Yahoo Tuning Groups Ultimate Backup tuning-math Algebraic functions for 5-limit temperaments

If b is the "brat" as defined on the tuning group, then the fifth and
the third, for any linear temperament, will be algebraic functions of
b (and vice versa.) If we choose a b value, and specialize one of the
polynomials below, then one of its real roots will be the fifth or
third we seek; similarly, inserting a value for a fifth or third
allows us to compute b.

Here's an example of what you might do with this. We know 2^(47/81)
is a good value for a meantone fifth, being poptimal. If we set f
equal to this value in the meantone fifth polynomial below and solve
for b, we get -6.01, which is awfully close to -6. If we set
b=-6 in the polynomial, we get 15f^4 - 10f - 60 (3f^4 - 2f - 12 if we
reduce to lowest terms), one of whose roots gives us a poptimal
meantone fifth with a brat of -6.

Comma 81/80 Meantone

Polynomial for fifth (-2*b+3)*f^4-10*f+10*b

Polynomial for major third 625/16*b^4+75/2*(2*b-3)^2*b^2*t^2-10*(2*b-
3)^3*b*t^3+(2*b-3)^4*t^4+(-125*b^4+375/2*b^3-625/4)*t

Comma 2048/2025 Diaschismic

Polynomial for fifth 25*f^6-50*f^5*b+25*f^4*b^2-8192*(2*b-3)^2

Polynomial for major third -1280000+16*(2*b-3)^4*t^6-160*b*(2*b-3)
^3*t^5+600*b^2*(2*b-3)^2*t^4-1000*b^3*(2*b-3)*t^3+625*t^2*b^4

Comma 15625/15552 Kleismic

Polynomial for fifth 15625*f^6+(-69120*b^4+18432*b^5-438*b+138240*b^3-
2048*b^6-23328-155520*b^2)*f^5+234375*f^4*b^2-
312500*f^3*b^3+234375*b^4*f^2-93750*b^5*f+15625*b^6

Polynomial for major third 6250*t^6+32*(2*b-3)^5*t^5-400*b*(2*b-3)
^4*t^4+2000*b^2*(2*b-3)^3*t^3-5000*b^3*(2*b-3)^2*t^2+6250*b^4*(2*b-3)
*t-3125*b^5

Comma 32805/32768 Schismic

Polynomial for fifth 5*f^9-5*f^8*b-192+128*b

Polynomial for major third -12500000+256*(2*b-3)^8*t^9-5120*b*(2*b-3)
^7*t^8+44800*b^2*(2*b-3)^6*t^7-224000*b^3*(2*b-3)^5*t^6+700000*b^4*
(2*b-3)^4*t^5-1400000*b^5*(2*b-3)^3*t^4+1750000*b^6*(2*b-3)^2*t^3-
1250000*b^7*(2*b-3)*t^2+390625*t*b^8

Comma 128/125 Augmented

Polynomial for fifth 125*f^3-375*b*f^2+375*b^2*f+3*b^3+864*b-432-
576*b^2

Polynomial for major third t^3-2

Comma 135/128 Pelogic

Polynomial for fifth 5*f^4-5*f^3*b-24+16*b

Polynomial for major third 500+8*(2*b-3)^3*t^4-60*b*(2*b-3)
^2*t^3+150*b^2*(2*b-3)*t^2-125*t*b^3

Comma 250/243 Porcupine

Polynomial for fifth 32*(2*b-3)^3*f^5+125*f^3-375*b*f^2+375*b^2*f-
125*b^3

Polynomial for major third 128*(2*b-3)^5*t^5-1600*b*(2*b-3)^4*t^4-(-
3125+216000*b^2-64000*b^5-432000*b^3+288000*b^4)*t^3-20000*b^3*(2*b-3)
^2*t^2+25000*b^4*(2*b-3)*t-12500*b^5

Comma 2109375/2097152 Orwell

Polynomial for fifth 3125*f^8-15625*f^7*b+31250*f^6*b^2-
31250*f^5*b^3+15625*f^4*b^4-3125*f^3*b^5+8192*(2*b-3)^5

Polynomial for major third 32000+8*(2*b-3)^3*t^8-60*b*(2*b-3)
^2*t^7+150*b^2*(2*b-3)*t^6-125*t^5*b^3

--- In tuning-math@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
<gwsmith@s...> wrote:
> If b is the "brat" as defined on the tuning group, then the fifth and
> the third, for any linear temperament, will be algebraic functions of
> b (and vice versa.) If we choose a b value, and specialize one of the
> polynomials below, then one of its real roots will be the fifth or
> third we seek; similarly, inserting a value for a fifth or third
> allows us to compute b.

I prefer to use numerical methods since a spreadsheet is all you need
for them.

e.g. for meantone just set up a column with decimal fractions from
1/11 (.091) to 1/3 (0.333) in increments of 0.001, representing the
fractions of a comma by which the 5th is tempered. Call this x. Then
in the next two columns calculate the corresponding tempering of the
major and minor thirds, y=4x-1 and z=1-3x. Then from those calculate
the three beat ratios in the major triad 3/2*z/y, 5/3*y/x, 5/2*z/x and
the three in the minor triad, z/y, 2*y/x, 2*z/x. Then scan down the
columns until you find the beat ratios you are looking for and note
the corresponding value in the comma fraction column (x).

Algebraic functions for 5-limit temperaments

🔗Gene Ward Smith <gwsmith@svpal.org> <gwsmith@svpal.org>

🔗Dave Keenan <d.keenan@uq.net.au> <d.keenan@uq.net.au>

🔗Gene Ward Smith <gwsmith@svpal.org> <gwsmith@svpal.org>