User:BudjarnLambeth/435zpi

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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.

It has a step size of 14.986 cents, and the octave (2/1) is tuned slightly impurely to 1198.9 cents.

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
Steps		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
Steps		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 full 41-limit tuning, or as a more accurate 2.3.5.11 or 2.3.5.11.19 subgroup tuning.

Scales

Any scales from 80edo should also be useable here.

Combination product sets

See also: Combination product set

CPS (10 of 1 3 5 9 11 15 19 25 27 29 33)

(11 tones per octave)

Untempered (JI):

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

Tempered to 435zpi:

• 149.9 - 314.7 - 404.6 - 494.5 - 644.4 - 704.4 - 809.3 - 869.2 - 989.1 - 1124.0 - 1198.9

Untempered (JI, different rotation):

• 12/11 - 9/8 - 6/5 - 36/29 - 4/3 - 36/25 - 3/2 - 18/11 - 9/5 - 36/19 - 2/1

Tempered to 435zpi:

• 149.9 - 209.8 - 314.7 - 374.7 - 494.5 - 629.4 - 704.4 - 854.2 - 1019.1 - 1109.0 - 1198.9

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

(12 tones per octave)

Untempered (JI]):

• 12/11 - 6/5 - 36/29 - 4/3 - 36/25 - 3/2 - 18/11 - 72/41 - 9/5 - 36/19 - 72/37 - 2/1

Tempered to 435zpi:

• 149.9 - 314.7 - 374.7 - 494.5 - 629.4 - 704.4 - 854.2 - 974.1 - 1019.1 - 1109.0 - 1153.9 - 1198.9

CPS (12 of 1 3 5 9 11 15 19 25 27 29 33 37 41)

(13 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 - 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 - 989.1 - 1124.0 - 1198.9

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)

(11 tones per octave)

Untempered (JI):

• 25/24 - 9/8 - 29/24 - 5/4 - 4/3 - 11/8 - 3/2 - 19/12 - 5/3 - 11/6 - 2/1

Tempered to 435zpi:

• 74.9 - 209.8 - 329.7 - 389.6 - 494.5 - 554.5 - 704.4 - 794.3 - 884.2 - 1049.0 - 1198.9

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

(12 tones per octave)

Untempered (JI]):

• 37/36 - 19/18 - 10/9 - 41/36 - 11/9 - 4/3 - 25/18 - 3/2 - 29/18 - 5/3 - 11/6 - 2/1

Tempered to 435zpi:

• 45.0 - 89.9. - 179.8 - 224.8 - 344.7 - 494.5 - 569.5 - 704.4 - 824.2 - 884.2 - 1049.0 - 1198.9

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

(13 tones per octave)

Untempered (JI):

• 25/24 - 9/8 - 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:

• 74.9 - 209.8 - 329.7 - 389.6 - 494.5 - 554.5 - 704.4 - 749.3 - 794.3 - 884.2 - 929.1 - 1049.0 - 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:

• 74.9 - 209.8 - 239.8 - 299.7 - 329.7 - 389.6 - 494.5 - 554.5 - 704.4 - 749.3 - 794.3 - 884.2 - 929.1 - 1049.0 - 1198.9

Other scales

Equipentatonic

• 239.8 - 479.6 - 719.3 - 959.1 - 1198.9

Instruments

All instruments listed under 80edo#Instruments also work for 435zpi.