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| Augmented 1sn = 6\37 = 195¢
| Augmented 1sn = 6\37 = 195¢
}}
}}
'''37edo''' is a scale derived from dividing the octave into 37 equal steps. It is the 12th [[prime_numbers|prime]] edo, following [[31edo]] and coming before [[41edo]].
'''37EDO''' is a scale derived from dividing the octave into 37 equal steps. It is the 12th [[prime_numbers|prime]] EDO, following [[31edo|31EDO]] and coming before [[41edo|41EDO]].


== Theory ==
== Theory ==
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Using its best (and sharp) fifth, 37edo tempers out 250/243, making it a variant of [[porcupine]] temperament. It is the optimal patent val for [[Porcupine_family#Porcupinefish|porcupinefish]], which is about as accurate as "13-limit porcupine" will be. Using its alternative flat fifth, it tempers out 16875/16384, making it a [[Negri|negri]] tuning. It also tempers out 2187/2000, resulting in a temperament where three minor whole tones make up a fifth ([[gorgo]]/[[laconic]]).
Using its best (and sharp) fifth, 37EDO tempers out 250/243, making it a variant of [[porcupine]] temperament. It is the optimal patent val for [[Porcupine family #Porcupinefish|porcupinefish]], which is about as accurate as "13-limit porcupine" will be. Using its alternative flat fifth, it tempers out 16875/16384, making it a [[Negri|negri]] tuning. It also tempers out 2187/2000, resulting in a temperament where three minor whole tones make up a fifth ([[gorgo]]/[[laconic]]).


37edo is also a very accurate equal tuning for [[undecimation]] temperament, which has a generator of about 519 cents; 2 generators lead to 29/16; 3 generators to 32/13; 6 generators to a 10 cent sharp 6/1; 8 generators to a very accurate 11/1 and 10 generators to 20/1. It has a 7L+2s nonatonic MOS, which in 37-edo scale degrees is 0, 1, 6, 11, 16, 17, 22, 27, 32, a scale structure reminiscent of mavila; as well as a 16 note MOS.
37EDO is also a very accurate equal tuning for [[undecimation]] temperament, which has a generator of about 519 cents; 2 generators lead to 29/16; 3 generators to 32/13; 6 generators to a 10 cent sharp 6/1; 8 generators to a very accurate 11/1 and 10 generators to 20/1. It has a 7L+2s nonatonic MOS, which in 37EDO scale degrees is 0, 1, 6, 11, 16, 17, 22, 27, 32, a scale structure reminiscent of mavila; as well as a 16 note MOS.


=== Subgroups ===
=== Subgroups ===
37edo offers close approximations to [[Overtone series|harmonics]] 5, 7, 11, and 13 [and a usable approximation of 9 as well].
37EDO offers close approximations to [[Overtone series|harmonics]] 5, 7, 11, and 13 [and a usable approximation of 9 as well].


12\37 = 389.2 cents
12\37 = 389.2 cents
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26\37 = 843.2 cents
26\37 = 843.2 cents


[6\37edo = 194.6 cents]
[6\37 = 194.6 cents]


This means 37 is quite accurate on the 2.5.7.11.13 subgroup, where it shares the same tuning as 111et. In fact, on the larger [[k*N_subgroups|3*37 subgroup]] 2.27.5.7.11.13.51.57 subgroup not only shares the same tuning as 19-limit 111et, it tempers out the same commas. A simpler but less accurate approach is to use the 2*37-subgroup, 2.9.7.11.13.17.19, on which it has the same tuning and commas as 74et.
This means 37 is quite accurate on the 2.5.7.11.13 subgroup, where it shares the same tuning as 111EDO. In fact, on the larger [[k*N_subgroups|3*37 subgroup]] 2.27.5.7.11.13.51.57 subgroup not only shares the same tuning as 19-limit 111EDO, it tempers out the same commas. A simpler but less accurate approach is to use the 2*37-subgroup, 2.9.7.11.13.17.19, on which it has the same tuning and commas as 74EDO.


=== The Two Fifths ===
=== The Two Fifths ===
The just [[perfect fifth]] of frequency ratio 3:2 is not well-approximated, and falls between two intervals in 37edo:
The just [[perfect fifth]] of frequency ratio 3:2 is not well-approximated, and falls between two intervals in 37EDO:


The flat fifth is 21\37 = 681.1 cents (37b val)
The flat fifth is 21\37 = 681.1 cents (37b val)
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If the minor third of 259.5 cents is mapped to 7/6, this superpythagorean scale can be thought of as a variant of [[The_Biosphere|Biome]] temperament.
If the minor third of 259.5 cents is mapped to 7/6, this superpythagorean scale can be thought of as a variant of [[The_Biosphere|Biome]] temperament.


Interestingly, the "major thirds" of both systems are not 12\37 = 389.2¢, the closest approximation to 5/4 available in 37edo.
Interestingly, the "major thirds" of both systems are not 12\37 = 389.2¢, the closest approximation to 5/4 available in 37EDO.


37edo has great potential as a near-just xenharmonic system, with high-prime chords such as 8:10:11:13:14 with no perfect fifths available for common terrestrial progressions. The 9/8 approximation is usable but introduces error. One may choose to treat either of the intervals close to 3/2 as 3/2, introducing additional approximations with considerable error (see interval table below).
37EDO has great potential as a near-just xenharmonic system, with high-prime chords such as 8:10:11:13:14 with no perfect fifths available for common terrestrial progressions. The 9/8 approximation is usable but introduces error. One may choose to treat either of the intervals close to 3/2 as 3/2, introducing additional approximations with considerable error (see interval table below).


