39edt: Difference between revisions

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== Theory ==

It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]]; in fact it has a better no-twos 13-[[odd limit]] relative error than any other edt up to [[914edt]]. Like [[26edt]] and [[52edt]], it is a multiple of 13edt and so contains the Bohlen-Pierce scale, being [[contorted]] in the no-twos 7-limit, tempering out the same BP commas, [[245/243]] and [[3125/3087]], as 13edt. In the [[11-limit]] it tempers out [[1331/1323]] and in the [[13-limit]] [[275/273]], [[1575/1573]], and [[847/845]]. An efficient traversal is therefore given by [[Mintra]] temperament, which in the 13-limit tempers out 275/273 and ~~1575~~/~~1573~~ alongside 245/243, and is generated by the interval of [[11/7]], which serves as a [[macrodiatonic]] "superpyth" fourth and splits the [[BPS]] generator of [[9/7]], up a tritave, in three.

It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]]; in fact it has a better no-twos 13-[[odd limit]] relative error than any other edt up to [[914edt]]. Like [[26edt]] and [[52edt]], it is a multiple of 13edt and so contains the Bohlen-Pierce scale, being [[contorted]] in the no-twos 7-limit, tempering out the same BP commas, [[245/243]] and [[3125/3087]], as 13edt. In the [[11-limit]] it tempers out [[1331/1323]] and in the [[13-limit]] [[275/273]], [[1575/1573]], and [[847/845]]. An efficient traversal is therefore given by [[Mintra]] temperament, which in the 13-limit tempers out 275/273 and 1331/1323 alongside 245/243, and is generated by the interval of [[11/7]], which serves as a [[macrodiatonic]] "superpyth" fourth and splits the [[BPS]] generator of [[9/7]], up a tritave, in three.

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.

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! [[Cent]]s

! [[Hekt]]s

! [[4L 5s (3/1-equivalent)|Enneatonic]] degree

! [[4L 5s (3/1-equivalent)|Enneatonic]] degree

! Corresponding~~ ~~3.5.7.11.13 subgroup intervals

! Corresponding 3.5.7.11.13 subgroup intervals

! [[Lambda ups and downs notation|Lambda]] (sLsLsLsLs, {{nowrap|J {{=}} 1/1}})

! Mintaka[7] (E macro-Phrygian)

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| 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)

@@ Line 5: / Line 5: @@
 == Theory ==
-It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]]; in fact it has a better no-twos 13-[[odd limit]] relative error than any other edt up to [[914edt]]. Like [[26edt]] and [[52edt]], it is a multiple of 13edt and so contains the Bohlen-Pierce scale, being [[contorted]] in the no-twos 7-limit, tempering out the same BP commas, [[245/243]] and [[3125/3087]], as 13edt. In the [[11-limit]] it tempers out [[1331/1323]] and in the [[13-limit]] [[275/273]], [[1575/1573]], and [[847/845]]. An efficient traversal is therefore given by [[Mintra]] temperament, which in the 13-limit tempers out 275/273 and 1575/1573 alongside 245/243, and is generated by the interval of [[11/7]], which serves as a [[macrodiatonic]] "superpyth" fourth and splits the [[BPS]] generator of [[9/7]], up a tritave, in three.
+It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]]; in fact it has a better no-twos 13-[[odd limit]] relative error than any other edt up to [[914edt]]. Like [[26edt]] and [[52edt]], it is a multiple of 13edt and so contains the Bohlen-Pierce scale, being [[contorted]] in the no-twos 7-limit, tempering out the same BP commas, [[245/243]] and [[3125/3087]], as 13edt. In the [[11-limit]] it tempers out [[1331/1323]] and in the [[13-limit]] [[275/273]], [[1575/1573]], and [[847/845]]. An efficient traversal is therefore given by [[Mintra]] temperament, which in the 13-limit tempers out 275/273 and 1331/1323 alongside 245/243, and is generated by the interval of [[11/7]], which serves as a [[macrodiatonic]] "superpyth" fourth and splits the [[BPS]] generator of [[9/7]], up a tritave, in three.
 If octaves are inserted, 39edt is related to the {{nowrap|49f &amp; 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}}
@@ Line 18: / Line 22: @@
 ! [[Cent]]s
 ! [[Hekt]]s
-! [[4L 5s (3/1-equivalent)|Enneatonic]] degree
+! [[4L 5s (3/1-equivalent)|Enneatonic]]<br />degree
-! Corresponding<br />3.5.7.11.13 subgroup<br />intervals
+! Corresponding 3.5.7.11.13 subgroup<br />intervals
 ! [[Lambda ups and downs notation|Lambda]]<br />(sLsLsLsLs,<br />{{nowrap|J {{=}} 1/1}})
 ! Mintaka[7]<br />(E macro-Phrygian)
@@ Line 343: / Line 347: @@
 | 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)