60edo: Difference between revisions

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The 60 equal division divides the octave into 60 parts of exactly 20 cents each. Since 60 = 5 * 12, it belongs to the family of edos which contain 12edo, and like the other small edos of this kind, it tempers out the [[Pythagorean_comma|Pythagorean comma]], 531441/524288 = |-19 12>. In the 5-limit, it tempers out both the magic comma, 3125/3072, and the amity comma, 1600000/1594323, and supplies the optimal patent val for 5-limit magic, tempering out 3125/3072. In the 7-limit it tempers out 875/864, 245/243, 225/224 and 10976/10935, and supports [[Magic|magic]], [[Compton|compton]] and [[Tritonic|tritonic]] temperaments. In the 11-limit, the 60e val scores lower in badness than the patent val, and makes for an excellent tritonic tuning. It tempers out 121/120 and 441/440, whereas the patent val tempers out 100/99, 385/384 and 540/539. The tuning of 13 is superb at half a cent flat, and the 60e val also works excellently for 13-limit tritonic. As a no-fives val, it is also excellent for the 2.3.7.11.13 [[Chromatic_pairs#Bleu|bleu temperament]].

The '''60 equal division''' divides the octave into 60 parts of exactly 20 cents each.

=~~Intervals~~=

== Theory ==

Since 60 = 5×12, 60edo belongs to the family of edos which contain [[12edo]], and like the other small edos of this kind, it tempers out the [[Pythagorean comma]], 531441/524288 = {{monzo|-19 12}}. In the [[5-limit]], it tempers out both the [[magic comma]], 3125/3072, and the [[amity comma]], 1600000/1594323, and supplies the optimal patent val for 5-limit [[magic]], tempering out 3125/3072. In the [[7-limit]] it tempers out [[875/864]], [[245/243]], [[225/224]] and [[10976/10935]], and supports [[magic]], [[compton]] and [[tritonic]] temperaments. In the [[11-limit]], the 60e [[val]] scores lower in [[badness]] than the [[patent val]], and makes for an excellent tritonic tuning. It tempers out [[121/120]] and [[441/440]], whereas the patent val tempers out [[100/99]], [[385/384]] and [[540/539]]. The tuning of 13 is superb at half a cent flat, and the 60e val also works excellently for [[13-limit]] tritonic. As a no-fives val, it is also excellent for the 2.3.7.11.13 [[Chromatic_pairs#Bleu|bleu temperament]].

{| class="wikitable"

== Intervals ==

{| class="wikitable center-all right-2 left-3 left-4"

|-

~~| | degrees of 60edo~~

! Degrees

~~| | cents value~~

! Cents

~~| | approximate ratios~~ in the 2.3.5.13 subgroup

! Approximate Ratios<br>in the 2.3.5.13 subgroup

~~| | additional ratios~~ of 7 and 11 (~~assuming~~ flat ~~values for primes~~)

! Additional Ratios<br>of 7 and 11 (tending flat, i.e. 60e val)

|-

| 0

|0

| 0

| |

|

| |

|

|-

| | 1

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|}

=Compositions=

== Compositions ==

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/rojqoq.html Rojqoq (So-Called Peace)] [http://micro.soonlabel.com/gene_ward_smith/Others/Sethares/Rojqoq.mp3 play] by William Sethares

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by Graham Breed

=Images=

== Images ==

[[File:60edo_wheel_with_cents_values.png|alt=60edo wheel with cents values.png|560x560px|60edo wheel with cents values.png]]

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Robin Perry guitar

[[Category:60edo]]

[[Category:edo]]

[[Category:listen]]

@@ Line 1: / Line 1: @@
-The 60 equal division divides the octave into 60 parts of exactly 20 cents each. Since 60 = 5 * 12, it belongs to the family of edos which contain 12edo, and like the other small edos of this kind, it tempers out the [[Pythagorean_comma|Pythagorean comma]], 531441/524288 = |-19 12&gt;. In the 5-limit, it tempers out both the magic comma, 3125/3072, and the amity comma, 1600000/1594323, and supplies the optimal patent val for 5-limit magic, tempering out 3125/3072. In the 7-limit it tempers out 875/864, 245/243, 225/224 and 10976/10935, and supports [[Magic|magic]], [[Compton|compton]] and [[Tritonic|tritonic]] temperaments. In the 11-limit, the 60e val scores lower in badness than the patent val, and makes for an excellent tritonic tuning. It tempers out 121/120 and 441/440, whereas the patent val tempers out 100/99, 385/384 and 540/539. The tuning of 13 is superb at half a cent flat, and the 60e val also works excellently for 13-limit tritonic. As a no-fives val, it is also excellent for the 2.3.7.11.13 [[Chromatic_pairs#Bleu|bleu temperament]].
+The '''60 equal division''' divides the octave into 60 parts of exactly 20 cents each.
-=Intervals=
+== Theory ==
+Since 60 = 5×12, 60edo belongs to the family of edos which contain [[12edo]], and like the other small edos of this kind, it tempers out the [[Pythagorean comma]], 531441/524288 = {{monzo|-19 12}}. In the [[5-limit]], it tempers out both the [[magic comma]], 3125/3072, and the [[amity comma]], 1600000/1594323, and supplies the optimal patent val for 5-limit [[magic]], tempering out 3125/3072. In the [[7-limit]] it tempers out [[875/864]], [[245/243]], [[225/224]] and [[10976/10935]], and supports [[magic]], [[compton]] and [[tritonic]] temperaments. In the [[11-limit]], the 60e [[val]] scores lower in [[badness]] than the [[patent val]], and makes for an excellent tritonic tuning. It tempers out [[121/120]] and [[441/440]], whereas the patent val tempers out [[100/99]], [[385/384]] and [[540/539]]. The tuning of 13 is superb at half a cent flat, and the 60e val also works excellently for [[13-limit]] tritonic. As a no-fives val, it is also excellent for the 2.3.7.11.13 [[Chromatic_pairs#Bleu|bleu temperament]].
-{| class="wikitable"
+== Intervals ==
+{| class="wikitable center-all right-2 left-3 left-4"
 |-
-| | degrees of 60edo
+! Degrees
-| | cents value
+! Cents
-| | approximate ratios in the 2.3.5.13 subgroup
+! Approximate Ratios<br>in the 2.3.5.13 subgroup
-| | additional ratios of 7 and 11 (assuming flat values for primes)
+! Additional Ratios<br>of 7 and 11 (tending flat, i.e. 60e val)
 |-
 | 0
-|0
+| 0
-| |
+|
-| |
+|
 |-
 | | 1
@@ Line 316: / Line 318: @@
 |}
-=Compositions=
+== Compositions ==
 [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/rojqoq.html Rojqoq (So-Called Peace)] [http://micro.soonlabel.com/gene_ward_smith/Others/Sethares/Rojqoq.mp3 play] by William Sethares
@@ Line 335: / Line 337: @@
 by Graham Breed
-=Images=
+== Images ==
 [[File:60edo_wheel_with_cents_values.png|alt=60edo wheel with cents values.png|560x560px|60edo wheel with cents values.png]]
@@ Line 343: / Line 345: @@
 Robin Perry guitar
 [[Category:60edo]]
 [[Category:edo]]
 [[Category:listen]]