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Delta-rational chord: Difference between revisions

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(TODO: Case analysis according to whether <math>r_{\mathbf{u}, \mathbf{v}}(x)</math> simplifies to a polynomial.)
(TODO: Case analysis according to whether <math>r_{\mathbf{u}, \mathbf{v}}(x)</math> simplifies to a polynomial.)
== DR and RTT ==
== DR and RTT ==
<!--Using the principle of proportional beating, it is possible to optimize [[regular temperaments]] for specific triads. This is done by setting up an algebraic equation relating the intervals in the chord to a generator and then solving for the generator that produces proportionally-beating triads. If we want to optimize a 4:5:6 triad in meantone, for instance, we want a 1:1 isoratio between the major third and minor third. The minor third can be written as g-g^4/4, and the major third can be written as g^4/4-1. Therefore we must find the roots of the polynomial g^4-2g-2 (the difference between the two, simplified to make all coefficients integers). This results in a generator of 1.49453, or about 695.63 cents.
Below is a list of temperaments and their various optimizations for proportionally beating chords. They are ordered by highest power in the polynomial, with ties broken by leading coefficients, then 2nd term coefficients, 3rd term coefficients, 4th term coefficients..., etc. In the case of negative coefficients, only the absolute value is considered.
{| class="wikitable"
|-
| colspan="10" | <span style="display: block; text-align: center;"><u>Coefficients of Terms:</u></span>
|
|
|
|
|-
| g^10
| g^9
| g^8
| g^7
| g^6
| g^5
| g^4
| g^3
| g^2
| g^1
| g^0
| <u>Chord</u>
| <u>Isoratio</u>
| <u>Corresponding Temperament</u>
| <u>Generator (in cents)</u>
| EDO(s)
|-
|
|
|
|
|
|
|
|
| 1
| -1
| -1
| 4:5:6
| 1:1
| |1,-2,1&gt;
| 833.09 (phi)
| 36
|-
|
|
|
|
|
|
|
|
| 2
| -1
| -2
| 4:5:6
| 1:1
| Father
| 428.42
| 14
|-
|
|
|
|
|
|
|
| 3
| -2
|
| -2
| 6:7:9
| 1:2
| Beep
| 258.65
| 33, 42, 51
|-
|
|
|
|
|
|
| 1
|
|
| -1
| -2
| 4:5:6
| 1:1
| Mavila
| 523.66
| 23, 39
|-
|
|
|
|
|
|
| 1
|
|
| -2
| -2
| 4:5:6
| 1:1
| Meantone
| 695.63
| 19, 31, 50
|-
|
|
|
|
|
| 1
|
|
| 2
|
| -4
| 4:5:6
| 1:1
| Porcupine
| 160.89
| 15
|-
|
|
|
|
|
| 1
| -4
|
|
|
| 12
| 5:6:9
| 1:3
| Mavila
| 674.90
| 16, 25
|-
|
|
|
|
| 1
|
|
|
|
| -4
| -4
| 4:5:6
| 1:1
| Avila
| 660.23
| 20
|-
|
|
|
|
| 1
| -2
|
|
|
|
| 2
| 4:5:6
| 1:1
| Hanson
| 317.96
| 19
|-
|
|
|
| 1
|
|
|
|
|
| -2
| -4
| 4:5:6
| 1:1
| Uncle
| 467.46
| 18
|-
|
|
| 3
|
|
| -4
|
|
|
|
| -16
| 4:5:7
| 1:2
| Mabila
| 527.66
| 25
|-
|
| 1
|
|
|
|
| -1
|
|
|
| -1
| 4:5:6
| 1:1
| Tetracot
| 176.54
| 34
|-
|
| 1
|
| -1
|
|
|
|
|
|
| -4
| 4:5:6
| 1:1
| Sensi
| 442.74
| 19, 65
|-
| 1
|
|
|
|
|
|
| 2
|
|
| -8
| 4:5:6
| 1:1
| Orson
| 271.51
| 22, 31, 53
|}-->


== Finding approximate DR chords in edos ==
== Finding approximate DR chords in edos ==
Retrieved from "https://en.xen.wiki/w/Delta-rational_chord"