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== Representations of Just Intervals ==
== Representations of Just Intervals ==
{| class="wikitable"
{| class="wikitable"
! Degree
! Cents
! Nearest Ratio
! Error
! 17-Limit Ratios <ref>based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament</ref>
|-
|-
! | Degree
! | Cents
! | Nearest Ratio
! | Error
! | 17-Limit Ratios*
|-
| 0
| 0
|0.00
| 0.00
| | 1/1
| 1/1
| | 0
| 0
| | 1/1
| 1/1
|-
|-
| | 1
| 1
| | 66.67
| 66.67
| | 27/26
| 27/26
| | +1.329
| +1.329
| | 78/75, 75/72
| 78/75, 75/72
|-
|-
| | 2
| 2
| | 133.33
| 133.33
| | 27/25
| 27/25
| | +0.096
| +0.096
| | 51/55, 42/39
| 51/55, 42/39
|-
|-
| | 3
| 3
| | 200
| 200
| | 9/8
| 9/8
| | -3.910
| -3.910
| | 9/8
| 9/8
|-
|-
| | 4
| 4
| | 266.67
| 266.67
| | 7/6
| 7/6
| | -0.204
| -0.204
| | 75/64
| 75/64
|-
|-
| | 5
| 5
| | 333.33
| 333.33
| | 17/14 or 40/33
| 17/14 or 40/33
| | -2.796 +0.293
| -2.796 +0.293
| | 39/32
| 39/32
|-
|-
| | 6
| 6
| | 400
| 400
| | 5/4 or 44/35
| 5/4 or 44/35
| | +13.686 +3.822
| +13.686 +3.822
| | 64/55
| 64/55
|-
|-
| | 7
| 7
| | 466.67
| 466.67
| | 21/16
| 21/16
| | -4.114
| -4.114
| | 21/16
| 21/16
|-
|-
| | 8
| 8
| | 533.33
| 533.33
| | 15/11
| 15/11
| | -3.617
| -3.617
| | 102/75
| 102/75
|-
|-
| | 9
| 9
| | 600
| 600
| | 17/12 or 24/17
| 17/12 or 24/17
| | -3.000 +3.000
| -3.000 +3.000
| | 17/12
| 17/12
|-
|-
| | 10
| 10
| | 666.67
| 666.67
| | 22/15
| 22/15
| | +3.617
| +3.617
| | 75/51
| 75/51
|-
|-
| | 11
| 11
| | 733.33
| 733.33
| | 32/21
| 32/21
| | +4.114
| +4.114
| | 32/21
| 32/21
|-
|-
| | 12
| 12
| | 800
| 800
| | 8/5 or 35/22
| 8/5 or 35/22
| | -13.686 -3.822
| -13.686 -3.822
| | 51/32
| 51/32
|-
|-
| | 13
| 13
| | 866.67
| 866.67
| | 28/17 or 33/20
| 28/17 or 33/20
| | +2.796 -0.293
| +2.796 -0.293
| | 64/39
| 64/39
|-
|-
| | 14
| 14
| | 933.33
| 933.33
| | 12/7
| 12/7
| | +0.204
| +0.204
| | 55/32
| 55/32
|-
|-
| | 15
| 15
| | 1000
| 1000
| | 16/9
| 16/9
| | +3.910
| +3.910
| | 16/9
| 16/9
|-
|-
| | 16
| 16
| | 1066.67
| 1066.67
| | 50/27
| 50/27
| | -0.096
| -0.096
| | 39/21
| 39/21
|-
|-
| | 17
| 17
| | 1133.33
| 1133.33
| | 52/27
| 52/27
| | -1.329
| -1.329
| | 75/39
| 75/39
|-
|-
| | 18
| 18
| | 1200
| 1200
| | 2/1
| 2/1
| | 0
| 0
| | 2/1**
| 2/1**
|}
|}
*based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament
<references/>


[[File:18-ED2-JI-approximations-2.png|alt=18-ED2-JI-approximations-2.png|18-ED2-JI-approximations-2.png]]
[[File:18-ED2-JI-approximations-2.png|alt=18-ED2-JI-approximations-2.png|18-ED2-JI-approximations-2.png]]


== Commas ==
== Commas ==
18 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes the [[val|val]] &lt; 18 29 42 51 62 67 |.)


{| class="wikitable"
18 EDO [[tempering out|tempers out]] the following [[Comma|commas]]. (Note: This assumes the [[val]] &lt; 18 29 42 51 62 67 |.)
|-
 
