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.
+{{Editable user page}}
-edo tunes almost all simple harmonics slightly sharp by roughly the same amount, 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'''.
+It has a step size of 14.986 [[cent]]s, and the [[octave]] ([[2/1]]) is tuned slightly impurely to 1198.9 cents.
+edo 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==
@@ Line 7: / Line 11: @@
 {{Harmonics in cet|14.986|columns=7|start=8|intervals=prime|title=Approximation of prime harmonics in 435zpi}}
-For primes up to 43:
+For [[prime]]s up to 43:
 zpi approximates these with less than 3 cents error (<20% relative error):
@@ Line 18: / Line 22: @@
 * 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==
+All [[80edo#Scales|scales from 80edo]] also occur in 435zpi.
-===Combination product sets===
+==Instruments==
-''See also: [[combination product set]]
+All instruments listed under [[80edo#Instruments]] also work for 435zpi.
-; 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:
+==Music==
-* 149.9
+; [[Budjarn Lambeth]]
-* 269.8
+* [https://youtu.be/6N_8QM2UK5I Improvisation in compressed 80edo (435zpi)] (2025)
-* 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
-[[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]]