== Intervals ==
== Intervals ==
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| 1
| 1
| 32.43
| 32.43
|
| [[55/54]], [[56/55]]
|
|
|
|
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| 97.30
| 97.30
| [[55/52]]
| [[55/52]]
|
| [[16/15]]
|
|
|
|
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| 129.73
| 129.73
| [[14/13]]
| [[14/13]]
| [[13/12]]
| [[13/12]], [[15/14]]
| [[12/11]]
| [[12/11]]
|
|
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| 259.46
| 259.46
|
|
| [[7/6]]
| [[7/6]], [[15/13]]
|
|
|
|
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| 551.35
| 551.35
| [[11/8]]
| [[11/8]]
|
| [[15/11]]
|
|
| [[18/13]]
| [[18/13]]
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| 648.65
| 648.65
| [[16/11]]
| [[16/11]]
|
| [[22/15]]
|
|
| [[13/9]]
| [[13/9]]
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| 940.54
| 940.54
|
|
| [[12/7]]
| [[12/7]], [[26/15]]
|
|
|
|
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| 1070.27
| 1070.27
| [[13/7]]
| [[13/7]]
| [[24/13]]
| [[24/13]], [[28/15]]
| [[11/6]]
| [[11/6]]
|
|
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| 1102.70
| 1102.70
| [[104/55]]
| [[104/55]]
|
| [[15/8]]
|
|
|
|
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=== Temperament measures ===
=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 37et.  
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 37EDO.  
{| class="wikitable center-all"
{| class="wikitable center-all"
! colspan="2" |
! colspan="2" |
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|}
|}


* 37et is most prominent in the no-3 11-, 13-, 17-, 19- and 23-limit subgroups. The next ET that does better in these subgroups is 109, 581, 103, 124 and 93, respectively.  
* 37EDO is most prominent in the no-3 11-, 13-, 17-, 19- and 23-limit subgroups. The next EDO that does better in these subgroups is 109, 581, 103, 124 and 93, respectively.  


== Scales ==
== Scales ==
* [[MOS_Scales_of_37edo|MOS Scales of 37edo]]
* [[MOS_Scales_of_37edo|MOS Scales of 37edo]]
* [[roulette6]]
* [[roulette6]]
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|-
|-
| 5\37
| 5\37
| [[Porcupine]]/[[The_Biosphere#Oceanfront-Oceanfront Children-Porcupinefish|porcupinefish]]
| [[Porcupine]]/[[The Biosphere #Oceanfront-Oceanfront Children-Porcupinefish|porcupinefish]]
|  
|  
|-
|-
| 6\37
| 6\37
| colspan="2" | [[Chromatic_pairs#Roulette|Roulette]]
| colspan="2" | [[Chromatic pairs #Roulette|Roulette]]
|-
|-
| 7\37
| 7\37
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| 9\37
| 9\37
|  
|  
| [[Chromatic_pairs#Gariberttet|Gariberttet]]
| [[Chromatic pairs #Gariberttet|Gariberttet]]
|-
|-
| 10\37
| 10\37
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|-
|-
| 13\37
| 13\37
| [[Meantone_family#Squares|Squares]]
| [[Meantone family #Squares|Squares]]
|  
|  
|-
|-
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|-
|-
| 15\37
| 15\37
| [[The_Biosphere#Oceanfront-Oceanfront Children-Ultrapyth|Ultrapyth]], '''not''' [[superpyth]]
| [[The Biosphere#Oceanfront-Oceanfront Children-Ultrapyth|Ultrapyth]], '''not''' [[superpyth]]
|  
|  
|-
|-
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== Music ==
== Music ==
* [http://www.akjmusic.com/audio/toccata_bianca_37edo.mp3 Toccata Bianca 37edo] by [http://www.akjmusic.com/ Aaron Krister Johnson]
* [http://www.akjmusic.com/audio/toccata_bianca_37edo.mp3 Toccata Bianca 37EDO] by [http://www.akjmusic.com/ Aaron Krister Johnson]
* [http://andrewheathwaite.bandcamp.com/track/shorn-brown Shorn Brown] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2002%20Shorn%20Brown.mp3 play] and [http://andrewheathwaite.bandcamp.com/track/jellybear Jellybear] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2003%20Jellybear.mp3 play] by [[Andrew Heathwaite]]
* [http://andrewheathwaite.bandcamp.com/track/shorn-brown Shorn Brown] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2002%20Shorn%20Brown.mp3 play] and [http://andrewheathwaite.bandcamp.com/track/jellybear Jellybear] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2003%20Jellybear.mp3 play] by [[Andrew Heathwaite]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Monzo/monzo_kog-sisters_2014-0405.mp3 The Kog Sisters] by [[Joe Monzo]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Monzo/monzo_kog-sisters_2014-0405.mp3 The Kog Sisters] by [[Joe Monzo]]
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== Links ==
== Links ==
* [http://tonalsoft.com/enc/number/37-edo/37edo.aspx 37edo at Tonalsoft] [[Category:37edo| ]] <!-- main article -->
* [http://tonalsoft.com/enc/number/37-edo/37edo.aspx 37edo at Tonalsoft]
[[Category:37edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
[[Category:Prime EDO]]
[[Category:Prime EDO]]
[[Category:Subgroup]]
[[Category:Subgroup]]
Retrieved from "https://en.xen.wiki/w/37edo"