60edo: Difference between revisions

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== Theory ==

Since {{nowrap| 60 {{=}} 5 × 12 }}, 60edo belongs to the family of edos which contain [[12edo]], and like the other small edos of this kind, it [[tempering out|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 [[225/224]], [[245/243]], [[875/864]], and [[10976/10935]], and [[support]]s [[magic]], [[compton]] and [[tritonic]] temperaments. In the [[11-limit]], the 60e [[val]] {{val| 60 95 139 168 '''207''' }} scores lower in [[badness]] than the [[patent val]] {{val| 60 95 139 168 '''208''' }} 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 [[subgroup temperament]], it is also excellent for the 2.3.7.11.13-subgroup [[bleu]] temperament, using the 60d val.

Since {{nowrap| 60 {{=}} 5 × 12 }}, 60edo belongs to the family of edos which contain [[12edo]], and like the other small edos of this kind, it [[tempering out|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]]. In the [[7-limit]] it tempers out [[225/224]], [[245/243]], [[875/864]], and [[10976/10935]], and [[support]]s [[magic]], [[compton]] and [[tritonic]] temperaments. In the [[11-limit]], the 60e [[val]] {{val| 60 95 139 168 '''207''' }} scores lower in [[badness]] than the [[patent val]] {{val| 60 95 139 168 '''208''' }} 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 [[subgroup temperament]], it is also excellent for the 2.3.7.11.13-subgroup [[bleu]] temperament, using the 60d val.

=== Odd harmonics ===

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| 280

| [[20/17]]

| ~~''[[13/11]]'',~~ [[33/28]]

| [[33/28]]

|-

| 15

| 300

| [[32/27]]

|

| ''[[13/11]]''

|-

| 16

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| 400

| [[81/64]]

|

| ''[[33/26]]''

|-

| 21

| 420

|

| [[14/11]]~~, ''[[33/26]]''~~

| [[14/11]]

|-

| 22

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| 780

|

| ~~''[[52/33]]'',~~ [[11/7]]

| [[11/7]]

|-

| 40

| 800

| [[128/81]]

|

| ''[[52/33]]''

|-

| 41

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| 900

| [[27/16]]

|

| ''[[22/13]]''

|-

| 46

| 920

| [[17/10]]

| ~~''[[22/13]]'',~~ [[56/33]]

| [[56/33]]

|-

| 47

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== Notation ==

===~~Ups~~ and downs notation===

=== Stein–Zimmermann–Gould notation ===

60edo can be notated with [[ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).

[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:

=== Kite's ups and downs notation ===

60edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).

Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:{{Sharpness-sharp5|60}}

=== Sagittal notation ===

This notation is a superset of the notations for ~~EDOs~~ [[12edo#Sagittal notation|12]] and [[6edo#Sagittal notation|6]].

This notation is a superset of the notations for edos [[12edo #Sagittal notation|12]] and [[6edo #Sagittal notation|6]].

==== Evo flavor ====

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default [[File:60-EDO_Revo_Sagittal.svg]]

</imagemap>

== Approximation to JI ==

=== Interval mappings ===

{{Q-odd-limit intervals|59.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 60e val mapping}}

== Regular temperament properties ==

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| [[Degrees]] (60e)

|}

<nowiki/>* [[Normal ~~lists~~|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~|minimal form]] in parentheses if distinct

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Diagrams ==

@@ Line 3: / Line 3: @@
 == Theory ==
-Since {{nowrap| 60 {{=}} 5 × 12 }}, 60edo belongs to the family of edos which contain [[12edo]], and like the other small edos of this kind, it [[tempering out|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 [[225/224]], [[245/243]], [[875/864]], and [[10976/10935]], and [[support]]s [[magic]], [[compton]] and [[tritonic]] temperaments. In the [[11-limit]], the 60e [[val]] {{val| 60 95 139 168 '''207''' }} scores lower in [[badness]] than the [[patent val]] {{val| 60 95 139 168 '''208''' }} 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 [[subgroup temperament]], it is also excellent for the 2.3.7.11.13-subgroup [[bleu]] temperament, using the 60d val.
+Since {{nowrap| 60 {{=}} 5 × 12 }}, 60edo belongs to the family of edos which contain [[12edo]], and like the other small edos of this kind, it [[tempering out|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]]. In the [[7-limit]] it tempers out [[225/224]], [[245/243]], [[875/864]], and [[10976/10935]], and [[support]]s [[magic]], [[compton]] and [[tritonic]] temperaments. In the [[11-limit]], the 60e [[val]] {{val| 60 95 139 168 '''207''' }} scores lower in [[badness]] than the [[patent val]] {{val| 60 95 139 168 '''208''' }} 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 [[subgroup temperament]], it is also excellent for the 2.3.7.11.13-subgroup [[bleu]] temperament, using the 60d val.
 === Odd harmonics ===
@@ Line 94: / Line 94: @@
 | 280
 | [[20/17]]
-| ''[[13/11]]'', [[33/28]]
+| [[33/28]]
 |-
 | 15
 | 300
 | [[32/27]]
-|
+| ''[[13/11]]''
 |-
 | 16
@@ Line 124: / Line 124: @@
 | 400
 | [[81/64]]
-|
+| ''[[33/26]]''
 |-
 | 21
 | 420
 |
-| [[14/11]], ''[[33/26]]''
+| [[14/11]]
 |-
 | 22
@@ Line 219: / Line 219: @@
 | 780
 |
-| ''[[52/33]]'', [[11/7]]
+| [[11/7]]
 |-
 | 40
 | 800
 | [[128/81]]
-|
+| ''[[52/33]]''
 |-
 | 41
@@ Line 249: / Line 249: @@
 | 900
 | [[27/16]]
-|
+| ''[[22/13]]''
 |-
 | 46
 | 920
 | [[17/10]]
-| ''[[22/13]]'', [[56/33]]
+| [[56/33]]
 |-
 | 47
@@ Line 328: / Line 328: @@
 == Notation ==
-===Ups and downs notation===
+=== Stein–Zimmermann–Gould notation ===
-edo can be notated with [[ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
+[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
+{{Sharpness-sharp5-szg|60}}
+=== Kite's ups and downs notation ===
+edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
 {{Sharpness-sharp5a}}
-Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:{{Sharpness-sharp5|60}}
 === Sagittal notation ===
-This notation is a superset of the notations for EDOs [[12edo#Sagittal notation|12]] and [[6edo#Sagittal notation|6]].
+This notation is a superset of the notations for edos [[12edo #Sagittal notation|12]] and [[6edo #Sagittal notation|6]].
 ==== Evo flavor ====
@@ Line 357: / Line 360: @@
 default [[File:60-EDO_Revo_Sagittal.svg]]
 </imagemap>
+== Approximation to JI ==
+=== Interval mappings ===
+{{Q-odd-limit intervals|60}}
+{{Q-odd-limit intervals|59.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 60e val mapping}}
 == Regular temperament properties ==
@@ Line 540: / Line 548: @@
 | [[Degrees]] (60e)
 |}
-<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 == Diagrams ==