720edo: Difference between revisions

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== Theory ==

~~{{Harmonics in equal|720}}~~

720edo is only [[consistent]] to the [[5-odd-limit]], but it has a reasonable approximation of the full 17-limit using the [[patent val]]. It tempers out the [[schisma]] in the 5-limit. In the 11-limit, it is a tuning for the [[Schismatic family #Octant|octant]] temperament.

720edo is ~~the 14th~~ [[~~superabundant EDO~~]]~~, and also~~ the 6th factorial EDO (720 = 1*2*3*4*5*6 = 6!), ~~which means~~ it ~~contains~~ a ~~massive amount~~ of ~~sub~~-~~EDOs, limited modes of transposition, and fraction~~-~~octave MOSses~~. ~~With 720edo, it's better to use various vals mimicking smaller EDOs instead of~~ the ~~patent val~~, ~~because~~ it ~~sounds as if~~ the ~~patent val is ''creating'' commas, not tempering them out~~.

~~=== Simple interpretations ===~~

The patent val can also be thought of as a 2.3.17.23.31.43 subgroup-suited val, because these harmonics have error of less than 1 standard deviaiton away from step. In it, it supports the 195 & 720 temperament, period 15 with comma basis 1377/1376, 19683/19652, 67797/67712, 177147/176824.

~~Nonetheless, in low-complexity tones, it is consistent in the~~ 2.3.5.11 subgroup ~~and provides satisfactory representation~~ of the ~~17-limit~~.

~~In the 11-limit, it is a tuning for the [[Schismatic family#Octant~~|~~octant]] temperament, period 8. This also means that 720edo tempers out the schisma and is a tuning for helmholtz when its patent val fifth is used as a generator.~~

=== Prime harmonics ===

=== ~~Highly melodic theory~~ ===

=== Subsets and supersets ===

~~Since~~ 720 = ~~72 x 10~~, ~~its possible to conceptualize~~ it as a ~~superset~~ of ~~[[72edo]]~~ and [[~~10edo~~]], ~~which are interesting in their own right~~.

720edo is the 14th [[superabundant edo]], and also the 6th factorial edo (720 = 1 × 2 × 3 × 4 × 5 × 6 = 6!), which means it contains a massive amount of subsets, limited modes of transposition, and fraction-octave [[mos]]ses. With 720edo, it is better to use various vals mimicking smaller edos instead of the patent val, because it sounds as if the patent val is ''creating'' commas, not tempering them out{{clarify}}.

However, the patent val's 5/4 of 720edo comes from [[90edo]], and not 72edo.

Since 720 = 72 × 10, its possible to conceptualize it as a superset of [[72edo]] and [[10edo]], which are interesting in their own right. However, the patent val's 5/4 of 720edo comes from [[90edo]], and not 72edo.

=== ~~Other~~ ===

== Regular temperament properties ==

720edo patent val can be thought of as a 2.3.17.23.31.43 subgroup-suited val, because these harmonics have error of less than 1 standard deviaiton away from step. In it, it supports the 195 & 720 temperament, period 15 with comma basis 1377/1376, 19683/19652, 67797/67712, 177147/176824.

=== Rank-2 temperaments ===

== Rank-2 temperaments ~~by generator~~ ==

{| class="wikitable center-all left-5"

!Periods per 8ve

|+Table of rank-2 temperaments by generator

!Generator

! Periods per 8ve

!Cents

! Generator*

!Associated ~~ratio~~

! Cents*

!Temperaments

! Associated Ratio

! Temperaments

|-

|1

| 1

|421\720

| 421\720

|701.667

| 701.667

|4/3

| 4/3

|[[Helmholtz]]

| [[Helmholtz]]

|-

|8

| 8

|421\720 (61\720)

| 421\720 (61\720)

|701.667 (101.667)

| 701.667 (101.667)

|4/3 (36/35)

| 4/3 (36/35)

|[[Octant]]

| [[Octant]]

|-

|80

| 80

|421\720 (7\720)

| 421\720 (7\720)

|701.667 (11.667)

| 701.667 (11.667)

|4/3 (?)

| 4/3 (?)

|[[Octogintic]]

| [[Octogintic]]

|-

|80

| 80

|283\720 (4\720)

| 283\720 (4\720)

|471.667 (6.667)

| 471.667 (6.667)

|130/99 (?)

| 130/99 (?)

|[[Tetraicosic]]

| [[Tetraicosic]]

|}

[[~~Category:Highly composite~~]]

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

[[Category:Schismatic]]

