16ed5/3: Difference between revisions

'''16ed5/3''' is the [[Ed5/3|equal division of the just major sixth]] into sixteen parts of 55.2724 [[cent]]s each, corresponding to 21.7106[[edo]]. It is very closely related to the [[Escapade family|escapade temperament]]. It is vaguely equivalent to [[22edo]].

16ed5/3 can be notated using steps 7 (~5/4) and 9 (~4/3) as generators, as these are accurate to within 0.6¢. The resulting scale is a heptatonic 2L 5s (similar to the octave repeating antidiatonic). It can also be notated using the fifth-generated [[Blackcomb]] temperament as discussed in [[#Temperaments]], which lines up quite nicely with diatonic notation, aside from the "minor second" being in neutral second range and "perfect fourth" being in superfourth range.

! 5/3.4/3.11/6.31/18 subgroup interval

| '''1/1'''

| 31/30, 32/31, 33/32

| 36/35

| 16/15, 33/31

| 21/20

| 75/64, 88/75

| 7/6, 20/17

| aug second

| 6/5, 17/14

| 25/18, 18/13

| 44/31, 64/45

| 10/7, 17/12

| aug fourth

| 72/49

| dim fifth

| 3/2

| aug fifth

| dim sixth

| 16/9, 55/31

| 7/4

| Cx, Dbb

! [[9ed4/3]] (Noleta)

| 55.2429

| 110.4859

| 165.7288

| 220.9718

| 276.2147

| 331.4576

| 386.7006

| 441.9435

| 497.1865

| 552.4294

| 607.6723

| 662.9153

| 718.1582

| 773.4011

| 828.6441

| 883.8870

| 1L Ns

| Bb

Depending on your mapping, 16ed5/3 can be said to temper a number of commas, including the [[diaschisma]], the [[marvel comma]], [[64/63|Archytas' comma]], and the [[jubilisma]], all discussed in the temperaments section. In addition, being an even division of the 5/3, it tempers the [[sensamagic comma]], as the half mosoctave is midway between [[9/7]] and [[35/27]]. This is analogous to the tritone in 2n edo systems. The [[keema]] is tempered due to the septimal interpretation of the diatonic sevenths, and the [[mothwellsma]] is tempered by two major mos3rds ([[7/6]]) resulting in an augmented mos5th ([[11/8]]).

The tempered marvel comma also means that the two large [[tritone]]s ([[64/45|pental]] and [[10/7|septimal]]) are addressed by the same scale step. The tempered diaschisma, on the other hand, means that both pental tritones are also addressed by the same scale step.

 

Both of the 7-limit approaches also temper Archytas' comma as a result of equating the [[16/9]] with [[7/4]], and the jubilisma ([[50/49]]) due to tritone equivalence. These are relatively large commas, given the step size (about half, and 7/11ths respectively).

This shows the close relationships with [[srutal]] and [[pajara]] octave temperaments. In 16ed5/3's case, there is a close equivalence to [[22edo]]'s pajara tuning.

16ed5/3 primes can be mapped on the 31-limit to the val ⟨65 103 151 183 225 241 266 276 294 316 322], using every 3 steps of a shrinked [[65edo]] (-2.431¢ per octave). It differs from the patent val of 65edo in the mapping of prime 7 (val 65d).

As 3 semitones make a period-reduced octave, and it alludes to tritone tempering, [[User:Ayceman|Ayceman]] proposes the name '''tristone''' for the basic [[Diaschismic family|diaschismic temperament]], based on the 16/15 to 6/5 relationship, as well as the following variants and extensions:

[[Optimal ET sequence]]: 9ed5/3, 16ed5/3, 25ed5/3

[[Optimal ET sequence]]: 9ed5/3, 16ed5/3  

[[Optimal ET sequence]]: 9ed5/3, 16ed5/3, 25ed5/3  

[[Optimal ET sequence]]: 9ed5/3, 16ed5/3  

 

 

* [[Alpha, beta, and gamma family of equal divisions]]