60edo: Difference between revisions

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 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 [[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 ~~val~~, it is also excellent for the 2.3.7.11.13 [[~~Chromatic_pairs#Bleu|~~bleu ~~temperament~~]].

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.

=== Odd harmonics ===

Line 343:

| 2.3.5

| 3125/3072, 531441/524288

| [{{~~val~~| 60 95 139 }}]

| {{mapping| 60 95 139 }}

| +1.32

| 1.11

Line 350:

| 2.3.5.7

| 225/224, 245/243, 64827/64000

| [{{~~val~~| 60 95 139 168 }}]

| {{mapping| 60 95 139 168 }}

| +1.78

| 1.25

Line 357:

| 2.3.5.7.13

| 105/104, 196/195, 245/243, 8281/8192

| [{{~~val~~| 60 95 139 168 222 }}]

| {{mapping| 60 95 139 168 222 }}

| +1.45

| 1.29

| 6.46

|-

|-style="border-top: double;"

| style="border-top: double;" | 2.3.5.7.11

| 2.3.5.7.11

~~| style="border-top: double;"~~ | 121/120, 225/224, 245/243, 441/440

| 121/120, 225/224, 245/243, 441/440

| ~~style="border-top: double;" | [~~{{~~val~~| 60 95 139 168 207 }}] (60e)

| {{mapping| 60 95 139 168 207 }} (60e)

~~| style="border-top: double;"~~ | +2.08

| +2.08

~~| style="border-top: double;"~~ | 1.27

| 1.27

~~| style="border-top: double;"~~ | 6.33

| 6.33

|-

| 2.3.5.7.11.13

| 105/104, 121/120, 196/195, 275/273, 325/324

| [{{~~val~~| 60 95 139 168 207 222 }}] (60e)

| {{mapping| 60 95 139 168 207 222 }} (60e)

| +1.75

| 1.36

| 6.80

|-

|-style="border-top: double;"

| style="border-top: double;" | 2.3.5.7.11

| 2.3.5.7.11

~~| style="border-top: double;"~~ | 100/99, 225/224, 385/384, 3087/3025

| 100/99, 225/224, 385/384, 3087/3025

| ~~style="border-top: double;" | [~~{{~~val~~| 60 95 139 168 208 }}] (60)

| {{mapping| 60 95 139 168 208 }} (60)

~~| style="border-top: double;"~~ | +0.91

| +0.91

~~| style="border-top: double;"~~ | 2.05

| 2.05

~~| style="border-top: double;"~~ | 10.22

| 10.22

|-

| 2.3.5.7.11.13

| 100/99, 105/104, 144/143, 196/195, 1352/1331

| [{{~~val~~| 60 95 139 168 208 222 }}] (60)

| {{mapping| 60 95 139 168 208 222 }} (60)

| +0.79

| 1.89

@@ 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 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 [[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 val, it is also excellent for the 2.3.7.11.13 [[Chromatic_pairs#Bleu|bleu temperament]].
+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.
 === Odd harmonics ===
@@ Line 343: / Line 343: @@
 | 2.3.5
 | 3125/3072, 531441/524288
-| [{{val| 60 95 139 }}]
+| {{mapping| 60 95 139 }}
 | +1.32
 | 1.11
@@ Line 350: / Line 350: @@
 | 2.3.5.7
 | 225/224, 245/243, 64827/64000
-| [{{val| 60 95 139 168 }}]
+| {{mapping| 60 95 139 168 }}
 | +1.78
 | 1.25
@@ Line 357: / Line 357: @@
 | 2.3.5.7.13
 | 105/104, 196/195, 245/243, 8281/8192
-| [{{val| 60 95 139 168 222 }}]
+| {{mapping| 60 95 139 168 222 }}
 | +1.45
 | 1.29
 | 6.46
-|-
+|-style="border-top: double;"
-| style="border-top: double;" | 2.3.5.7.11
+| 2.3.5.7.11
-| style="border-top: double;" | 121/120, 225/224, 245/243, 441/440
+| 121/120, 225/224, 245/243, 441/440
-| style="border-top: double;" | [{{val| 60 95 139 168 207 }}] (60e)
+| {{mapping| 60 95 139 168 207 }} (60e)
-| style="border-top: double;" | +2.08
+| +2.08
-| style="border-top: double;" | 1.27
+| 1.27
-| style="border-top: double;" | 6.33
+| 6.33
 |-
 | 2.3.5.7.11.13
 | 105/104, 121/120, 196/195, 275/273, 325/324
-| [{{val| 60 95 139 168 207 222 }}] (60e)
+| {{mapping| 60 95 139 168 207 222 }} (60e)
 | +1.75
 | 1.36
 | 6.80
-|-
+|-style="border-top: double;"
-| style="border-top: double;" | 2.3.5.7.11
+| 2.3.5.7.11
-| style="border-top: double;" | 100/99, 225/224, 385/384, 3087/3025
+| 100/99, 225/224, 385/384, 3087/3025
-| style="border-top: double;" | [{{val| 60 95 139 168 208 }}] (60)
+| {{mapping| 60 95 139 168 208 }} (60)
-| style="border-top: double;" | +0.91
+| +0.91
-| style="border-top: double;" | 2.05
+| 2.05
-| style="border-top: double;" | 10.22
+| 10.22
 |-
 | 2.3.5.7.11.13
 | 100/99, 105/104, 144/143, 196/195, 1352/1331
-| [{{val| 60 95 139 168 208 222 }}] (60)
+| {{mapping| 60 95 139 168 208 222 }} (60)
 | +0.79
 | 1.89