Pele

Pele

Subgroups	2.3.5.7.11; 2.3.5.7.11.13
Comma basis	441/440, 896/891; 196/195, 352/351, 364/363
Reduced mapping	⟨1; 1 0 -6 -10 -13; 0 1 1 1 1]
ET join	41 & 46 & 58
Generators (CWE)	~3/2 = 703.4 ¢, ~5/4 = 387.8 ¢
MOS scales	n/a
Ploidacot	n/a
Minimax error	11-odd-limit: 3.51 ¢; 13-limit 21-odd-limit: 4.43 ¢
Target scale size	11-odd-limit: ? notes; 13-limit 21-odd-limit: ? notes

Pele is a rank-3 temperament generated by a perfect fifth of ~3/2 and a step for the syntonic~septimal comma to reach the interval classes of 5, 7, and higher primes. Using an arrow to represent this comma step, we have 5/4 at the down major third (C–vE), 7/4 at the down minor seventh (C–vBb), and 11/8 at the down diminished fifth (C–vGb), tempering out 441/440 and 896/891, which makes it a member of both werckismic temperaments and pentacircle clan.

The canonical extension to the 13-limit finds 13/8 at the down diminished seventh (C–vBbb), tempering out 196/195, 352/351 and 847/845, and a 17-limit extension is available by recognizing 17/16 at the up minor second (C–^Db), 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 up augmented second (C–^D#).

See Hemifamity family #Pele for technical details.

Interval lattice

• 
  13-limit pele
• 
  17-limit pele

This lattice shows pele as an extension of parapyth, generated by ~2, ~3/2, and ~7/4.

Chords and harmony

Pele enables essentially tempered chords of werckismic and pentacircle in the 11-odd-limit, in addition to mynucumic, major minthmic, minor minthmic and 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

11-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~3/2 = 703.2829 ¢, ~5/4 = 386.5647 ¢	CWE: ~3/2 = 703.2804 ¢, ~5/4 = 387.3911 ¢	POTE: ~3/2 = 703.2791 ¢, ~5/4 = 387.7906 ¢

13-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~3/2 = 703.4398 ¢, ~5/4 = 386.8933 ¢	CWE: ~3/2 = 703.4225 ¢, ~5/4 = 387.7761 ¢	POTE: ~3/2 = 703.4143 ¢, ~5/4 = 388.1971 ¢

17-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~3/2 = 703.5544 ¢, ~5/4 = 387.9654 ¢	CWE: ~3/2 = 703.4518 ¢, ~5/4 = 388.4909 ¢	POTE: ~3/2 = 703.4265 ¢, ~5/4 = 388.6202 ¢