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).

--- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>"

<d.keenan@u...> wrote:

> I prefer to use numerical methods since a spreadsheet is all you

need

> for them.

If you have the tools to use it, my method rules, since, your method

takes longer and fails to deliver the precise answer. However, if you

want to extend this business to the 7-limit and need therefore to

look at approximations, this is something to try.