720edo: Difference between revisions

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{{ED intro}}

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 [[tempering out|tempers out]] the [[schisma]] in the 5-limit. It [[support]]s [[octant]] up to the 11-limit and [[tetraicosic]] up to the 19-limit.  

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.  

=== Prime harmonics ===

{{Harmonics in equal|720}}

720edo is the 14th [[superabundant edo]], and also the 6th factorial edo ({{nowrap|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 (for example, the 421\720 patent val fifth vs. the 420\720 fifth stemming from [[12edo]]).  

Since {{nowrap|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.

=== Rank-2 temperaments ===

! Periods<br />per 8ve

! Generator*

! Cents*

! Associated<br />ratio*

! Temperaments

| 1

| 421\720

| 701.667

| 4/3

| [[Helmholtz (temperament)|Helmholtz]]

| 8

| 421\720<br />(61\720)

| 701.667<br />(101.667)

| 4/3<br />(36/35)

| [[Octant]]

| 80

| 421\720<br />(7\720)

| 701.667<br />(11.667)

| 4/3<br />(?)

| 130/99<br />(?)

| [[Tetraicosic]]

<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

 

[[Category:Schismic]]