Equal-step tuning: Difference between revisions
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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 == | ||