User:BudjarnLambeth/74zpi: Difference between revisions

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Created page with "{{Editable user page}} '''74 zeta peak index''' (abbreviated '''74zpi'''), is the equal-step tuning system obtained from the 74th peak of the Riemann zeta function. {{ZPI | zpi = 42 | steps = 13.9002525327005 | step size = 86.3293668353859 | height = 4.592177 | integral = 0.984037 | gap = 14.097244 | edo = 14edo | octave = 1208.61113569540 | consistent = 7 | distinct = 5 }} == Theory..."
 
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{{Editable user page}}
{{Editable user page}}
'''74 zeta peak index''' (abbreviated '''74zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 74th [[Zeta peak index|peak]] of the [[The Riemann zeta function and tuning|Riemann zeta function]].
'''74 zeta peak index''' (abbreviated '''74zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 74th [[Zeta peak index|peak]] of the [[The Riemann zeta function and tuning|Riemann zeta function]].
{{ZPI
| zpi = 42
| steps = 13.9002525327005
| step size = 86.3293668353859
| height = 4.592177
| integral = 0.984037
| gap = 14.097244
| edo = 14edo
| octave = 1208.61113569540
| consistent = 7
| distinct = 5
}}


== Theory ==
== Theory ==
Line 27: Line 14:
{{Harmonics in cet|57.7271958810858|columns=15|title=Approximation of harmonics in 74zpi}}
{{Harmonics in cet|57.7271958810858|columns=15|title=Approximation of harmonics in 74zpi}}
{{Harmonics in cet|57.7271958810858|columns=16|start=16|title=Approximation of harmonics in 74zpi}}
{{Harmonics in cet|57.7271958810858|columns=16|start=16|title=Approximation of harmonics in 74zpi}}


{{Harmonics in cet|57.1429|columns=15|title=21edo for comparison|collapsed=1}}
{{Harmonics in cet|57.1429|columns=15|title=21edo for comparison|collapsed=1}}
Line 42: Line 28:
  21
  21
!
!
  86.329
  57.727
  172.659
  115.454
  258.988
  173.182
  345.317
  230.909
  431.647
  288.636
  517.976
  346.363
  604.306
  404.090
  690.635
  461.818
  776.964
519.545
  863.294
577.272
  949.623
634.999
  1035.952
692.726
  1122.282
750.454
  1208.611
808.181
865.908
  923.635
  981.362
  1039.090
  1096.817
  1154.544
  1212.271
</pre>
</pre>


Line 95: Line 88:
* 1212.271
* 1212.271


[[Category:Zeta peak indexes]][[Category:Pages with Scala files]][[Category:Tempered scales]][[Category:14-tone scales]]
[[Category:Zeta peak indexes]][[Category:Pages with Scala files]][[Category:Tempered scales]][[Category:21-tone scales]][[Category:21edo]]

Latest revision as of 09:04, 17 November 2025

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74 zeta peak index (abbreviated 74zpi), is the equal-step tuning system obtained from the 74th peak of the Riemann zeta function.

Theory

74zpi is like 21edo but octave stretched by about 12 cents.

It sounds best with aperiodic or metallophone timbres (similar to gamelan instruments).

Compared to 21edo:

  • It improves upon primes 3, 5, 7, 19 and 23
  • It worsens primes 2, 11, 13 and 17

Harmonic series

Approximation of harmonics in 74zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Error Absolute (¢) +12.3 +3.0 +24.5 -15.4 +15.3 -20.6 -20.9 +6.1 -3.1 +5.0 +27.6 +4.5 -8.4 -12.4 -8.6
Relative (%) +21.3 +5.3 +42.5 -26.7 +26.5 -35.8 -36.2 +10.5 -5.4 +8.7 +47.8 +7.7 -14.5 -21.4 -15.0
Step 21 33 42 48 54 58 62 66 69 72 75 77 79 81 83
Approximation of harmonics in 74zpi
Harmonic 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
Error Absolute (¢) +1.9 +18.4 -17.5 +9.1 -17.6 +17.3 -1.9 -17.9 +26.9 +16.7 +9.1 +3.9 +0.9 -0.1 +0.9 +3.6
Relative (%) +3.2 +31.8 -30.3 +15.8 -30.5 +30.0 -3.3 -31.0 +46.6 +29.0 +15.8 +6.7 +1.5 -0.2 +1.5 +6.3
Step 85 87 88 90 91 93 94 95 97 98 99 100 101 102 103 104


21edo for comparison
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Error Absolute (¢) +0.0 -16.2 +0.0 +13.7 -16.2 +2.6 +0.0 +24.7 +13.7 +20.1 -16.2 +16.6 +2.6 -2.6 +0.0
Relative (%) +0.0 -28.4 +0.0 +24.0 -28.4 +4.6 +0.0 +43.2 +24.0 +35.2 -28.4 +29.1 +4.6 -4.5 +0.0
Step 21 33 42 49 54 59 63 67 70 73 75 78 80 82 84
21edo for comparison
Harmonic 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
Error Absolute (¢) +9.3 +24.7 -11.8 +13.7 -13.6 +20.1 +0.3 -16.2 +27.4 +16.6 +8.4 +2.6 -1.0 -2.6 -2.2 +0.0
Relative (%) +16.3 +43.2 -20.6 +24.0 -23.9 +35.2 +0.5 -28.4 +47.9 +29.1 +14.7 +4.6 -1.8 -4.5 -3.8 +0.0
Step 86 88 89 91 92 94 95 96 98 99 100 101 102 103 104 105

Scala file

! 74zpi.scl
! Created using Scale Workshop 3.1.0
!
! https://scaleworkshop.plainsound.org/scale/wbuk2CLvA
!
74th zeta peak index
 21
!
 57.727
 115.454
 173.182
 230.909
 288.636
 346.363
 404.090
 461.818
 519.545
 577.272
 634.999
 692.726
 750.454
 808.181
 865.908
 923.635
 981.362
 1039.090
 1096.817
 1154.544
 1212.271

Scales

See also: 21edo#Scales

Values given in cents.

Pseudo-equiheptatonic
  • 173.182
  • 346.363
  • 519.545
  • 692.726
  • 865.908
  • 1039.09
  • 1212.271
Pseudo-equipentatonic
  • 230.909
  • 519.545
  • 692.726
  • 981.362
  • 1212.271
5-tone pseudo-pelog
  • 115.454
  • 288.636
  • 692.726
  • 808.181
  • 1212.271
7-tone pseudo-pelog
  • 115.454
  • 288.636
  • 519.545
  • 692.726
  • 808.181
  • 1039.09
  • 1212.271
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