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 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.

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-[[throdd 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.

39edt also supports the temperaments: [[suhail]] (generators ~634.1c, ~49.7c), [[erigone]] (3/1, ~682.4c), [[electra]] (3/1, ~536.1c), [[bohlenic]] (1\13edt, ~11/1) and [[deneb]] (3/1, ~892.6c).

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]].

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== Approximation to JI ==

==~~= No-~~2 ~~zeta peak ===~~

According to the finite Euler product with sigma = 1, the 3.5.7.11.13 subgroup gets its maxima at 48.82085 ¢. With sigma = 1/2, the maxima is 48.82100 ¢.

~~{| class="wikitable"~~

|+

The Tenney–Euclidean regular temperement in the 3.5.7.11.13 subgroup mapped with [⟨39 57 69 85 91]] gives 48.82201 ¢.

~~!Steps~~

~~per octave~~

[[69ed7]], with a step size of 48.82356 ¢, is an equal division that approximates this area better than 39edt.

~~!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 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.
+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-[[throdd 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.
+edt also supports the temperaments: [[suhail]] (generators ~634.1c, ~49.7c), [[erigone]] (3/1, ~682.4c), [[electra]] (3/1, ~536.1c), [[bohlenic]] (1\13edt, ~11/1) and [[deneb]] (3/1, ~892.6c).
 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]].
@@ Line 350: / Line 352: @@
 == Approximation to JI ==
-=== No-2 zeta peak ===
+According to the finite Euler product with sigma = 1, the 3.5.7.11.13 subgroup gets its maxima at 48.82085 ¢. With sigma = 1/2, the maxima is 48.82100 ¢.
-{| class="wikitable"
-|+
+The Tenney–Euclidean regular temperement in the 3.5.7.11.13 subgroup mapped with [⟨39 57 69 85 91]] gives 48.82201 ¢.
-!Steps
-per octave
+[[69ed7]], with a step size of 48.82356 ¢, is an equal division that approximates this area better than 39edt.
-!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)