Syntonic–diatonic equivalence continuum: Difference between revisions

* Superpyth ({{nowrap| ''n'' {{=}} 1 }}) is generated by a fifth;

* Immunity ({{nowrap| ''n'' {{=}} 2 }}) splits its twelfth in two;

* Rodan ({{nowrap| ''n'' {{=}} 3 }}) splits its fifth in three;

At {{nowrap| ''n'' {{=}} 5 }}, the corresponding temperament splits the ''octave'' into five instead, as after a stack of five syntonic commas, both the orders of 3 and 5 are multiples of 5 again.  

If we let {{nowrap| ''k'' {{=}} ''n'' + 1 }} so that {{nowrap| ''k'' {{=}} 0 }} means {{nowrap|''n'' {{=}} −1}}, {{nowrap| ''k'' {{=}} 1 }} means {{nowrap| ''n'' {{=}} 0 }}, etc. then the continuum corresponds to {{nowrap| (81/80)^''k'' {{=}} 16/15 }}. Some prefer this way of conceptualising it because:

* 16/15 is the classic diatonic semitone, notable in the 5-limit as the difference between 4/3 and 5/4, so this shifted continuum could also logically be termed the "syntonic–diatonic equivalence continuum". This means that at {{nowrap| ''k'' {{=}} 0 }}, 4/3 and 5/4 are mapped to the same interval while 81/80 becomes independent of 16/15 (meaning 81/80 may or may not be tempered out) because the relation becomes {{nowrap| (81/80)⁰ ~ 1/1 ~ 16/15 }}.

* {{nowrap| ''k'' {{=}} 1 }} and upwards (up to a point) represent temperaments with the potential for reasonably good accuracy as equating at least one 81/80 with 16/15 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be rodan ({{nowrap| ''k'' {{=}} 4 }}), with the only exception being meantone ({{nowrap| ''n'' {{=}} ''k'' {{=}} ∞ }}). (Temperaments corresponding to {{nowrap| ''k'' {{=}} 0, −1, −2, … }} are comparatively low-accuracy to the point of developing various intriguing structures and consequences.)

* 16/15 is the simplest ratio to be tempered out in the continuum.  

| {{Monzo| -8 11 -4 }}

| {{Monzo| -4 7 -3 }}

| {{Monzo| 0 3 -2 }}

| {{Monzo| 4 -1 -1 }}

| {{Monzo| 8 -5 }}

| {{Monzo| 12 -9 1 }}

| {{Monzo| 16 -13 2 }}

| {{Monzo| 20 -17 3 }}

| {{Monzo| 24 -21 4 }}

| {{Monzo| -28 25 -5 }}

| {{Monzo| -32 29 -6 }}

| {{Monzo| -4 4 -1 }}

We may invert the continuum by setting ''m'' such that {{nowrap| 1/''m'' + 1/''n'' {{=}} 1 }}. This may be called the ''superpyth–diatonic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.3130…. The [[superpyth comma]] is both larger and more complex than the syntonic comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.  

| {{Monzo| 20 -14 1 }}

| {{Monzo| 8 -5 }}

| {{Monzo| -4 4 -1 }}

| {{Monzo| 16 -13 2 }}

| {{Monzo| 28 -22 3 }}

| {{Monzo| 12 -9 1 }}

| −3/2 = −1.5 || 3/5 = 0.6 || [[University]] || {{Monzo| 4 2 -3 }}

| −1/2 = −0.5 || 1/3 = 0.{{overline|3}} || [[Uncle]] || {{Monzo| 12 -6 -1 }}

| 1/3 = 0.{{overline|3}} || −1/2 = −0.5 || [[Dirt]] || {{Monzo| 28 -19 1 }}

| 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Counterpental]] || {{Monzo| 36 -30 5 }}

| 7/2 = 3.5 || 7/5 = 1.4 || [[Septiquarter]] || {{Monzo| 44 -38 7 }}

| 21/5 = 4.2 || 21/16 = 1.3125 || 559 & 2513 || {{Monzo| -124 109 -21 }}

| 9/2 = 4.5 || 9/7 = 1.{{overline|285714}} || 5 & 118 || {{Monzo| -52 46 -9 }}

| 11/2 = 5.5 || 11/9 = 1.{{overline|2}} || 5 & 137 || {{Monzo| -60 54 -11 }}

In the 5-limit, superpyth tempers out [[20480/19683]]. It has a fifth generator of {{nowrap| ~3/2 {{=}} ~710{{c}} }} and ~5/4 is found at +9 generator steps, as an augmented second (C–D#). It corresponds to {{nowrap| ''n'' {{=}} 1 }}, meaning that the syntonic comma is equated with the diatonic semitone.  

The 5-limit version of ultrapyth tempers out the [[ultrapyth comma]]. It is generated by a perfect fifth. The interval class of 5 is found at +14 fifths as a double-augmented unison (C–Cx). It corresponds to {{nowrap| ''m'' {{=}} -1 }} and {{nowrap| ''n'' {{=}} 1/2 }}.  

Named by [[John Moriarty]], university is the {{nowrap| 5 & 6b }} temperament, and tempers out [[144/125]], the triptolemaic diminished third. It corresponds to {{nowrap| ''n'' {{=}} −3/2 }} and {{nowrap| ''m'' {{=}} 3/5 }}. In this temperament, two instances of [[6/5]] make a [[5/4]], and three make a [[3/2]]. Equating 6/5 with [[8/7]] (which makes sense since it is already very flat in the most accurate tunings of this temperament) leads to [[Gamelismic clan #Gidorah|gidorah]], and 6/5 with [[7/6]] leads to [[Mint temperaments #Penta|penta]].

== Trisatriyo (5 & 56) ==

[[Comma list]]: {{monzo| 28 -22 3 }} (33554432000/31381059609)

== Quinla-tritrigu (5 & 118) ==

== Tribilalegu (5 & 137) ==

== 559 & 2513 ==