| [[9/8]], [[10/9]]
| |
|-
|-
| | 6
| 6
| | 233.9850
| 234.0
| |
| [[8/7]]
| |
|-
|-
| | 7
| 7
| | 272.9825
| 273.0
| |
| [[7/6]]
| |
|-
|-
| | 8
| 8
| | 311.9800
| 312.0
| |
| [[6/5]]
| |
|-
|-
| | 9
| 9
| | 350.9775
| 351.0
| | 60/49, 49/40
| [[11/9]], [[16/13]]
| |
|-
|-
| | 10
| 10
| | 389.9750
| 390.0
| |
| [[5/4]]
| |
|-
|-
| | 11
| 11
| | 428.9725
| 429.0
| |
| [[9/7]], [[14/11]]
| |
|-
|-
| | 12
| 12
| | 467.9700
| 468.0
| |
| [[13/10]], [[21/16]]
| |
|-
|-
| | 13
| 13
| | 506.9675
| 507.0
| | 75/56
| [[4/3]]
| |
|-
|-
| | 14
| 14
| | 545.9650
| 546.0
| |
| [[11/8]], [[15/11]]
| |
|-
|-
| | 15
| 15
| | 584.9625
| 585.0
| |
| [[7/5]]
| |
|-
|-
| | 16
| 16
| | 623.9600
| 624.0
| |
| [[10/7]]
| |
|-
|-
| | 17
| 17
| | 662.9575
| 663.0
| | [[22/15]]
| [[16/11]], [[22/15]]
| |
|-
|-
| | 18
| 18
| | 701.9550
| 702.0
| | '''exact [[3/2]]'''
| [[3/2]]
| | just perfect fifth
|-
| 19
| 741.0
| [[20/13]], [[32/21]]
|-
| 20
| 780.0
| [[11/7]], [[14/9]]
|-
| 21
| 818.9
| [[8/5]]
|-
| 22
| 857.9
| [[18/11]]
|-
| 23
| 896.9
| [[5/3]]
|-
| 24
| 935.9
| [[12/7]]
|-
| 25
| 974.9
| [[7/4]]
|-
| 26
| 1013.9
| [[9/5]]
|-
| 27
| 1052.9
| [[11/6]]
|-
| 28
| 1091.9
| [[15/8]]
|-
| 29
| 1130.9
| [[27/14]]
|-
| 30
| 1169.9
| [[35/18]], [[49/25]], [[63/32]]
|-
| 31
| 1208.9
| [[2/1]]
|-
| 32
| 1247.9
| [[33/16]], [[45/22]], [[49/24]], [[55/27]]
|-
| 33
| 1286.9
| [[21/10]], [[25/12]]
|-
| 34
| 1325.9
| [[15/7]]
|-
| 35
| 1364.9
| [[11/5]]
|-
| 36
| 1403.9
| [[9/4]]
|}
|}


==Related regular temperaments==
== Related regular temperaments ==
The rank-two regular temperament supported by 31edo and 369edo has three equal divisions of the interval which equals an octave minus the step interval of 18EDF as a generator.
The rank-two regular temperament supported by 31edo and 369edo has three equal divisions of the interval which equals an octave minus the step interval of 18EDF as a generator.


===7-limit 31&amp;369===
=== 7-limit 31 &amp; 369 ===
Commas: 2401/2400, 8589934592/8544921875
Commas: 2401/2400, 8589934592/8544921875


POTE generator: ~5/4 = 386.997
POTE generator: ~5/4 = 386.997


Map: [&lt;1 19 2 7|, &lt;0 -54 1 -13|]
Mapping: [{{map| 1 19 2 7 }}, {{map| 0 -54 1 -13 }}]


EDOs: 31, 369, 400, 431, 462
EDOs: {{EDOs|31, 369, 400, 431, 462}}


===11-limit 31&amp;369===
=== 11-limit 31 &amp; 369 ===
Commas: 2401/2400, 5632/5625, 46656/46585
Commas: 2401/2400, 5632/5625, 46656/46585


POTE generator: ~5/4 = 386.999
POTE generator: ~5/4 = 386.999


Map: [&lt;1 19 2 7 37|, &lt;0 -54 1 -13 -104|]
Mapping: [{{map| 1 19 2 7 37 }}, {{map| 0 -54 1 -13 -104 }}]


EDOs: 31, 369, 400, 431, 462
EDOs: 31, 369, 400, 431, 462


===13-limit 31&amp;369===
=== 13-limit 31 &amp; 369 ===
Commas: 1001/1000, 1716/1715, 4096/4095, 46656/46585
Commas: 1001/1000, 1716/1715, 4096/4095, 46656/46585


POTE generator: ~5/4 = 387.003
POTE generator: ~5/4 = 387.003


Map: [&lt;1 19 2 7 37 -35|, &lt;0 -54 1 -13 -104 120|]
Mapping: [{{map| 1 19 2 7 37 -35 }}, {{map| 0 -54 1 -13 -104 120 }}]


EDOs: 31, 369, 400, 431, 462
EDOs: 31, 369, 400, 431, 462


[[Category:Edf]]
{{Todo|cleanup|expand|inline=1|comment=say what the temperaments are like and why one would want to use them, and for what}}
[[Category:Edonoi]]
 
== 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]]
Retrieved from "https://en.xen.wiki/w/18edf"