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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. | | '''435zpi''', the 435th [[zeta peak index]], is a [[Octave shrinking|compressed-octaves]] version of [[80edo]]. It can be thought of as '''80ed1198.9c''' or as '''14.986cet'''. |
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| 80edo tunes almost all simple harmonics slightly sharp by roughly the same amount, so 435zpi is one possible way of correcting for this. | | It has a step size of 14.986 [[cent]]s, and the [[octave]] ([[2/1]]) is tuned slightly impurely to 1198.9 cents. |
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| | 80edo tunes almost all simple [[harmonic]]s slightly sharp by roughly the same amount, so 435zpi is one possible way of correcting for this. |
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| ==Prime harmonics== | | ==Prime harmonics== |
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| {{Harmonics in cet|14.986|columns=7|start=8|intervals=prime|title=Approximation of prime harmonics in 435zpi}} | | {{Harmonics in cet|14.986|columns=7|start=8|intervals=prime|title=Approximation of prime harmonics in 435zpi}} |
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| For primes up to 43: | | For [[prime]]s up to 43: |
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| 435zpi approximates these with less than 3 cents error (<20% relative error): | | 435zpi approximates these with less than 3 cents error (<20% relative error): |
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| * 43 | | * 43 |
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| 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. | | 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. |
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| ==Scales== | | ==Scales== |
| Any [[80edo#Scales|scales from 80edo]] should also be useable here.
| | All [[80edo#Scales|scales from 80edo]] also occur in 435zpi. |
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| ===Combination product sets===
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| ''See also: [[Combination product set]]
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| ; CPS (10 of 1 3 5 9 11 15 19 25 27 29 33)
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| ; (11 tones per octave)
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| Untempered ([[JI]]):
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| * 12/11 - 6/5 - 24/19 - 4/3 - 16/11 - 3/2 - 8/5 - 48/29 - 16/9 - 48/25 - 2/1
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| Tempered to 435zpi:
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| * 149.9 - 314.7 - 404.6 - 494.5 - 644.4 - 704.4 - 809.3 - 869.2 - 989.1 - 1124.0 - 1198.9
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| Untempered (JI, different rotation):
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| * 12/11 - 9/8 - 6/5 - 36/29 - 4/3 - 36/25 - 3/2 - 18/11 - 9/5 - 36/19 - 2/1
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| Tempered to 435zpi:
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| * 149.9 - 209.8 - 314.7 - 374.7 - 494.5 - 629.4 - 704.4 - 854.2 - 1019.1 - 1109.0 - 1198.9
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| ; CPS (1 of 1 3 5 9 11 15 19 25 27 29 33)
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| ; (11 tones per octave)
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| Untempered (JI):
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| * 25/24 - 9/8 - 29/24 - 5/4 - 4/3 - 11/8 - 3/2 - 19/12 - 5/3 - 11/6 - 2/1
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| Tempered to 435zpi:
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| * 74.9 - 209.8 - 329.7 - 389.6 - 494.5 - 554.5 - 704.4 - 794.3 - 884.2 -
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| ; CPS (12 of 1 3 5 9 11 15 19 25 27 29 33 37 41)
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| ; (13 tones per octave)
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| Untempered (JI):
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| * 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
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| Tempered to 435zpi:
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| * 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
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| ; CPS (1 of 1 3 5 9 11 15 19 25 27 29 33 37 41)
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| ; (13 tones per octave)
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| Untempered (JI):
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| * 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
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| Tempered to 435zpi:
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| * 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
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| ; CPS (14 of 1 3 5 9 11 15 19 25 27 29 33 37 41 55 57)
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| ; (15 tones per octave)
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| Untempered (JI):
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| * 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
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| Tempered to 435zpi:
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| * 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
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| ; CPS (1 of 1 3 5 9 11 15 19 25 27 29 33 37 41 55 57)
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| ; (15 tones per octave)
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| Untempered (JI):
| | ==Instruments== |
| * 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
| | All instruments listed under [[80edo#Instruments]] also work for 435zpi. |
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| Tempered to 435zpi:
| | ==Music== |
| * 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 | | ; [[Budjarn Lambeth]] |
| | * [https://youtu.be/6N_8QM2UK5I Improvisation in compressed 80edo (435zpi)] (2025) |
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| [[Category:Zeta peak indexes]] [[Category:80edo]] [[Category:80-tone scales]] [[Category:15-tone scales]] [[Category:13-tone scales]] [[Category:11-tone scales]] | | [[Category:Zeta peak indexes]] [[Category:80edo]] [[Category:80-tone scales]] [[Category:15-tone scales]] [[Category:13-tone scales]] [[Category:12-tone scales]] [[Category:11-tone scales]] |