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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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| * [[Ed8]] (… of the 8th harmonic) | | * [[Ed8]] (… of the 8th harmonic) |
| * [[Ed12]] (… of the 12th harmonic) | | * [[Ed12]] (… of the 12th harmonic) |
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| === 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. |
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| == Alpha-beta-gamma family of equal divisions ==
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| 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 [[9edf|9ed3/2]], [[11edf|11ed3/2]], and [[20edf|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 the first members of this family:
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| {| class="wikitable"
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| |+The Alpha-Beta-Gamma family
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| ! colspan="4" |Tuning
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| ! colspan="3" |Intervals
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| ! rowspan="2" |Comment
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| |-
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| !Equal division
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| !Type
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| !Cents<br>per steps
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| !Steps<br>per octave
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| !Ratio divided
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| !Successive superparticular<br>complementary pair folding<br>in the ratio divided
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| !Approximation<br>in cents of these<br>three intervals
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| |-
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| |[[3edt|3ed3/1]]
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| |Alpha
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| |633.985
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| |1.893
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| | rowspan="3" |3/1
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| | rowspan="3" |3/2, 2/1
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| |0, -67.970, 67.970
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| | rowspan="2" |Too much off to be useful
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| |-
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| |[[5edt|5ed3/1]]
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| |Beta
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| |380.391
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| |3.155
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| |0, 58.827, -58.827
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| |-
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| |[[8edt|8ed3/1]]
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| |Gamma
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| |237.744
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| |5.047
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| |0, 11.278, -11.278
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| | rowspan="3" |Pairs are off but still recognizable
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| |-
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| |[[5edo|5ed2/1]]
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| |Alpha
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| |240
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| |5
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| | rowspan="3" |2/1
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| | rowspan="3" |4/3, 3/2
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| |0, -18.045, 18.045
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| |-
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| |[[7edo|7ed2/1]]
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| |Beta
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| |171.429
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| |7
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| |0, 16.241, -16.241
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| |-
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| |[[12edo|12ed2/1]]
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| |Gamma
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| |100
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| |12
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| |0, 1.955, -1.955
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| | rowspan="13" |Happy divisions musically useful
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| |-
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| |[[7ed5/3]]
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| |Alpha
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| |126.337
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| |9.498
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| | rowspan="3" |5/3
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| | rowspan="3" |5/4, 4/3
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| |0, -7.303, 7.303
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| |-
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| |[[9ed5/3]]
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| |Beta
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| |98.262
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| |12.212
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| |0, 6.735, -6.735
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| |-
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| |[[16ed5/3]]
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| |Gamma
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| |55.272
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| |21.711
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| |0, 0.593, -0.593
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| |-
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| |[[9edf|9ed3/2]]
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| |Alpha
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| |77.995
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| |15.386
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| | rowspan="3" |3/2
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| | rowspan="3" |6/5, 5/4
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| |0, -3.661, 3.661
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| |-
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| |[[11edf|11ed3/2]]
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| |Beta
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| |63.814
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| |18.805
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| |0, 3.429, -3.429
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| |-
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| |[[20edf|20ed3/2]]
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| |Gamma
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| |35.098
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| |34.190
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| |0, 0.238, -0.238
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| |-
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| |[[11ed7/5]]
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| |Alpha
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| |52.956
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| |22.660
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| | rowspan="3" |7/5
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| | rowspan="3" |7/6, 6/5
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| |0, -2.093, 2.093
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| |-
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| |[[13ed7/5]]
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| |Beta
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| |44.809
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| |26.781
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| |0, 1.981, -1.981
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| |-
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| |[[24ed7/5]]
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| |Gamma
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| |24.271
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| |49.441
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| |0, 0.114, -0.114
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| |-
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| |[[13ed4/3]]
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| |Alpha
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| |38.311
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| |31.322
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| | rowspan="3" |4/3
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| | rowspan="3" |8/7, 7/6
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| |0, -1.307, 1.307
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| |-
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| |[[15ed4/3]]
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| |Beta
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| |33.203
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| |36.141
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| |0, 1.247, -1.247
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| |-
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| |[[28ed4/3]]
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| |Gamma
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| |17.787
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| |67.464
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| |0, 0.061, -0.061
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| |}
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| A pair of small and big successive superparticulars
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| <math>S_n=\dfrac{n+1}{n}</math> and <math>B_n=\dfrac{n}{n-1}</math>
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| has product
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| <math>\dfrac{n+1}{n}\cdot\dfrac{n}{n-1}=\dfrac{n+1}{n-1}</math>.
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| Thus they are complementary in the ratio <math>R_n=\dfrac{n+1}{n-1}</math>.
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|
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| For each <math>n\ge 2</math> consider the three equal divisions of <math>R_n</math> where low errors appear for <math>S_n</math> and <math>B_n</math> as a converging sequence and pattern:
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| * '''Alpha:''' <math>k_\alpha=2n-1</math>
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| * '''Beta:''' <math>k_\beta=2n+1</math>
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| * '''Gamma:''' <math>k_\gamma=4n=k_\alpha+k_\beta</math>
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|
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| {| class="wikitable"
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| |+Converging sequence and pattern
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| ! rowspan="2" |<math>n</math>
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| ! rowspan="2" |Ratio divided
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| ! colspan="2" |SSCP
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| ! colspan="3" |Number of divisions
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| |-
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| !Small
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| !Big
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| !Alpha
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| !Beta
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| !Gamma
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| |-
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| |2
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| |3/1
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| |3/2
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| |2/1
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| |3
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| |5
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| |8
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| |-
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| |3
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| |2/1
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| |4/3
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| |3/2
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| |5
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| |7
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| |12
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| |-
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| |4
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| |5/3
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| |5/4
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| |4/3
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| |7
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| |9
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| |16
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| |-
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| |5
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| |3/2
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| |6/5
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| |5/4
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| |9
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| |11
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| |20
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| |-
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| |6
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| |7/5
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| |7/6
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| |6/5
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| |11
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| |13
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| |24
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| |-
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| |7
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| |4/3
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| |8/7
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| |7/6
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| |13
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| |15
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| |28
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| |}
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| 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.
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| The Alpha, Beta, and Gamma types bring their interval pairs increasingly close to just intonation.
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| ''A bigger picture of the converging Alpha-Beta-Gamma sequence: [[User:Contribution/Successive superparticular complementary pair#The converging Alpha-Beta-Gamma sequence|The_converging_Alpha-Beta-Gamma_sequence]]''
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| == See also == | | == See also == |
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| [[Category:Equal-step tuning| ]] <!-- main article --> | | [[Category:Equal-step tuning| ]] <!-- main article --> |
| <!-- main article -->
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| [[Category:Terms]] | | [[Category:Terms]] |
| [[Category:Acronyms]] | | [[Category:Acronyms]] |
| [[Category:Tuning]] | | [[Category:Tuning]] |