60edo: Difference between revisions

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== 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 [[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.

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.

=== Odd harmonics ===

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! Degrees

! Cents

! Approximate ~~Ratios~~ in the 2.3.5.7.13.17 subgroup

! Approximate ratios in the 2.3.5.7.13.17 subgroup

! Additional ~~Ratios~~ of 11 (tending flat, 60e val)

! Additional ratios of 11 (tending flat, 60e val)

|-

| 0

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Multiple vals are listed since they all provide good temperaments.

{~~| class="wikitable center-4 center-5 center-6"~~

|-

~~! rowspan="2" | [[Subgroup]]~~

~~! rowspan="2" | [[Comma list|Comma List]]~~

~~! rowspan="2" | [[Mapping]]~~

~~! rowspan="2" | Optimal 8ve Stretch (¢)~~

~~! colspan="2" | Tuning Error~~

|-

~~! [[TE error|Absolute]] (¢)~~

~~! [[TE simple badness|Relative]] (%)~~

|-

| 2.3.5

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| 1.89

| 9.44

|}

=== Rank-2 temperaments ===

{~~| class="wikitable center-all right-3 left-5"~~

|-

~~! Periods per 8ve~~

! Generator*

! Cents*

! Associated Ratio*

~~! Temperaments~~

|-

| 1

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|-

| 2

| 19\60 (11\60)

| 19\60 (11\60)

| 380.0 (220.0)

| 380.0 (220.0)

| 5/4 (25/22)

| 5/4 (25/22)

| [[Astrology]] (60de) / [[divination]] (60e)

|-

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|-

| 5

| 19\60 (5\60)

| 19\60 (5\60)

| 380.0 (100.0)

| 380.0 (100.0)

| 5/4 (256/245)

| 5/4 (256/245)

| [[Warlock]]

|-

| 5

| 25\60 (1\60)

| 25\60 (1\60)

| 500.0 (20.0)

| 500.0 (20.0)

| 4/3 (81/80)

| 4/3 (81/80)

| [[Pental (temperament)|Pental]] (60)

|-

| 6

| 17\60 (3\60)

| 17\60 (3\60)

| 340.0 (60.0)

| 340.0 (60.0)

| 375/308 (1760/1701)

| 375/308 (1760/1701)

| [[Semiseptichrome]] (11-limit, 60e)

|-

| 10

| 25\60 (1\60)

| 25\60 (1\60)

| 500.0 (20.0)

| 500.0 (20.0)

| 4/3 (91/90)

| 4/3 (91/90)

| [[Decal]] (60e) [[Decic]] (60) / splendecic (60e) / prodecic (60e)

| [[Decal]] (60e) [[Decic]] (60) / splendecic (60e) / prodecic (60e)

|-

| 12

| 19\60 (1\60)

| 19\60 (1\60)

| 380.0 (20.0)

| 380.0 (20.0)

| 5/4 (81/80)

| 5/4 (81/80)

| [[Compton]] / comptone (60e)

|-

| 12

| 12\60 (2\60)

| 12\60 (2\60)

| 240.0 (40.0)

| 240.0 (40.0)

| 8/7 (40/39)

| 8/7 (40/39)

| [[Catnip]] (60cf)

|-

| 15

| 25\60 (3\60)

| 25\60 (3\60)

| 500.0 (20.0)

| 500.0 (20.0)

| 4/3 (126/125)

| 4/3 (126/125)

| [[Pentadecal]] (60) / quindecal (60e)

|-

| 20

| 25\60 (2\60)

| 25\60 (2\60)

| 500.0 (20.0)

| 500.0 (20.0)

| 4/3 (99/98)

| 4/3 (99/98)

| [[Degrees]] (60e)

