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Equal-step tuning: Difference between revisions

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{{interwiki
{{interwiki
| de = Gleichstufige_Tonsysteme
| de = Gleichstufige_Tonsysteme
| en = Equal-step_tuning
| en = Equal-step tuning
| ja = 平均律
| ja = 平均律
}}
}}
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(Unlimited resolution version: [[:File:equal.svg|equal.svg]])
(Unlimited resolution version: [[:File:equal.svg|equal.svg]])


For the mathematically inclined, this kind of diagram is closely related to [[The Riemann zeta function and tuning|the Riemann zeta function]].
For the mathematically inclined, this kind of diagram is closely related to the [[Riemann zeta function]].


== Catalog of equal-step tunings ==
== Catalog of equal-step tunings ==
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* [[Ed8]] (… of the 8th harmonic)
* [[Ed8]] (… of the 8th harmonic)
* [[Ed12]] (… of the 12th harmonic)
* [[Ed12]] (… of the 12th harmonic)


=== Equal multiplications ===
=== Equal multiplications ===
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Other edonoi contain no approximation of an octave or a compound octave (at least, not for a while), and continue generating new tones as they continue upward or downward. Such scales lack a very familiar compositional redundancy, that of [[octave equivalence]] – this might necessitate special attention.
Other edonoi contain no approximation of an octave or a compound octave (at least, not for a while), and continue generating new tones as they continue upward or downward. Such scales lack a very familiar compositional redundancy, that of [[octave equivalence]] – this might necessitate special attention.
== Alpha-beta-gamma family ==
Wendy Carlos invented in the 1980's the Alpha, Beta, and Gamma scales. These are scales that divides 3/2 with steps very near to the successive superparticular complementary pair folding in 3/2, namely 6/5 and 5/4. The happy equal divisions are 9ed3/2, 11ed3/2, and 20ed3/2. However, these scales belong to a much vaster family, where also applies the same principle of divisions of a ratio with steps very near to the successive superparticular complementary pair folding into it. Below is a table showing a segment of this family:
{| class="wikitable"
|+The Alpha-Beta-Gamma family
! colspan="4" |Tuning
! colspan="3" |Intervals
|-
!Equal division
!Type
!Cents
per steps
!Steps
per octave
!Ratio divided
!Successive superparticular
complementary pair folding
in the ratio divided
!Approximation
in cents of these
three intervals
|-
|[[7ed5/3]]
|Alpha
|126.34
|9.4984
| rowspan="3" |5/3
| rowspan="3" |5/4, 4/3
|0, -7.303, 7.303
|-
|[[9ed5/3]]
|Beta
|98.262
|12.212
|0, 6.735, -6.735
|-
|[[16ed5/3]]
|Gamma
|55.272
|21.711
|0, 0.593, -0.593
|-
|[[9edf|9ed3/2]]
|Alpha
|77.995
|15.386
| rowspan="3" |3/2
| rowspan="3" |6/5, 5/4
|0, -3.661, 3.661
|-
|[[11edf|11ed3/2]]
|Beta
|63.814
|18.805
|0, 3.429, -3.429
|-
|[[20edf|20ed3/2]]
|Gamma
|35.098
|34.190
|0, 0.238, -0.238
|-
|[[11ed7/5]]
|Alpha
|52.956
|22.660
| rowspan="3" |7/5
| rowspan="3" |7/6, 6/5
|0, -2.093, 2.093
|-
|[[13ed7/5]]
|Beta
|44.809
|26.781
|0, 1.981, -1.981
|-
|[[24ed7/5]]
|Gamma
|24.271
|49.441
|0, 0.114, -0.114
|-
|[[13ed4/3]]
|Alpha
|38.311
|31.322
| rowspan="3" |4/3
| rowspan="3" |8/7, 7/6
|0, -1.307, 1.307
|-
|[[15ed4/3]]
|Beta
|33.203
|36.141
|0, 1.247, -1.247
|-
|[[28ed4/3]]
|Gamma
|17.787
|67.464
|0, 0.061, -0.061
|}
As you can see, some patterns appear:
* b
* Alpha types flatten the smaller interval and sharpen the larger; Beta types do the reverse; Gamma types again flatten the smaller and sharpen the larger.
* The Alpha, Beta, and Gamma types bring their interval pairs increasingly close to just intonation.


== See also ==  
== See also ==  
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[[Category:Equal-step tuning| ]] <!-- main article -->
[[Category:Equal-step tuning| ]] <!-- main article -->
<!-- main article -->
[[Category:Terms]]
[[Category:Terms]]
[[Category:Acronyms]]
[[Category:Acronyms]]
[[Category:Tuning]]
[[Category:Tuning]]
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