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.~~

~~80edo tunes almost all simple harmonics slightly sharp by roughly~~ the ~~same amount (roughly 3 cents)~~, ~~so 435zpi~~ is ~~one possible way~~ of ~~correcting for this~~.

'''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'''.

~~==Harmonics==~~

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

~~===Odd===~~

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

~~{{Harmonics in cet|14.986|columns=13|intervals=odd|title=Approximation~~ of ~~odd harmonics in 435zpi}}~~

~~{{Harmonics in cet|14~~.~~986|columns=13|start=14|intervals=odd|title=Approximation of odd harmonics in 435zpi}}~~

===Prime===

==Prime harmonics==

{{Harmonics in cet|14.986|columns=13|intervals=prime|title=Approximation of prime harmonics in 435zpi}}

{{Harmonics in cet|14.986|columns=7|intervals=prime|title=Approximation of prime harmonics in 435zpi}}

{{Harmonics in cet|14.986|columns=13|start=14|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}}

[[Category:Zeta peak indexes]]

For [[prime]]s 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 [[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:13-tone scales]] [[Category:12-tone scales]] [[Category:11-tone scales]]

@@ Line 1: / Line 1: @@
-'''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.
+{{Editable user page}}
-edo tunes almost all simple harmonics slightly sharp by roughly the same amount (roughly 3 cents), so 435zpi is one possible way of correcting for this.
+'''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'''.
-==Harmonics==
+It has a step size of 14.986 [[cent]]s, and the [[octave]] ([[2/1]]) is tuned slightly impurely to 1198.9 cents.
-===Odd===
+edo tunes almost all simple [[harmonic]]s slightly sharp by roughly the same amount, so 435zpi is one possible way of correcting for this.
-{{Harmonics in cet|14.986|columns=13|intervals=odd|title=Approximation of odd harmonics in 435zpi}}
-{{Harmonics in cet|14.986|columns=13|start=14|intervals=odd|title=Approximation of odd harmonics in 435zpi}}
-===Prime===
+==Prime harmonics==
-{{Harmonics in cet|14.986|columns=13|intervals=prime|title=Approximation of prime harmonics in 435zpi}}
+{{Harmonics in cet|14.986|columns=7|intervals=prime|title=Approximation of prime harmonics in 435zpi}}
-{{Harmonics in cet|14.986|columns=13|start=14|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}}
-[[Category:Zeta peak indexes]]
+For [[prime]]s up to 43:
+zpi 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 [[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:13-tone scales]] [[Category:12-tone scales]] [[Category:11-tone scales]]