37edo: Difference between revisions

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<span style="display: block; text-align: right;">[[:de:37edo|Deutsch]]</span>

37edo is a scale derived from dividing the octave into 37 equal steps of approximately 32.43 cents each. It is the 12th [[prime_numbers|prime]] edo, following [[31edo|31edo]] and coming before [[41edo|41edo]].

'''37edo''' is a scale derived from dividing the octave into 37 equal steps of approximately 32.43 cents each. It is the 12th [[prime_numbers|prime]] edo, following [[31edo|31edo]] and coming before [[41edo|41edo]].

Using its best (and sharp) fifth, 37edo tempers out 250/243, making it a variant of [[Porcupine|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|gorgo]]/[[laconic|laconic]]).

== Theory ==

Using its best (and sharp) fifth, 37edo tempers out 250/243, making it a variant of [[Porcupine|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|gorgo]]/[[laconic|laconic]]).

~~37 edo~~ 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 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.

__FORCETOC__

~~-----~~

===Subgroups===

=Subgroups=

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.

=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:

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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).

=Intervals=

==Intervals==

{| class="wikitable"

|-

! | Degrees ~~of 37edo~~

! | Degrees

! | Cents ~~Value~~

! | Cents

! | Approximate Ratios

Line 93:

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|-

| | 2

| | 64.~~865~~

| | 64.86

| | 28/27, 27/26

| |

Line 101:

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|-

| | 3

| | 97.3

| | 97.30

| |

Line 125:

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|-

| | 6

| | 194.~~595~~

| | 194.59

| |

Line 325:

Line 324:

|-

| | 31

| | 1005.~~405~~

| | 1005.41

| |

Line 357:

Line 356:

|-

| | 35

| | 1135.~~135~~

| | 1135.14

| | 27/14, 52/27

| |

Line 373:

Line 372:

|-

|37

|1200

|1200.00

|2/1

|

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|}

=Scales=

==Scales==

[[MOS_Scales_of_37edo|MOS Scales of 37edo]]

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[[square_root_of_13_over_10|The Square Root of 13/10]]

=Linear temperaments=

==Linear temperaments==

[[List_of_37et_rank_two_temperaments_by_badness|List of 37et rank two temperaments by badness]]

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|}

=Music ~~in 37edo~~=

==Music==

[http://www.akjmusic.com/audio/toccata_bianca_37edo.mp3 Toccata Bianca 37edo] by [http://www.akjmusic.com/ Aaron Krister Johnson]

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[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]

@@ Line 1: / Line 1: @@
 <span style="display: block; text-align: right;">[[:de:37edo|Deutsch]]</span>
-edo is a scale derived from dividing the octave into 37 equal steps of approximately 32.43 cents each. It is the 12th [[prime_numbers|prime]] edo, following [[31edo|31edo]] and coming before [[41edo|41edo]].
+'''37edo''' is a scale derived from dividing the octave into 37 equal steps of approximately 32.43 cents each. It is the 12th [[prime_numbers|prime]] edo, following [[31edo|31edo]] and coming before [[41edo|41edo]].
-Using its best (and sharp) fifth, 37edo tempers out 250/243, making it a variant of [[Porcupine|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|gorgo]]/[[laconic|laconic]]).
+== Theory ==
+Using its best (and sharp) fifth, 37edo tempers out 250/243, making it a variant of [[Porcupine|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|gorgo]]/[[laconic|laconic]]).
-edo 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.
+edo 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.
 __FORCETOC__
------
+===Subgroups===
-=Subgroups=
 edo offers close approximations to [[OverToneSeries|harmonics]] 5, 7, 11, and 13 [and a usable approximation of 9 as well].
@@ Line 25: / Line 24: @@
 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:
@@ Line 50: / Line 49: @@
 edo 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"
 |-
-! | Degrees of 37edo
+! | Degrees
-! | Cents Value
+! | Cents
 ! | Approximate Ratios
@@ Line 93: / Line 92: @@
 |-
 | | 2
-| | 64.865
+| | 64.86
 | | 28/27, 27/26
 | |
@@ Line 101: / Line 100: @@
 |-
 | | 3
-| | 97.3
+| | 97.30
 | |
 | |
@@ Line 125: / Line 124: @@
 |-
 | | 6
-| | 194.595
+| | 194.59
 | |
 | |
@@ Line 325: / Line 324: @@
 |-
 | | 31
-| | 1005.405
+| | 1005.41
 | |
 | |
@@ Line 357: / Line 356: @@
 |-
 | | 35
-| | 1135.135
+| | 1135.14
 | | 27/14, 52/27
 | |
@@ Line 373: / Line 372: @@
 |-
 |37
-|1200
+|1200.00
 |2/1
 |
@@ Line 381: / Line 380: @@
 |}
-=Scales=
+==Scales==
 [[MOS_Scales_of_37edo|MOS Scales of 37edo]]
@@ Line 399: / Line 398: @@
 [[square_root_of_13_over_10|The Square Root of 13/10]]
-=Linear temperaments=
+==Linear temperaments==
 [[List_of_37et_rank_two_temperaments_by_badness|List of 37et rank two temperaments by badness]]
@@ Line 480: / Line 479: @@
 |}
-=Music in 37edo=
+==Music==
 [http://www.akjmusic.com/audio/toccata_bianca_37edo.mp3 Toccata Bianca 37edo] by [http://www.akjmusic.com/ Aaron Krister Johnson]
@@ Line 487: / Line 486: @@
 [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]