! | [[Ratio]]
{| class="wikitable" style="text-align:center;"
! | [[Monzo]]
! [[Ratio]]
! | [[Cents]]
! [[Monzo]]
![[Color notation/Temperament Names|Color Name]]
! [[Cents]]
! | Name 1
! [[Color notation/Temperament Names|Color Name]]
! | Name 2
! Name 1
! Name 2
|-
|-
| style="text-align:center;" | 128/125
| 128/125
| |<nowiki> | 7 0 -3 </nowiki>&gt;
| style="text-align:left;" | <nowiki> | 7 0 -3 </nowiki>&gt;
| style="text-align:right;" | 41.06
| style="text-align:right;" | 41.06
| style="text-align:center;" |Trigu
| Trigu
| style="text-align:center;" | Diesis
| Diesis
| style="text-align:center;" | Augmented Comma
| Augmented Comma
|-
|-
| style="text-align:center;" |  
| style="text-align:center;" | ???
| |<nowiki> | 23 6 -14 </nowiki>&gt;
| style="text-align:left;" | <nowiki> | 23 6 -14 </nowiki>&gt;
| style="text-align:right;" | 3.34
| style="text-align:right;" | 3.34
| style="text-align:center;" |Sasa-sepbigu
| Sasa-sepbigu
| style="text-align:center;" | Vishnuzma
| Vishnuzma
| style="text-align:center;" | Semisuper
| Semisuper
|-
|-
| style="text-align:center;" | 50/49
| 50/49
| |<nowiki> | 1 0 2 -2 </nowiki>&gt;
| style="text-align:left;" | <nowiki> | 1 0 2 -2 </nowiki>&gt;
| style="text-align:right;" | 34.98
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| Biruyo
| style="text-align:center;" | Tritonic Diesis
| Tritonic Diesis
| style="text-align:center;" | Jubilisma
| Jubilisma
|-
|-
| style="text-align:center;" | 686/675
| 686/675
| |<nowiki> | 1 -3 -2 3 </nowiki>&gt;
| style="text-align:left;" | <nowiki> | 1 -3 -2 3 </nowiki>&gt;
| style="text-align:right;" | 27.99
| style="text-align:right;" | 27.99
| style="text-align:center;" |Trizo-agugu
| Trizo-agugu
| style="text-align:center;" | Senga
| Senga
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 875/864
| 875/864
| |<nowiki> | -5 -3 3 1 </nowiki>&gt;
| style="text-align:left;" | {{Monzo| -5 -3 3 1 }}
| style="text-align:right;" | 21.90
| style="text-align:right;" | 21.90
| style="text-align:center;" |Zotriyo
| Zotriyo
| style="text-align:center;" | Keema
| Keema
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 1728/1715
| 1728/1715
| |<nowiki> | 6 3 -1 -3 </nowiki>&gt;
| style="text-align:left;" | {{Monzo| 6 3 -1 -3 }}
| style="text-align:right;" | 13.07
| style="text-align:right;" | 13.07
| style="text-align:center;" |Triru-agu
| Triru-agu
| style="text-align:center;" | Orwellisma
| Orwellisma
| style="text-align:center;" | Orwell Comma
| Orwell Comma
|-
|-
| style="text-align:center;" | 16875/16807
| 16875/16807
| |<nowiki> | 0 3 4 -5 </nowiki>&gt;
| style="text-align:left;" | {{Monzo| 0 3 4 -5 }}
| style="text-align:right;" | 6.99
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| Mirkwai
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 3136/3125
| 3136/3125
| |<nowiki> | 6 0 -5 2 </nowiki>&gt;
| style="text-align:left;" | {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:right;" | 6.08
| style="text-align:center;" |Zozoquingu
| Zozoquingu
| style="text-align:center;" | Hemimean
| Hemimean
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 99/98
| 99/98
| |<nowiki> | -1 2 0 -2 1 </nowiki>&gt;
| style="text-align:left;" | {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:right;" | 17.58
| style="text-align:center;" |Loruru
| Loruru
| style="text-align:center;" | Mothwellsma
| Mothwellsma
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 100/99
| 100/99
| |<nowiki> | 2 -2 2 0 -1 </nowiki>&gt;
| style="text-align:left;" | {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:right;" | 17.40
| style="text-align:center;" |Luyoyo
| Luyoyo
| style="text-align:center;" | Ptolemisma
| Ptolemisma
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 65536/65219
| 65536/65219
| |<nowiki> | 16 0 0 -2 -3 </nowiki>&gt;
| style="text-align:left;" | {{Monzo| 16 0 0 -2 -3 }}
| style="text-align:right;" | 8.39
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| Satrilu-aruru
| style="text-align:center;" | Orgonisma
| Orgonisma
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 385/384
| 385/384
| |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
| style="text-align:left;" | {{Monzo| -7 -1 1 1 1 }}
| style="text-align:right;" | 4.50
| style="text-align:right;" | 4.50
| style="text-align:center;" |Lozoyo
| Lozoyo
| style="text-align:center;" | Keenanisma
| Keenanisma
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 9801/9800
| 9801/9800
| |<nowiki> | -3 4 -2 -2 2 </nowiki>&gt;
| style="text-align:left;" | {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:right;" | 0.18
| style="text-align:center;" |Bilorugu
| Bilorugu
| style="text-align:center;" | Kalisma
| Kalisma
| style="text-align:center;" | Gauss' Comma
| Gauss' Comma
|-
|-
| style="text-align:center;" | 91/90
| 91/90
| |<nowiki> | -1 -2 -1 1 0 1 </nowiki>&gt;
| style="text-align:left;" | {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| Thozogu
| style="text-align:center;" | Superleap
| Superleap
| style="text-align:center;" |
|
|}
|}


== Useful Moment-of-Symmetry Scales ==
== Useful Moment-of-Symmetry Scales ==
Note: This list excludes scales found in 9-EDO.


