Carlos Beta: Difference between revisions

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'''Carlos Beta''' is a non-octave [[equal temperament]] with step size about 63.833 [[cent~~]]s<ref>Wendy Carlos, "Tuning: At the Crossroads", Computer Music Journal vol. 11 no. 1, 1987, pp. 29-43</ref>~~, or almost exactly every fifth step of [[94edo]]. ~~It was invented by~~ [[~~Wendy Carlos~~]].

'''Carlos Beta''' is a non-octave [[equal temperament]] invented by [[Wendy Carlos]], with step size about 63.833{{cent}}, or almost exactly every fifth step of [[94edo]]. In this temperament, the interval of 11 steps approximates [[3/2]], that of 6 steps approximates [[5/4]], and that of 5 steps approximates [[6/5]].

~~In this temperament~~, the ~~interval~~ of ~~11 steps approximates [[~~3/2]], ~~that of 6 steps approximates [[~~5/4]], and ~~that of 5 steps approximates [[~~6/5~~]].~~

== Theory ==

Carlos provided the tuning of 63.8{{cent}}<ref>Wendy Carlos, "Tuning: At the Crossroads", ''Computer Music Journal'', vol. 11 no. 1, 1987, pp. 29-43</ref><ref>Wendy Carlos, ''Three Asymmetric Divisions of the Octave''. </ref>. Based on her work, Dave Benson optimized the temperament for 3/2, 5/4, and 6/5, such that the tuning divides the octave in

~~== Theory ==~~

<math>\displaystyle

~~Carlos optimized the tuning on 3/2, 5/4, and 6/5, such that the tuning divides the octave in~~ <math>\frac{11^2 + 6^2 + 5^2}{11\log_2(3/2) + 6\log_2(5/4) + 5\log_2(6/5)}</math> ~~≃ 18.799074~~ equal steps and the fifth in 10.996753 equal steps of 63.832933 ~~cents~~ each. It is thus very close to the [[EDF|equal division of the just perfect fifth]] into eleven parts of 63.8141 ~~cents~~ each (11ed3/2), corresponding to 18.8046[[edo]].

\frac{11^2 + 6^2 + 5^2}{11\log_2(3/2) + 6\log_2(5/4) + 5\log_2(6/5)} ≃ 18.799074

</math>

equal steps and the fifth in 10.996753 equal steps of 63.832933{{cent}} each<ref>Dave Benson, ''Music: A Mathematical Offering'', pp. 232-233.</ref>. It is thus very close to the [[EDF|equal division of the just perfect fifth]] into eleven parts of 63.8141{{cent}} each (11ed3/2), corresponding to 18.8046[[edo]].

Carlos Beta is related to the [[~~Sycamore~~ family~~|sycamore temperaments~~]]; betic and [[5-limit]] sycamore in particular.

Carlos Beta is related to the [[sycamore family]] of temperaments – betic and [[5-limit]] sycamore in particular.

Lookalikes: [[19edo]], [[11edf]], [[30edt]], every 5th step of [[94edo]], every 4th step of [[75edo]]

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* [[Carlos super|Carlos Super Just]]

== ~~Reference~~ ==

== References ==

== External links ==

* [http://www.wendycarlos.com/resources/pitch.html Wendy Carlos Pitch article] ([http://www.webcitation.org/5xelDVVBj Permalink])

* [http://www.wendycarlos.com/resources/pitch.html Wendy Carlos Pitch article | ''Three Asymmetric Divisions of the Octave''] ([http://www.webcitation.org/5xelDVVBj Permalink])

[[Category:Wendy Carlos]]

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 {{Wikipedia|Beta scale}}
-'''Carlos Beta''' is a non-octave [[equal temperament]] with step size about 63.833 [[cent]]s<ref>Wendy Carlos, "Tuning: At the Crossroads", Computer Music Journal vol. 11 no. 1, 1987, pp. 29-43</ref>, or almost exactly every fifth step of [[94edo]]. It was invented by [[Wendy Carlos]].
+'''Carlos Beta''' is a non-octave [[equal temperament]] invented by [[Wendy Carlos]], with step size about 63.833{{cent}}, or almost exactly every fifth step of [[94edo]]. In this temperament, the interval of 11 steps approximates [[3/2]], that of 6 steps approximates [[5/4]], and that of 5 steps approximates [[6/5]].
-In this temperament, the interval of 11 steps approximates [[3/2]], that of 6 steps approximates [[5/4]], and that of 5 steps approximates [[6/5]].
+== Theory ==
+Carlos provided the tuning of 63.8{{cent}}<ref>Wendy Carlos, "Tuning: At the Crossroads", ''Computer Music Journal'', vol. 11 no. 1, 1987, pp. 29-43</ref><ref>Wendy Carlos, ''Three Asymmetric Divisions of the Octave''. </ref>. Based on her work, Dave Benson optimized the temperament for 3/2, 5/4, and 6/5, such that the tuning divides the octave in
-== Theory ==
+<math>\displaystyle
-Carlos optimized the tuning on 3/2, 5/4, and 6/5, such that the tuning divides the octave in <math>\frac{11^2 + 6^2 + 5^2}{11\log_2(3/2) + 6\log_2(5/4) + 5\log_2(6/5)}</math> ≃ 18.799074 equal steps and the fifth in 10.996753 equal steps of 63.832933 cents each. It is thus very close to the [[EDF|equal division of the just perfect fifth]] into eleven parts of 63.8141 cents each (11ed3/2), corresponding to 18.8046[[edo]].
+\frac{11^2 + 6^2 + 5^2}{11\log_2(3/2) + 6\log_2(5/4) + 5\log_2(6/5)} ≃ 18.799074
+</math>
+equal steps and the fifth in 10.996753 equal steps of 63.832933{{cent}} each<ref>Dave Benson, ''Music: A Mathematical Offering'', pp. 232-233.</ref>. It is thus very close to the [[EDF|equal division of the just perfect fifth]] into eleven parts of 63.8141{{cent}} each (11ed3/2), corresponding to 18.8046[[edo]].
-Carlos Beta is related to the [[Sycamore family|sycamore temperaments]]; betic and [[5-limit]] sycamore in particular.
+Carlos Beta is related to the [[sycamore family]] of temperaments – betic and [[5-limit]] sycamore in particular.
 Lookalikes: [[19edo]], [[11edf]], [[30edt]], every 5th step of [[94edo]], every 4th step of [[75edo]]
@@ Line 196: / Line 200: @@
 * [[Carlos super|Carlos Super Just]]
-== Reference ==
+== References ==
 <references/>
 == External links ==
-* [http://www.wendycarlos.com/resources/pitch.html Wendy Carlos Pitch article] ([http://www.webcitation.org/5xelDVVBj Permalink])
+* [http://www.wendycarlos.com/resources/pitch.html Wendy Carlos Pitch article | ''Three Asymmetric Divisions of the Octave''] ([http://www.webcitation.org/5xelDVVBj Permalink])
 [[Category:Wendy Carlos]]