39edt: Difference between revisions
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If octaves are inserted, 39edt is related to the {{nowrap|49f & 172f}} temperament in the full 13-limit, known as [[Sensamagic clan#Triboh|triboh]], tempering out 245/243, 275/273, 847/845 and 1575/1573, which has mapping [{{val|1 0 0 0 0 0}}, {{val|0 39 57 69 85 91}}]. This has a POTE generator which is an approximate 77/75 of 48.822 cents. 39edt is the ninth [[The Riemann zeta function and tuning#Removing primes|no-twos zeta peak edt]]. | If octaves are inserted, 39edt is related to the {{nowrap|49f & 172f}} temperament in the full 13-limit, known as [[Sensamagic clan#Triboh|triboh]], tempering out 245/243, 275/273, 847/845 and 1575/1573, which has mapping [{{val|1 0 0 0 0 0}}, {{val|0 39 57 69 85 91}}]. This has a POTE generator which is an approximate 77/75 of 48.822 cents. 39edt is the ninth [[The Riemann zeta function and tuning#Removing primes|no-twos zeta peak edt]]. | ||
When treated as an octave-repeating tuning with the sharp octave of 25 steps (about 1219 cents), and the other primes chosen by their best octave-reduced mappings, it functions as a tuning of [[mavila]] temperament, analogous to [[25edo]]'s mavila. | |||
Mavila is one of the few places where octave-stretching makes sense, due to how flat the fifth and often the major third are; this fifth of 683 cents is much more recognizable as a perfect fifth of 3/2 than the 672-cent tuning with just octaves. | |||
{{Harmonics in equal|39|3|1|intervals=prime|columns=12}} | {{Harmonics in equal|39|3|1|intervals=prime|columns=12}} | ||
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! [[Cent]]s | ! [[Cent]]s | ||
! [[Hekt]]s | ! [[Hekt]]s | ||
! [[4L 5s (3/1-equivalent)|Enneatonic]] degree | ! [[4L 5s (3/1-equivalent)|Enneatonic]]<br />degree | ||
! Corresponding | ! Corresponding 3.5.7.11.13 subgroup<br />intervals | ||
! [[Lambda ups and downs notation|Lambda]]<br />(sLsLsLsLs,<br />{{nowrap|J {{=}} 1/1}}) | ! [[Lambda ups and downs notation|Lambda]]<br />(sLsLsLsLs,<br />{{nowrap|J {{=}} 1/1}}) | ||
! Mintaka[7]<br />(E macro-Phrygian) | ! Mintaka[7]<br />(E macro-Phrygian) | ||
| Line 343: | Line 347: | ||
| E | | E | ||
|} | |} | ||
== Approximation to JI == | |||
=== No-2 zeta peak === | |||
{| class="wikitable" | |||
|+ | |||
!Steps | |||
per octave | |||
!Steps | |||
per tritave | |||
!Step size | |||
(cents) | |||
!Height | |||
!Tritave size | |||
(cents) | |||
!Tritave stretch | |||
(cents) | |||
|- | |||
|24.573831630 | |||
|38.948601633 | |||
|48.832433543 | |||
|4.665720 | |||
|1904.464908194 | |||
|2.509907328 | |||
|} | |||
Every 7 steps of the [[172edo|172f]] val is an excellent approximation of the ninth no-2 zeta peak in the 15-limit. | |||
== Music == | |||
; [[Francium]] | |||
* [https://www.youtube.com/watch?v=jstg4_B0jfY ''Strange Juice''] (2025) | |||
;[https://www.youtube.com/@PhanomiumMusic Phanomium] | |||
* ''[https://www.youtube.com/watch?v=GX79ZX1Z8C8 Polygonal]'' (2025) | |||