<span style="font-size: 1.1em;">'''<u>Pentatonic:</u>'''</span>
Note: This list excludes scales found in [[9-EDO]].
 
=== Pentatonic ===


3L2s Father Pentatonic: 4 4 3 4 3
3L2s Father Pentatonic: 4 4 3 4 3


<span style="font-size: 1.1em;">'''<u>Hexatonic:</u>'''</span>
=== Hexatonic ===


4L2s Bicycle: 4 4 1 4 4 1
4L2s Bicycle: 4 4 1 4 4 1
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2L4s Rice Hexatonic: 2 5 2 2 5 2
2L4s Rice Hexatonic: 2 5 2 2 5 2


<span style="font-size: 1.1em;">'''<u>Heptatonic:</u>'''</span>
=== Heptatonic ===


4L3s Amity/Mish Heptatonic: 3 2 3 2 3 3 2
4L3s Amity/Mish Heptatonic: 3 2 3 2 3 3 2


<span style="font-size: 1.1em;">'''<u>Octatonic:</u>'''</span>
=== Octatonic ===


5L3s Father Octatonic: 3 1 3 3 1 3 3 1
5L3s Father Octatonic: 3 1 3 3 1 3 3 1
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2L6s Rice Octatonic: 2 2 3 2 2 2 3 2
2L6s Rice Octatonic: 2 2 3 2 2 2 3 2


<span style="font-size: 1.1em;">'''<u>Decatonic:</u>'''</span>
=== Decatonic ===


8L2s Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2
8L2s Biggie Decatonic: 2 2 1 2 2 2 2 1 2 2


<span style="font-size: 1.1em;">'''<u>Dodecatonic:</u>'''</span>
=== Dodecatonic ===


6L 6s Hexe: 2 1 2 1 2 1 2 1 2 1 2 1
6L 6s Hexe: 2 1 2 1 2 1 2 1 2 1 2 1


== Application to Guitar ==
== Application to Guitar ==
18-EDO is an ideal scale for the first-time refretter, because you can retain all the even-number frets from 12-tET--essentially 1/3 of your work is done for you!
18-EDO is an ideal scale for the first-time refretter, because you can retain all the even-number frets from 12-tET--essentially 1/3 of your work is done for you!


Line 532: Line 534:
* [http://www.h-pi.com/mp3/18ETPrelude.mp3 18ETPrelude] by [[Aaron Andrew Hunt]] {{dead link}}
* [http://www.h-pi.com/mp3/18ETPrelude.mp3 18ETPrelude] by [[Aaron Andrew Hunt]] {{dead link}}
* [https://soundcloud.com/uz1kt3k/fuga-a3-in-18et Fuga a3 in 18ET] by Aaron Andrew Hunt  
* [https://soundcloud.com/uz1kt3k/fuga-a3-in-18et Fuga a3 in 18ET] by Aaron Andrew Hunt  
* [http://micro.soonlabel.com/18-ET/prelude-in-18et.mp3 Prelude in 18et] by [http://www.chrisvaisvil.com Chris Vaisvil] =&gt; [http://chrisvaisvil.com/?p=3 composer notes]
* [http://micro.soonlabel.com/18-ET/prelude-in-18et.mp3 Prelude in 18et] by [http://www.chrisvaisvil.com Chris Vaisvil] &rarr; [http://chrisvaisvil.com/?p=3 composer notes]
* [http://micro.soonlabel.com/18-ET/daily20110401-18c-flippertronics.mp3 Flippertronics] by Chris Vaisvil
* [http://micro.soonlabel.com/18-ET/daily20110401-18c-flippertronics.mp3 Flippertronics] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/9-edo/daily20111008b_gerbils_at_the_wheel_of_government.mp3 Gerbils at the Wheel of Government] by [http://chrisvaisvil.com/?p=1402 Chris Vaisvil (in 9 and 18 edo simultaneously)]
* [http://micro.soonlabel.com/9-edo/daily20111008b_gerbils_at_the_wheel_of_government.mp3 Gerbils at the Wheel of Government] by [http://chrisvaisvil.com/?p=1402 Chris Vaisvil (in 9 and 18 edo simultaneously)]
* [http://www.seraph.it/dep/det/DoAndroidsDreamof18ED2.mp3.mp3 Do Androids Dream Of 18ED2?] by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/fb0306486b51c270607f90a0c795d531-202.html blog entry])
* [http://www.seraph.it/dep/det/DoAndroidsDreamof18ED2.mp3.mp3 Do Androids Dream Of 18ED2?] by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/fb0306486b51c270607f90a0c795d531-202.html blog entry])
* [https://soundcloud.com/tomprice719/composition-of-june-2015 Composition of June 2015 by TomPrice719]
* [https://soundcloud.com/tomprice719/composition-of-june-2015 Composition of June 2015 by TomPrice719]


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