User:BudjarnLambeth/435zpi: Difference between revisions
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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'''. | ||
80edo tunes almost all simple | It has a step size of 14.986 [[cent]]s, and the [[octave]] ([[2/1]]) is tuned slightly impurely to 1198.9 cents. | ||
80edo tunes almost all simple [[harmonic]]s slightly sharp by roughly the same amount, so 435zpi is one possible way of correcting for this. | |||
==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}} | ||
For | For [[prime]]s up to 43: | ||
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 | ||
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. | ||
==Scales== | ==Scales== | ||
All [[80edo#Scales|scales from 80edo]] also occur in 435zpi. | |||
== | ==Instruments== | ||
All instruments listed under [[80edo#Instruments]] also work for 435zpi. | |||
==Music== | |||
* | ; [[Budjarn Lambeth]] | ||
* [https://youtu.be/6N_8QM2UK5I Improvisation in compressed 80edo (435zpi)] (2025) | |||
[[Category:Zeta peak indexes]] [[Category:80edo]] [[Category:80-tone scales]] [[Category:15-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]] | ||
Latest revision as of 04:37, 19 August 2025
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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
| 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 | |
| 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 full 41-limit tuning, or as a more accurate 2.3.5.11 or 2.3.5.11.19 subgroup tuning.
Scales
All scales from 80edo also occur in 435zpi.
Instruments
All instruments listed under 80edo#Instruments also work for 435zpi.