Ultrapyth

Ultrapyth is an alternative extension of the archy chain of fifths to superpyth. Like superpyth, it is a temperament generated by a perfect fifth, where stacking two of them reaches the interval of 8/7~9/8, tempering out 64/63. The difference is that instead of extending to 2.3.5.7 by mapping 5 to +9 generators, it extends to the 2.3.7.13/5 subgroup (known as oceanfront) by mapping the ultramajor third 13/10 to +4 generators (which is also the diatonic major third), tempering out 91/90. This makes sense because the tunings of 2.3.7 archy that optimize for the simplest 2.3.7 intervals (8/7 and 7/6) are sharp of the optimal tuning for 9/7, making that third more ultramajor than supermajor. If intervals of 5 and 13 independently are desired (i.e. 5/4, 13/8), then oceanfront may be extended to ultrapyth by mapping 5 to +14 fifths (a double-augmented unison) and 13 to +18 fifths (a double-augmented third). The best tunings for ultrapyth are between 712 and 714 cents.

If intervals of 11 are desired, 14/11 may be mapped to +9 generators, implying 16/11 is (fittingly) mapped to +11 generators and 11/9 is tempered together with 6/5 (a feature common to many systems with sharp fifths).

The oceanfront mos scales take the form of 5L (5n + 2)s, for n up to 7. Most of these scales are extremely close to 5edo. 37edo makes a good tuning of oceanfront or ultrapyth.

For technical data, see The Biosphere #Oceanfront and Archytas clan #Ultrapyth.

Generator chain

#	Cents	Approximate ratios (Oceanfront)	Approximate ratios (added by Ultrapyth)	Approximate ratios (added in the 11-limit)
0	0	1/1
1	713.4	3/2
2	226.8	8/7, 9/8, 91/80
3	940.2	12/7, 26/15
4	453.6	13/10, 9/7
5	1167	39/20, 27/14
6	680.4	72/49, 52/35
7	193.8	54/49		112/99, 160/143
8	907.2	117/70		56/33
9	420.6			14/11
10	1134		160/81	64/33
11	647.4		40/27	16/11
12	160.8		10/9	12/11
13	874.2		5/3	128/77
14	387.6		5/4	96/77
15	1101		15/8, 52/27	144/77
16	614.4		45/32, 13/9
17	127.8		13/12, 135/128
18	841.2		13/8
19	354.6		39/32
20	1068		117/64