Schismic–commatic equivalence continuum: Difference between revisions

The '''schismic–commatic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equate a number of [[32805/32768|schismas (32805/32768)]] with [[Pythagorean comma|Pythagorean comma ({{monzo| -19 12 }})]]. This continuum is theoretically interesting in that these are all 5-limit temperaments [[support]]ed by [[12edo]].

All temperaments in the continuum satisfy {{nowrap|(32805/32768)^''n'' ~ {{monzo| -19 12 }}}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 12edo due to it being the unique equal temperament that [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 12.0078623975…, and temperaments having ''n'' near this value tend to be the most accurate ones – indeed, the fact that this number is so close to 12 reflects how small [[Kirnberger's atom]] (the difference between 12 schismas and the Pythagorean comma) is.

 

* [[Quindromeda]]  ({{nowrap| ''n'' {{=}} 5 }}) does not split the octave but splits the fourth in five, as 5 is coprime with 12.  

|+ style="font-size: 105%;" | Temperaments with integer ''n''

| [[Gracecordial]]

| (22 digits)

| {{Monzo| -34 20 1 }}

| [[Compton]]

| {{Monzo| -19 12 }}

| [[Meantone]]

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

| [[Diaschismic]]

| {{Monzo| 11 -4 -2 }}

| [[Misty]]

| {{Monzo| 26 -12 -3 }}

| [[Undim]]

| (26 digits)

| {{Monzo| 41 -20 -4 }}

| [[Quindromeda]]

| (34 digits)

| {{Monzo| 56 -28 -5 }}

| (44 digits)

| {{Monzo| 71 -36 -6 }}

| [[Heptacot]]

| (52 digits)

| {{Monzo| 86 -44 -7 }}

| [[World calendar]] restriction

| (62 digits)

| {{Monzo| 101 -52 -8 }}

| Quinbisa-tritrigu (12 & 441)

| (70 digits)

| {{Monzo| 116 -60 -9 }}

| Lesa-quinbigu (12 & 494)

| (80 digits)

| {{Monzo| 131 -68 -10 }}

| Quadtrisa-legu (12 & 559)

| (88 digits)

| {{Monzo| 146 -76 -11 }}

| [[Atomic]]

| (98 digits)

| [[Kirnberger's atom|{{Monzo| 161 -84 -12 }}]]

| Quintrila-theyo (12 & 677)

| (106 digits)

| {{Monzo| -176 92 13 }}

| {{Monzo| -15 8 1 }}

We may invert the continuum by setting ''m'' such that {{nowrap| 1/''m'' + 1/''n'' {{=}} 1 }}. This may be called the ''syntonic–commatic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.0908441588…. The [[syntonic comma]] is way larger but much simpler than the schisma. As such, this continuum does not contain as many [[microtemperament]]s, but has more useful lower-complexity temperaments.

 

{| class="wikitable center-1 center-2"

|+ style="font-size: 105%;" | Temperaments with integer ''m'' (and thus ''k'')

| {{Monzo| -23 16 -1 }}

| [[Compton]]

| {{Monzo| -19 12 }}

| {{Monzo| -15 8 1 }}

| [[Diaschismic]]

| {{Monzo| 11 -4 -2 }}

| [[Augmented (temperament)|Augmented]]

| {{Monzo| 7 0 -3 }}

| [[Diminished (temperament)|Diminished]]

| {{Monzo| 3 4 -4 }}

| {{Monzo| -1 8 -5 }}

| {{Monzo| -5 12 -6 }}

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

|+ style="font-size: 105%;" | Temperaments with fractional ''n'' and ''m''

! ''n'' !! ''m'' !! ''k'' !! Temperament !! Comma

| 5/3 = 1.{{overline|6}} || 5/2 = 2.5 || 1/2 || [[Passion]] || {{monzo| 18 -4 -5 }}

| 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || 4/3 || [[Quintaleap]] || {{monzo| 37 -16 -5 }}

Python is generated by a fifth, which is typically flatter than 7\12. The ~5/4 is reached by +16 fifths octave reduced, which is a double-augmented second (C–Dx). It can be described as {{nowrap| 12 & 91 }}, and [[103edo]] is a good tuning. It corresponds to {{nowrap| ''m'' {{=}} -1 }} and {{nowrap| ''n'' {{=}} 1/2 }}.  

[[Comma list]]: 43046721/41943040

[[Comma list]]: 571919811374025/562949953421312

{{Mapping|legend=1| 2 0 49 | 0 1 -14 }}

* [[WE]]: ~23914845/16777216 = 600.153{{c}}, ~3/2 = 701.211{{c}}

* [[CWE]]: ~23914845/16777216 = 600.000{{c}}, ~3/2 = 701.019{{c}}

{{Optimal ET sequence|legend=1| 12, 166, 178, 190, 392, 582 }}

[[Badness]] (Sintel): 15.7

 

 

 

 

 

 

 

 

== Sextile (5-limit) ==

{{See also| Garischismic clan #Sextile }}

 

 

[[Badness]] (Sintel): 13.0

 

 

 

 

 

: Mapping generators: ~10/9, ~3

* [[WE]]: ~10/9 = 200.1488{{c}}, ~3/2 = 695.5574{{c}}

* [[CWE]]: ~10/9 = 1200.0000{{c}}, ~3/2 = 695.6109{{c}}

: error map: {{val| 0.000 -6.344 +4.908 }}

{{Optimal ET sequence|legend=1| 12, 66b, 78b, 90b, 102b }}

[[Badness]] (Sintel): 8.02

== Heptacot (5-limit) ==

The 5-limit version of heptacot tempers out the [[heptacot comma]] and divides the [[3/2|perfect fifth]] into seven equal parts, the most notable example being [[12edo]] (7\12). It corresponds to {{nowrap| ''n'' {{=}} 7 }}, meaning the Pythagorean comma is equated with a stack of seven schismas. It was named by [[Tristan Bay]] in 2024, for it splits the perfect fifth into seven.  

{{Mapping|legend=1| 1 1 6 | 0 7 -44 }}

* [[WE]]: ~2 = 1199.9328{{c}}, ~{{monzo| -37 19 3 }} = 100.3012{{c}}

[[Badness]] (Sintel): 16.0