Father

This page is about the regular temperament. For the scale structure sometimes associated with it, see 5L 3s.

Father

Subgroups	2.3.5, 2.3.5.7
Comma basis	16/15 (5-limit); 16/15, 28/27 (7-limit)
Reduced mapping	⟨1; 1 -1 3]
ET join	2(d) & 3(d)
Generators (CWE)	~3/2 = 738.443 ¢
MOS scales	1L 1s, 2L 1s, 3L 2s
Ploidacot	monocot
Pergen	(P8, P5)
Minimax error	5-odd-limit: 55.9 ¢; 7-odd-limit: 68.1 ¢
Target scale size	5-odd-limit: 3 notes; 7-odd-limit: 5 notes

Father is a very coarse, simplistic, and inaccurate exotemperament. It tempers out 16/15, the classical diatonic semitone. This means the classical major third (5/4) is conflated with the perfect fourth (4/3), making it one that challenges the very notion of JI approximation, and playing harmony in it, it sounds only remotely reminiscent of the 5-limit no matter how it is tuned. If one could get their head around this way of hearing intervals, they may as well take a look at the 7-limit interpretation, where it tempers out 28/27 and 36/35.

The main interest in this temperament is its mos scales, as antipentic (3L 2s) and oneirotonic (5L 3s) are often chosen first, and only later is each step associated with a ratio consistent with this temperament. Another potential reason to choose this temperament is to equate suspended chords and more conventional tertian chords (though options like trienstonian (4/3~9/7), blackwood (4/3~81/64), and fendo (4/3~13/10) are more accurate).

As an exotemperament, it has a large range of acceptable tunings, from roughly 3\5 (720 ¢) to 2\3 (800 ¢). However, only tunings between 3\5 and 5\8 (750 ¢) generate oneirotonic scales, which are the primary structure it represents as an exotemperament.

See Father family #Father for technical details.

Interval chain

In the following table, odd harmonics 1–9 are labeled in bold.

#	Cents*	Approximate ratios
0	0.0	1/1
1	738.4	3/2, 8/5, 14/9
2	276.9	6/5, 7/6, 9/8
3	1015.3	7/4, 9/5
4	553.8	7/5
5	92.2	21/20

* In 7-limit CWE tuning

Tunings

5-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~3/2 = 737.469 ¢	CWE: ~3/2 = 742.290 ¢	POTE: ~3/2 = 743.986 ¢

7-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~3/2 = 727.855 ¢	CWE: ~3/2 = 738.443 ¢	POTE: ~3/2 = 742.002 ¢

Tuning spectrum

Edo generator	Eigenmonzo (Unchanged-interval)*	Generator (¢)	Comments
1\2		600.0	Lower bound of 5-odd-limit diamond monotone
	3/2	702.0	Pythagorean tuning
3\5		720.0	Lower bound of 7-odd-limit diamond monotone 9-odd-limit diamond monotone (singleton)
	7/4	722.9
	7/6	733.4
8\13		738.5
	9/5	739.2	1/3-comma
	7/5	745.6	7-odd-limit minimax
5\8		750.0	Upper bound of 7-odd-limit diamond monotone
	5/3	757.8	1/2-comma, 5-odd-limit minimax, 5-limit CEE & CSEE
	9/7	764.9	9-odd-limit minimax
2\3		800.0	Upper bound of 5-odd-limit diamond monotone
	5/4	813.7	Full-comma

* Besides the octave

Music

• Noodles adorno foucault