User:BudjarnLambeth/42zpi: Difference between revisions
Jump to navigation
Jump to search
Created page with "'''42 zeta peak index''' (abbreviated '''42zpi'''), is the equal-step tuning system obtained from the 42nd 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 == 42zpi is the closes..." |
m remove stub since it's a user page (not priority) |
||
| (2 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
{{Editable user page}} | |||
'''42 zeta peak index''' (abbreviated '''42zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 42nd [[Zeta peak index|peak]] of the [[The Riemann zeta function and tuning|Riemann zeta function]]. | '''42 zeta peak index''' (abbreviated '''42zpi'''), is the [[Equal-step tuning|equal-step]] [[tuning system]] obtained from the 42nd [[Zeta peak index|peak]] of the [[The Riemann zeta function and tuning|Riemann zeta function]]. | ||
| Line 46: | Line 47: | ||
1208.611 | 1208.611 | ||
</pre> | </pre> | ||
[[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:14-tone scales]] | ||
Latest revision as of 09:03, 17 November 2025
|
This user page is editable by any wiki editor.
As a general rule, most users expect their user space to be edited only by themselves, except for minor edits (e.g. maintenance), undoing obviously harmful edits such as vandalism or disruptive editing, and user talk pages. However, by including this message box, the author of this user page has indicated that this page is open to contributions from other users (e.g. content-related edits). |
42 zeta peak index (abbreviated 42zpi), is the equal-step tuning system obtained from the 42nd peak of the Riemann zeta function.
| Tuning | Strength | Closest edo | Integer limit | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| ZPI | Steps per 8ve |
Step size (cents) |
Height | Integral | Gap | Edo | Octave (cents) | Consistent | Distinct | |
| Size | Stretch | |||||||||
| 42zpi | 13.900253 | 86.329367 | 4.592177 | 0.984037 | 14.097244 | 14edo | 1208.611136 | 8.611136 | 7 | 5 |
Theory
42zpi is the closest zeta peak index to 14edo. It is effectively a stretched version of 14edo which improves on its tuning of most primes.
Harmonic series
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +8.6 | -2.7 | +17.2 | -23.8 | +5.9 | -2.0 | +25.8 | -5.4 | -15.2 | -7.5 | +14.5 | -37.7 | +6.6 | -26.5 | +34.4 |
| Relative (%) | +10.0 | -3.1 | +19.9 | -27.5 | +6.8 | -2.3 | +29.9 | -6.3 | -17.6 | -8.7 | +16.8 | -43.7 | +7.7 | -30.7 | +39.9 | |
| Step | 14 | 22 | 28 | 32 | 36 | 39 | 42 | 44 | 46 | 48 | 50 | 51 | 53 | 54 | 56 | |
| Harmonic | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +15.8 | +3.2 | -4.1 | -6.6 | -4.7 | +1.1 | +10.5 | +23.1 | +38.8 | -29.1 | -8.1 | +15.2 | +40.8 | -17.9 | +11.7 | +43.1 |
| Relative (%) | +18.3 | +3.7 | -4.7 | -7.6 | -5.4 | +1.3 | +12.1 | +26.8 | +44.9 | -33.7 | -9.4 | +17.7 | +47.3 | -20.7 | +13.5 | +49.9 | |
| Step | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 65 | 66 | 67 | 68 | 68 | 69 | 70 | |
Scala file
! 42zpi.scl ! Created using Scale Workshop 3.1.0 ! ! https://scaleworkshop.plainsound.org/scale/g8qoINpGf ! 42nd zeta peak index 14 ! 86.329 172.659 258.988 345.317 431.647 517.976 604.306 690.635 776.964 863.294 949.623 1035.952 1122.282 1208.611