Pele: Difference between revisions
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{{Infobox regtemp | |||
| Title = Pele | |||
| Subgroups = 2.3.5.7.11; 2.3.5.7.11.13 | |||
| Comma basis = [[441/440]], [[896/891]]; <br>[[196/195]], [[352/351]], [[364/363]] | |||
| Edo join 1 = 41 | Edo join 2 = 46 | Edo join 3 = 58 | |||
| Mapping = 1; 1 0 -6 -10 -13; 0 1 1 1 1 | |||
| Generators = 3/2; 5/4 | Generators tuning = 703.4; 387.8 | |||
| Optimization method = CWE | |||
| Odd limit 1 = 11 | Mistuning 1 = 3.51 | Complexity 1 = ? | |||
| Odd limit 2 = 13-limit 21 | Mistuning 2 = 4.43 | Complexity 2 = ? | |||
}} | |||
The '''pele''' [[rank-3 temperament]] is one of the [[11-limit]] [[extension]]s of [[aberschismic]], inheriting the [[chain of fifths|chain-of-fifths-based]] structure and the generic comma step that results from equating the [[81/80|syntonic]] and [[64/63|septimal commas]]. It [[tempering out|tempers out]] [[441/440]] and [[896/891]], which makes it a member of both [[werckismic temperaments]] and [[pentacircle clan]], and as a consequence, [[11/8]] is mapped to the comma-down diminished fifth (C–vG♭), | |||
[[Category: | The canonical [[extension]] to the [[13-limit]] maps [[13/11]] to the diatonic minor third, which is the exact mean of [[6/5]] and [[7/6]], and implies [[13/8]] is the comma-down diminished seventh (C–vB𝄫) and that [[196/195]], [[352/351]] and [[847/845]] are tempered out. Additionally, a [[17-limit]] extension is available by recognizing [[17/16]] at the comma-up minor second (C–^D♭), the same interval as [[16/15]], and thus tempering out [[256/255]]. | ||
Another way to view this temperament is to look at it relative to [[parapyth]], for which it is an extension that addresses the missing prime 5. If we use an arrow to represent the quartertone spacer of parapyth, we have 5/4 at the quartertone-up augmented second (C–^D♯). | |||
See [[Aberschismic family #Pele]] for technical data. | |||
== Interval lattice == | |||
<gallery> | |||
File:Lattice Pele.png|13-limit pele | |||
File:Lattice Pele17.png|17-limit pele | |||
</gallery> | |||
This lattice shows pele as an extension of parapyth, generated by ~2, ~3/2, and ~7/4. | |||
== Chords and harmony == | |||
Pele enables [[essentially tempered chord]]s of [[werckismic chords|werckismic]] and [[pentacircle chords|pentacircle]] in the [[11-odd-limit]], in addition to [[mynucumic chords|mynucumic]], [[major minthmic chords|major minthmic]], [[minor minthmic chords|minor minthmic]] and [[cuthbert chords|cuthbert]] in the [[13-odd-limit]]. | |||
== Scales == | |||
* [[SNS ((2/1, 3/2)-12, 64/63: 441/440, 896/891)-24]] | |||
* [[SNS ((2/1, 3/2)-12, 64/63: 441/440, 896/891)-36]] | |||
* [[Dekany pele]] – a transversal scale | |||
* The [[compdye]] scale pattern | |||
== Tunings == | |||
{| class="wikitable mw-collapsible mw-collapsed" | |||
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings | |||
|- | |||
! rowspan="2" | | |||
! colspan="3" | Euclidean | |||
|- | |||
! Constrained | |||
! Constrained & skewed | |||
! Destretched | |||
|- | |||
! Tenney | |||
| CTE: ~3/2 = 703.2829{{c}}, ~5/4 = 386.5647{{c}} | |||
| CWE: ~3/2 = 703.2804{{c}}, ~5/4 = 387.3911{{c}} | |||
| POTE: ~3/2 = 703.2791{{c}}, ~5/4 = 387.7906{{c}} | |||
|} | |||
{| class="wikitable mw-collapsible mw-collapsed" | |||
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings | |||
|- | |||
! rowspan="2" | | |||
! colspan="3" | Euclidean | |||
|- | |||
! Constrained | |||
! Constrained & skewed | |||
! Destretched | |||
|- | |||
! Tenney | |||
| CTE: ~3/2 = 703.4398{{c}}, ~5/4 = 386.8933{{c}} | |||
| CWE: ~3/2 = 703.4225{{c}}, ~5/4 = 387.7761{{c}} | |||
| POTE: ~3/2 = 703.4143{{c}}, ~5/4 = 388.1971{{c}} | |||
|} | |||
{| class="wikitable mw-collapsible mw-collapsed" | |||
|+ style="font-size: 105%; white-space: nowrap;" | 17-limit norm-based tunings | |||
|- | |||
! rowspan="2" | | |||
! colspan="3" | Euclidean | |||
|- | |||
! Constrained | |||
! Constrained & skewed | |||
! Destretched | |||
|- | |||
! Tenney | |||
| CTE: ~3/2 = 703.5544{{c}}, ~5/4 = 387.9654{{c}} | |||
| CWE: ~3/2 = 703.4518{{c}}, ~5/4 = 388.4909{{c}} | |||
| POTE: ~3/2 = 703.4265{{c}}, ~5/4 = 388.6202{{c}} | |||
|} | |||
[[Category:Pele| ]] <!-- main article --> | |||
[[Category:Rank-3 temperaments]] | |||
[[Category:Aberschismic family]] | |||
[[Category:Pentacircle clan]] | |||
[[Category:Werckismic temperaments]] | [[Category:Werckismic temperaments]] | ||