User:BudjarnLambeth/435zpi

435zpi, the 435th zeta peak index, is a compressed-octaves version of 80edo. It can be thought of as 80ed1198.9c or as 14.986cet.

80edo tunes almost all simple harmonics slightly sharp by roughly the same amount, so 435zpi is one possible way of correcting for this.

Prime harmonics

Approximation of prime harmonics in 435zpi

Harmonic		2	3	5	7	11	13	17
Error	Absolute (¢)	-1.12	+1.27	+1.08	+3.02	-0.20	-4.67	-4.53
	Relative (%)	-7.5	+8.5	+7.2	+20.2	-1.3	-31.2	-30.3
Step		80	127	186	225	277	296	327

Approximation of prime harmonics in 435zpi

Harmonic		19	23	29	31	37	41	43
Error	Absolute (¢)	-2.27	-3.34	-0.02	+4.41	-2.18	-0.07	+7.39
	Relative (%)	-15.2	-22.3	-0.2	+29.4	-14.6	-0.5	+49.3
Step		340	362	389	397	417	429	435

For primes up to 43:

435zpi approximates these with less than 3 cents error (<20% relative error):

• 2, 3, 5, 11, 19, 29, 37, 41

...these with 3-5 cents error (20-33% relative error):

• 7, 13, 17, 23, 31

...and these with more than 5 cents error (>33% relative error):

• 43

This makes it usable as a 41-limit tuning, or as a more accurate 2.3.5.11 or 2.3.5.11.19 subgroup tuning.

Scales

Combination product sets

See also: combination product set

CPS (14 of 1 3 5 9 11 15 19 25 27 29 33 37 41 55 57)

(15 tones per octave)

Untempered (JI):

• 12/11
• 48/41
• 6/5
• 24/19
• 48/37
• 4/3
• 16/11
• 3/2
• 8/5
• 48/29
• 32/19
• 96/55
• 16/9
• 48/25
• 2/1

Tempered to 435zpi:

• 149.9
• 269.8
• 314.7
• 404.6
• 449.6
• 494.5
• 644.4
• 704.4
• 809.3
• 869.2
• 899.2
• 959.1
• 989.1
• 1124.0
• 1198.9

CPS (1 of 1 3 5 9 11 15 19 25 27 29 33 37 41 55 57)

(15 tones per octave)

Untempered (JI):

• 25/24
• 9/8
• 55/48
• 19/16
• 29/24
• 5/4
• 4/3
• 11/8
• 3/2
• 37/24
• 19/12
• 5/3
• 41/24
• 11/6
• 2/1

Tempered to 435zpi:

• 149.9
• 269.8
• 314.7
• 404.6
• 449.6
• 494.5
• 644.4
• 704.4
• 809.3
• 869.2
• 899.2
• 959.1
• 989.1
• 1124.0
• 1198.9