39edt: Difference between revisions

Line 4:

39edt can be described as approximately 24.606[[edo]]. This implies that each step of 39edt can be approximated by 5 steps of [[123edo]]. 39edt contains within it a close approximation of [[4ed11/5]]: every seventh step of 39edt equates to a step of 4ed11/5.

It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]]~~, and like~~ [[26edt]] and [[52edt]], it is a multiple of 13edt and so contains the Bohlen-Pierce scale~~. It is~~ [[contorted]] in the 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, 847/845 and 1575/1573. It is related to the 49f&172f temperament 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]].

== Theory ==

It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]]. 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.

If octaves are inserted, 39edt is related to the 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]].

~~== Harmonics ==~~

== Intervals ==

@@ Line 4: / Line 4: @@
 edt can be described as approximately 24.606[[edo]]. This implies that each step of 39edt can be approximated by 5 steps of [[123edo]]. 39edt contains within it a close approximation of [[4ed11/5]]: every seventh step of 39edt equates to a step of 4ed11/5.
-It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]], and like [[26edt]] and [[52edt]], it is a multiple of 13edt and so contains the Bohlen-Pierce scale. It is [[contorted]] in the 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, 847/845 and 1575/1573. It is related to the 49f&172f temperament 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]].
+== Theory ==
+It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]]. 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.
+If octaves are inserted, 39edt is related to the 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]].
-== Harmonics ==
 {{Harmonics in equal|39|3|1|intervals=prime}}
 == Intervals ==

39edt: Difference between revisions

Lériendil (talk | contribs)