[[Category:Octant]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
 {{EDO intro|720}}
 == Theory ==
-{{Harmonics in equal|720}}
+edo is only [[consistent]] to the [[5-odd-limit]], but it has a reasonable approximation of the full 17-limit using the [[patent val]]. It tempers out the [[schisma]] in the 5-limit. In the 11-limit, it is a tuning for the [[Schismatic family #Octant|octant]] temperament.
-edo is the 14th [[superabundant EDO]], and also the 6th factorial EDO (720 = 1*2*3*4*5*6 = 6!), which means it contains a massive amount of sub-EDOs, limited modes of transposition, and fraction-octave MOSses. With 720edo, it's better to use various vals mimicking smaller EDOs instead of the patent val, because it sounds as if the patent val is ''creating'' commas, not tempering them out.
-=== Simple interpretations ===
+The patent val can also be thought of as a 2.3.17.23.31.43 subgroup-suited val, because these harmonics have error of less than 1 standard deviaiton away from step. In it, it supports the 195 & 720 temperament, period 15 with comma basis 1377/1376, 19683/19652, 67797/67712, 177147/176824.
-Nonetheless, in low-complexity tones, it is consistent in the 2.3.5.11 subgroup and provides satisfactory representation of the 17-limit.
-In the 11-limit, it is a tuning for the [[Schismatic family#Octant|octant]] temperament, period 8. This also means that 720edo tempers out the schisma and is a tuning for helmholtz when its patent val fifth is used as a generator.
+=== Prime harmonics ===
+{{Harmonics in equal|720}}
-=== Highly melodic theory ===
+=== Subsets and supersets ===
-Since 720 = 72 x 10, its possible to conceptualize it as a superset of [[72edo]] and [[10edo]], which are interesting in their own right.
+edo is the 14th [[superabundant edo]], and also the 6th factorial edo (720 = 1 × 2 × 3 × 4 × 5 × 6 = 6!), which means it contains a massive amount of subsets, limited modes of transposition, and fraction-octave [[mos]]ses. With 720edo, it is better to use various vals mimicking smaller edos instead of the patent val, because it sounds as if the patent val is ''creating'' commas, not tempering them out{{clarify}}.
-However, the patent val's 5/4 of 720edo comes from [[90edo]], and not 72edo.
+Since 720 = 72 × 10, its possible to conceptualize it as a superset of [[72edo]] and [[10edo]], which are interesting in their own right. However, the patent val's 5/4 of 720edo comes from [[90edo]], and not 72edo.
-=== Other ===
+== Regular temperament properties ==
-edo patent val can be thought of as a 2.3.17.23.31.43 subgroup-suited val, because these harmonics have error of less than 1 standard deviaiton away from step. In it, it supports the 195 & 720 temperament, period 15 with comma basis 1377/1376, 19683/19652, 67797/67712, 177147/176824.
+=== Rank-2 temperaments ===
-== Rank-2 temperaments by generator ==
 {| class="wikitable center-all left-5"
-!Periods<br>per 8ve
+|+Table of rank-2 temperaments by generator
-!Generator
+! Periods<br>per 8ve
-!Cents
+! Generator*
-!Associated<br>ratio
+! Cents*
-!Temperaments
+! Associated<br>Ratio
+! Temperaments
 |-
-|1
+| 1
-|421\720
+| 421\720
-|701.667
+| 701.667
-|4/3
+| 4/3
-|[[Helmholtz]]
+| [[Helmholtz]]
 |-
-|8
+| 8
-|421\720<br>(61\720)
+| 421\720<br>(61\720)
-|701.667<br>(101.667)
+| 701.667<br>(101.667)
-|4/3<br>(36/35)
+| 4/3<br>(36/35)
-|[[Octant]]
+| [[Octant]]
 |-
-|80
+| 80
-|421\720<br>(7\720)
+| 421\720<br>(7\720)
-|701.667<br>(11.667)
+| 701.667<br>(11.667)
-|4/3<br>(?)
+| 4/3<br>(?)
-|[[Octogintic]]
+| [[Octogintic]]
 |-
-|80
+| 80
-|283\720<br>(4\720)
+| 283\720<br>(4\720)
-|471.667<br>(6.667)
+| 471.667<br>(6.667)
-|130/99<br>(?)
+| 130/99<br>(?)
-|[[Tetraicosic]]
+| [[Tetraicosic]]
 |}
-[[Category:Highly composite]]
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 [[Category:Schismatic]]
 [[Category:Octant]]