Schismic–Pythagorean equivalence continuum: Difference between revisions

The '''schismic–Pythagorean 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.

 

 

Alternatively, because the the 5-limit otonal detemperament of 12edo is a 4×3 rectangle (known as the [[duodene]]), we may be interested in expressing the continuum in terms of the boundary commas of this detemper, that is, as {{nowrap| ([[81/80]])^''k'' ~ ([[128/125]]) }}. This corresponds to these commas' structural significance via 128/125 being entirely in the 2.5 subgroup while 81/80 explains 5 in the simplest way relative to the 3-limit. This choice of coordinates has the advantage of finding all temperaments discussed in a relatively intuitive and simple way so that less accurate but structurally simpler temperaments are found at integer points while microtemperaments converging to atomic are found as successive mediants towards the JIP. Specifically, its JIP is at 1.90915584… which is approximated very closely by the microtemperament atomic at {{nowrap| 21/11 {{=}} 1.90909… }} so that the main continuum can be seen as taking successive mediants towards 2/1 (schismic) starting from 1/1 (diaschismic). It is noted for its nontrivial relation to the other better-motivated (in terms of mapping) coordinates discussed.

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

| [[Gracecordial]]

| (22 digits)

| [[Compton]]

| [[Meantone]]

| [[Diaschismic]]

| [[Misty]]

| [[Undim]]

| (26 digits)

| [[Quindromeda]]

| (34 digits)

| (44 digits)

| 11/6

| (52 digits)

| (62 digits)

| Quinbisa-tritrigu (12 & 441)

| (70 digits)

| Lesa-quinbigu (12 & 494)

| (80 digits)

| Quadtrisa-legu (12 & 559)

| (88 digits)

| [[Atomic]]

| (98 digits)

| Quintrila-theyo (12 & 677)

| (106 digits)

We may invert the continuum by setting ''m'' such that {{nowrap| 1/''m'' + 1/''n'' {{=}} 1 }}. This may be called the ''syntonic–Pythagorean 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.

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

| [[Compton]]

| [[Diaschismic]]

| [[Augmented (temperament)|Augmented]]

| [[Diminished (temperament)|Diminished]]

|+ 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 ==

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 }}.  

[[Subgroup]]: 2.3.5

[[Comma list]]: 43046721/41943040

{{Mapping|legend=1| 1 0 -23 | 0 1 16 }}

: mapping generators: ~2, ~3

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.8769{{c}}, ~3/2 = 699.5409{{c}}

: [[error map]]: {{val| +0.876 -1.537 +0.203 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 699.0789{{c}}

: error map: {{val| 0.000 -2.876 -1.051 }}

[[Badness]] (Sintel): 6.92

== Gracecordial (5-limit) ==

The 5-limit version of gracecordial is generated by a fifth, which is typically sharp of 7\12 but flat of just. The ~5/4 is reached by -20 fifths octave reduced, which is a triple-diminished fifth (C–Gbbb). It can be described as {{nowrap| 12 & 125 }}, and [[137edo]] is a good tuning. It corresponds to {{nowrap| ''n'' {{=}} -1 }} and {{nowrap| ''m'' {{=}} 1/2 }}.  

[[Subgroup]]: 2.3.5

[[Comma list]]: 17433922005/17179869184

{{Mapping|legend=1| 1 0 34 | 0 1 -20 }}

* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 700.734{{c}}

{{Optimal ET sequence|legend=1| 12, 113, 125, 137, 1221bbcc }}

[[Badness]] (Sintel): 7.20

== Sextile (5-limit) ==

{{See also| Landscape microtemperaments #Sextile }}

The 5-limit version of sextile reaches the interval class of 5 by −6 perfect fifths (i.e. a diminished fifth) minus a period of 1/6-octave. It corresponds to {{nowrap| ''n'' {{=}} 6 }}, meaning the Pythagorean comma is equated with a stack of six schismas.  

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| 71 -36 -6 }}

{{Mapping|legend=1| 6 0 71 | 0 1 -6 }}]

: mapping generators: ~4096/3645, ~3

* [[WE]]: ~4096/3645 = 199.9836{{c}}, ~3/2 = 702.1782{{c}} (~4428675/4194304 = 97.7564{{c}})

: [[error map]]: {{val| -0.098 +0.125 +0.045 }}

* [[CWE]]: ~4096/3645 = 200.0000{{c}}, ~3/2 = 702.2434{{c}} (~4428675/4194304 = 97.7566{{c}})

: error map: {{val| 0.000 +0.288 +0.226 }}

[[Badness]] (Sintel): 13.0

Wronecki equates a stack of six ~10/9's with the octave. It reaches the interval class of 5 by +2 perfect fifths (i.e. a major second) plus a period of 1/6-octave. It corresponds to {{nowrap| ''m'' {{=}} 6 }}, meaning the Pythagorean comma is equated with a stack of six syntonic commas.  

[[Subgroup]]: 2.3.5

[[Comma list]]: 531441/500000

{{Mapping|legend=1| 6 0 -5 | 0 1 2 }}

* [[CTE]]: ~10/9 = 200.000{{c}}, ~3/2 = 696.229{{c}}

* [[POTE]]: ~10/9 = 200.000{{c}}, ~3/2 = 695.040{{c}}

[[Badness]] (Sintel): 8.02

Heptacot tempers out the [[heptacot comma]] and divides the 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.  

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| 86 -44 -7 }}

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

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

: [[error map]]: {{val| -0.067 +0.086 +0.029 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -37 19 3 }} = 100.3076{{c}}

: error map: {{val| 0.000 +0.198 +0.153 }}

{{Optimal ET sequence|legend=1| 12, …, 287, 299, 311, 323, 981, 1304, 5539bc, 6843bbcc }}

[[Badness]] (Sintel): 16.0