Linear chord
Linear Chords (also known as Proportionally Beating chords) are a generalization of Isoharmonic chords extended to include irrational intervals. So by definition, all Just chords are also Linear chords. Additionally, chords such as (1):(1+π):(1+2π):(1+3π) are allowed under this broader definition.
Let us consider a n-note chord. There are n*(n-1)/2 possible dyads between each of the notes in the chord. For each dyad A:B, this dyad will beat at A-B Hertz. If all of the frequencies within the chord can be expressed in the form of a linear equation, y=mx+b, then that means that all of the beating frequencies are synced with each other because they are harmonically related. An example is the chord formed from the notes 400 Hz, 500 Hz and 700 Hz, with beating frequencies of 100 Hz, 200 Hz and 300 Hz. The beating frequencies can therefore be written as a ratio of 1:2 between the two smallest dyads. We call this ratio the Isoharmonic Ratio, or isoratio for short.
Syncing the beating frequencies doesn't always work with irrational chords and harmonic timbres however, because the nonlinear beating frequencies between upper partials compete with the linearly related fundamentals. In consequence, the sync is most effective when the chord can be well-approximated by small integer ratios, or when the timbre is weak (such as a sine or triangle wave). Or perhaps your motive is not to sync every beating frequency within a chord, but to sync the most noticeable ones in order to minimize chaotic beating.
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.
Coefficients of Terms: | |||||||||||||||
g^10 | g^9 | g^8 | g^7 | g^6 | g^5 | g^4 | g^3 | g^2 | g^1 | g^0 | Chord | Isoratio | Corresponding Temperament | Generator (in cents) | EDO(s) |
1 | -1 | -1 | 4:5:6 | 1:1 | 1,-2,1> | 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 |
Note: Essentially tempered dyadic triads cannot be optimized for proportional beating because they cannot be uniquely defined in the harmonic series.