Carlos Alpha: Difference between revisions

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'''Carlos Alpha''' is a non-octave [[equal temperament]] with step size about 77.965{{cent}} ~~in the standard tuning<ref>Wendy Carlos~~, ~~"Tuning: At~~ the ~~Crossroads"~~, ~~Computer Music Journal vol. 11 no. 1~~, ~~1987, pp. 29-43</ref>. It was invented by~~ [[~~Wendy Carlos~~]].

'''Carlos Alpha''' is a non-octave [[equal temperament]] invented by [[Wendy Carlos]], with step size about 77.965{{cent}}. In this temperament, the interval of 9 steps approximates [[3/2]], that of 5 steps approximates [[5/4]], and that of 4 steps approximates [[6/5]].

~~In this temperament~~, the ~~interval~~ of ~~9 steps approximates [[~~3/2]], ~~that of 5 steps approximates [[~~5/4~~]],~~ and ~~that of 4 steps approximates [[~~6/5~~]].~~

== Theory ==

Carlos provided the tuning of 78.0{{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{9^2 + 5^2 + 4^2}{9\log_2(3/2) + 5\log_2(5/4) + 4\log_2(6/5)}</math> ~~≃ 15.391524~~ equal steps and the fifth in 9.003464 equal steps of 77.964990 ~~cents~~ each. It is thus very close to the [[EDF|equal division of the just perfect fifth]] into nine parts of 77.995 ~~cents~~ each ([[9edf|9ed3/2]]), corresponding to 15.3856[[edo]].

\frac{9^2 + 5^2 + 4^2}{9\log_2(3/2) + 5\log_2(5/4) + 4\log_2(6/5)} ≃ 15.391524

</math>

equal steps and the fifth in 9.003464 equal steps of 77.964990{{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 nine parts of 77.995{{cent}} each ([[9edf|9ed3/2]]), corresponding to 15.3856[[edo]].

Carlos Alpha is very closely related to [[Starling temperaments #Valentine temperament|valentine temperament]]. For a list of 11-limit [[dyadic chord]]s of Carlos Alpha, see [[Chords of valentine]].

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It has a very good representation of the 7:8:10:11:12 chord from the [[harmonic series]].

Sequence of ~~EDOs~~ with lower and lower relative error: {{EDOs| 15, 31, 46, 77, 354}}

Sequence of edos with lower and lower relative error: {{EDOs| 15, 31, 46, 77, 354 }}.

== Intervals ==

@@ Line 1: / Line 1: @@
 {{Wikipedia|Alpha scale}}
-'''Carlos Alpha''' is a non-octave [[equal temperament]] with step size about 77.965{{cent}} in the standard tuning<ref>Wendy Carlos, "Tuning: At the Crossroads", Computer Music Journal vol. 11 no. 1, 1987, pp. 29-43</ref>. It was invented by [[Wendy Carlos]].
+'''Carlos Alpha''' is a non-octave [[equal temperament]] invented by [[Wendy Carlos]], with step size about 77.965{{cent}}. In this temperament, the interval of 9 steps approximates [[3/2]], that of 5 steps approximates [[5/4]], and that of 4 steps approximates [[6/5]].
-In this temperament, the interval of 9 steps approximates [[3/2]], that of 5 steps approximates [[5/4]], and that of 4 steps approximates [[6/5]].
+== Theory ==
+Carlos provided the tuning of 78.0{{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{9^2 + 5^2 + 4^2}{9\log_2(3/2) + 5\log_2(5/4) + 4\log_2(6/5)}</math> ≃ 15.391524 equal steps and the fifth in 9.003464 equal steps of 77.964990 cents each. It is thus very close to the [[EDF|equal division of the just perfect fifth]] into nine parts of 77.995 cents each ([[9edf|9ed3/2]]), corresponding to 15.3856[[edo]].
+\frac{9^2 + 5^2 + 4^2}{9\log_2(3/2) + 5\log_2(5/4) + 4\log_2(6/5)} ≃ 15.391524
+</math>
+equal steps and the fifth in 9.003464 equal steps of 77.964990{{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 nine parts of 77.995{{cent}} each ([[9edf|9ed3/2]]), corresponding to 15.3856[[edo]].
 Carlos Alpha is very closely related to [[Starling temperaments #Valentine temperament|valentine temperament]]. For a list of 11-limit [[dyadic chord]]s of Carlos Alpha, see [[Chords of valentine]].
@@ Line 11: / Line 15: @@
 It has a very good representation of the 7:8:10:11:12 chord from the [[harmonic series]].
-Sequence of EDOs with lower and lower relative error: {{EDOs| 15, 31, 46, 77, 354}}
+Sequence of edos with lower and lower relative error: {{EDOs| 15, 31, 46, 77, 354 }}.
 == Intervals ==