Xenharmonic Wiki

37edo: Difference between revisions

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


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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 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.


===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:


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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 ==


{| class="wikitable"
{| class="wikitable center-1 right-2"
|-
|-
! | Degrees
! Degrees
! | Cents
! Cents
! | Approximate Ratios
! Approximate Ratios


of 2.5.7.11.13.27 subgroup
of 2.5.7.11.13.27 subgroup
! | Ratios of 3 with
! Additional Ratios of 3<br>with a sharp 3/2
 
! Additional Ratios of 3<br>with a flat 3/2
a sharp 3/2
! Additional Ratios of 9<br>with 194.59¢ 9/8
! | Ratios of 3 with
 
a flat 3/2
! | Ratios of 9 with
 
194.59¢ 9/8
! | Ratios of 9 with
 
227.03¢ 9/8
 
(two sharp
 
3/2's)
|-
|-
| 0
| 0
|0.00
| 0.00
| | 1/1
| | 1/1
| |
| |  
| |  
| |  
| |  
| |  
| |  
|-
|-
| | 1
| 1
| | 32.43
| 32.43
| |  
|  
| |  
|  
| |  
|  
| |
|  
| |  
|-
|-
| | 2
| 2
| | 64.86
| 64.86
| | 28/27, 27/26
| 28/27, 27/26
| |  
|  
| |  
|  
| |
|  
| |  
|-
|-
| | 3
| 3
| | 97.30
| 97.30
| |
|  
| |  
| |
| |  
|  
| |  
|  
| |  
|-
|-
| | 4
| 4
| | 129.73
| 129.73
| | 14/13
| 14/13
| | 13/12
| 13/12
| | 12/11
| 12/11
| |
|  
| |  
|-
|-
| | 5
| 5
| | 162.16
| 162.16
| | 11/10
| 11/10
| | 12/11
| 10/9, 12/11
| | 13/12
| 13/12
| |
|  
| | 10/9
|-
|-
| | 6
| 6
| | 194.59
| 194.59
| |  
|  
| |  
|  
| |  
|  
| | 9/8, 10/9
| 9/8, 10/9
| |
|-
|-
| | 7
| 7
| | 227.03
| 227.03
| | 8/7
| 8/7
| |
| 9/8
| |  
|  
| |
|  
| | 9/8
|-
|-
| | 8
| 8
| | 259.46
| 259.46
| |  
|  
| | 7/6
| 7/6
| |  
|  
| |
|  
| |  
|-
|-
| | 9
| 9
| | 291.89
| 291.89
| | 13/11, 32/27
| 13/11, 32/27
| |  
|  
| | 6/5, 7/6
| 6/5, 7/6
| |
|  
| |  
|-
|-
| | 10
| 10
| | 324.32
| 324.32
| |  
|  
| | 6/5
| 6/5, 11/9
| |  
|  
| |
|  
| | 11/9
|-
|-
| | 11
| 11
| | 356.76
| 356.76
| | 16/13, 27/22
| 16/13, 27/22
| |  
|  
| |  
|  
| | 11/9
| 11/9
| |
|-
|-
| | 12
| 12
| | 389.19
| 389.19
| | 5/4
| 5/4
| |  
|  
| |  
|  
| |
|  
| |  
|-
|-
| | 13
| 13
| | 421.62
| 421.62
| | 14/11
| 14/11
| |  
|  
| |  
|  
| | 9/7
| 9/7
| |
|-
|-
| | 14
| 14
| | 454.05
| 454.05
| | 13/10
| 13/10
| |
| 9/7
| |  
|  
| |
|  
| | 9/7
|-
|-
| | 15
| 15
| | 486.49
| 486.49
| |  
|  
| | 4/3
| 4/3
| |  
|  
| |
|  
| |  
|-
|-
| | 16
| 16
| | 518.92
| 518.92
| | 27/20
| 27/20
| |  
|  
| | 4/3
| 4/3
| |
|  
| |  
|-
|-
| | 17