|}

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

== Diagrams ==

@@ Line 3: / Line 3: @@
 == 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 [[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.
+Since {{nowrap|60 {{=}} 5 &times; 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.
 === Odd harmonics ===
@@ Line 27: / Line 27: @@
 ! Degrees
 ! Cents
-! Approximate Ratios<br>in the 2.3.5.7.13.17 subgroup
+! Approximate ratios<br />in the 2.3.5.7.13.17 subgroup
-! Additional Ratios<br>of 11 (tending flat, 60e val)
+! Additional ratios<br />of 11 (tending flat, 60e val)
 |-
 | 0
@@ Line 339: / Line 339: @@
 Multiple vals are listed since they all provide good temperaments.
-{| class="wikitable center-4 center-5 center-6"
+{{comma basis begin}}
-|-
-! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
-! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
-! colspan="2" | Tuning Error
-|-
-! [[TE error|Absolute]] (¢)
-! [[TE simple badness|Relative]] (%)
 |-
 | 2.3.5
@@ Line 398: / Line 389: @@
 | 1.89
 | 9.44
-|}
+{{comma basis end}}
 === Rank-2 temperaments ===
+{{rank-2 begin}}
-{| class="wikitable center-all right-3 left-5"
-|-
-! Periods<br>per 8ve
-! Generator*
-! Cents*
-! Associated<br>Ratio*
-! Temperaments
 |-
 | 1
@@ Line 447: / Line 431: @@
 |-
 | 2
-| 19\60<br>(11\60)
+| 19\60<br />(11\60)
-| 380.0<br>(220.0)
+| 380.0<br />(220.0)
-| 5/4<br>(25/22)
+| 5/4<br />(25/22)
 | [[Astrology]] (60de) / [[divination]] (60e)
 |-
@@ Line 465: / Line 449: @@
 |-
 | 5
-| 19\60<br>(5\60)
+| 19\60<br />(5\60)
-| 380.0<br>(100.0)
+| 380.0<br />(100.0)
-| 5/4<br>(256/245)
+| 5/4<br />(256/245)
 | [[Warlock]]
 |-
 | 5
-| 25\60<br>(1\60)
+| 25\60<br />(1\60)
-| 500.0<br>(20.0)
+| 500.0<br />(20.0)
-| 4/3<br>(81/80)
+| 4/3<br />(81/80)
 | [[Pental (temperament)|Pental]] (60)
 |-
 | 6
-| 17\60<br>(3\60)
+| 17\60<br />(3\60)
-| 340.0<br>(60.0)
+| 340.0<br />(60.0)
-| 375/308<br>(1760/1701)
+| 375/308<br />(1760/1701)
 | [[Semiseptichrome]] (11-limit, 60e)
 |-
 | 10
-| 25\60<br>(1\60)
+| 25\60<br />(1\60)
-| 500.0<br>(20.0)
+| 500.0<br />(20.0)
-| 4/3<br>(91/90)
+| 4/3<br />(91/90)
-| [[Decal]] (60e)<br>[[Decic]] (60) / splendecic (60e) / prodecic (60e)
+| [[Decal]] (60e)<br />[[Decic]] (60) / splendecic (60e) / prodecic (60e)
 |-
 | 12
-| 19\60<br>(1\60)
+| 19\60<br />(1\60)
-| 380.0<br>(20.0)
+| 380.0<br />(20.0)
-| 5/4<br>(81/80)
+| 5/4<br />(81/80)
 | [[Compton]] / comptone (60e)
 |-
 | 12
-| 12\60<br>(2\60)
+| 12\60<br />(2\60)
-| 240.0<br>(40.0)
+| 240.0<br />(40.0)
-| 8/7<br>(40/39)
+| 8/7<br />(40/39)
 | [[Catnip]] (60cf)
 |-
 | 15
-| 25\60<br>(3\60)
+| 25\60<br />(3\60)
-| 500.0<br>(20.0)
+| 500.0<br />(20.0)
-| 4/3<br>(126/125)
+| 4/3<br />(126/125)
 | [[Pentadecal]] (60) / quindecal (60e)
 |-
 | 20
-| 25\60<br>(2\60)
+| 25\60<br />(2\60)
-| 500.0<br>(20.0)
+| 500.0<br />(20.0)
-| 4/3<br>(99/98)
+| 4/3<br />(99/98)
 | [[Degrees]] (60e)
-|}
+{{rank-2 end}}
-<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+{{orf}}
 == Diagrams ==