| 17
| | 551.35
| 551.35
| | 11/8
| 11/8
| |  
| |
| |  
|  
| | 18/13
| 18/13
| |
|-
|-
| | 18
| 18
| | 583.78
| 583.78
| | 7/5
| 7/5
| |
| 18/13
| |  
|  
| |
|  
| | 18/13
|-
|-
| | 19
| 19
| | 616.22
| 616.22
| | 10/7
| 10/7
| |
| 13/9
| |  
|  
| |
|  
| | 13/9
|-
|-
| | 20
| 20
| | 648.65
| 648.65
| | 16/11
| 16/11
| |  
| |
| |  
|  
| | 13/9
| 13/9
| |
|-
|-
| | 21
| 21
| | 681.08
| 681.08
| | 40/27
| 40/27
| |  
|  
| | 3/2
| 3/2
| |
|  
| |  
|-
|-
| | 22
| 22
| | 713.51
| 713.51
| |  
|  
| | 3/2
| 3/2
| |  
|  
| |
|  
| |  
|-
|-
| | 23
| 23
| | 745.95
| 745.95
| | 20/13
| 20/13
| |
| 14/9
| |  
|  
| |
|  
| | 14/9
|-
|-
| | 24
| 24
| | 778.38
| 778.38
| | 11/7
| 11/7
| |  
|  
| |  
|  
| | 14/9
| 14/9
| |
|-
|-
| | 25
| 25
| | 810.81
| 810.81
| | 8/5
| 8/5
| |  
|  
| |  
|  
| |
|  
| |  
|-
|-
| | 26
| 26
| | 843.24
| 843.24
| | 13/8, 44/27
| 13/8, 44/27
| |  
|  
| |  
|  
| | 18/11
| 18/11
| |
|-
|-
| | 27
| 27
| | 875.68
| 875.68
| |  
|  
| | 5/3
| 5/3, 18/11
| |  
|  
| |
|  
| | 18/11
|-
|-
| | 28
| 28
| | 908.11
| 908.11
| | 22/13, 27/16
| 22/13, 27/16
| |  
|  
| | 5/3, 12/7
| 5/3, 12/7
| |
|  
| |  
|-
|-
| | 29
| 29
| | 940.54
| 940.54
| |  
|  
| | 12/7
| 12/7
| |  
|  
| |
|  
| |  
|-
|-
| | 30
| 30
| | 972.97
| 972.97
| | 7/4
| 7/4
| |
| 16/9
| |  
|  
| |
|  
| | 16/9
|-
|-
| | 31
| 31
| | 1005.41
| 1005.41
| |  
|  
| |  
|  
| |  
|  
| | 16/9, 9/5
| 16/9, 9/5
| |
|-
|-
| | 32
| 32
| | 1037.84
| 1037.84
| | 11/6
| 11/6
| | 24/13
| 9/5, 11/6
| |  
|  
| | 9/5
| |
|
|-
|-
| | 33
| 33
| | 1070.27
| 1070.27
| | 13/7
| 13/7
| | 24/13
| 24/13
| | 11/6
| 11/6
| |
|  
| |  
|-
|-
| | 34
| 34
| | 1102.7
| 1102.70
| |
|  
| |  
| |
| |  
|  
| |  
|  
| |  
|-
|-
| | 35
| 35
| | 1135.14
| 1135.14
| | 27/14, 52/27
| 27/14, 52/27
| |  
|  
| |  
|  
| |
|  
| |  
|-
|-
| | 36
| 36
| | 1167.57
| 1167.57
| |  
|  
| |  
|  
| |  
|  
| |
|  
| |  
|-
|-
|37
| 37
|1200.00
| 1200.00
|2/1
| 2/1
|
|
|
|
|
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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|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|Joe Monzo]]


==Links==
== Links ==
[http://tonalsoft.com/enc/number/37-edo/37edo.aspx 37edo at Tonalsoft]
* [http://tonalsoft.com/enc/number/37-edo/37edo.aspx 37edo at Tonalsoft]


[[Category:37edo| ]] <!-- main articla -->
[[Category:37edo| ]] <!-- main article -->
[[Category:Edo]]
[[Category:Edo]]
[[Category:Prime EDO]]
[[Category:Prime EDO]]
[[Category:Subgroup]]
[[Category:Subgroup]]
Retrieved from "https://en.xen.wiki/w/37edo"