Higher limit beat ratios

Wendell defined his beat ratio in terms of the major and minor thords.
If f is the fifth and t is the major third, then the brat is
(6t-5f)/(4t-5). It makes for an equivalent but more straightforward
theory to define it in terms of the major third and fifth instead. Let
us set b5 = (4t-5)/(2f-3). Then we have the relationships

brat = (3b5 - 5)/2b5
b5 = 5/(2-3*brat)

If the tuning of the approximate 3 is 3+x, then the tuning of the
approximate 5 will be 5+b5x. Let us define the higher limit beat
ratios so that for prime p, the tuning of the approximate p is p +
bp*x. So long as we are not dealing with a pure fifths tuning, where x
is zero, the list of beat ratios (b5 b7 ... bp) gives a complete
picture. In a pure fifths system, we would need to refer everything to
something else, such as the third.

We may derive the composite beat ratios from the ratios for primes;
for instance, if the "3" is 3+x, the "9" will be 9+2x+x^2, with a
ratio of 2+x, close to 2. The "15" will be 3b5+5+b5x, and so forth.

We may also derive an approximate formula for x, which immediately
gives us the tuning for a given set of beat ratios and a given comma.

If a comma c has monzo |c2 c3 ... cp>, then

c2ln(2) + c3*ln(3+x) + c5*ln(5+b5*x) + ... + cp*ln(p+bp*x) = 0

for a value x correctly chosen to temper out c. This can be rewritten

ln(c) + c3*ln(1+x/3) + c5ln(1+b5*x/5) + ... + cp*ln(1+bp*x/p) = 0

Linearizing the logarithm, we get the approximate formula

x ~= -ln(c)/(c3/3 + b5*c5/5 +...+ bp*cp/p)

This gives an excellent approximation to the tuning, particularly for
more accurate temperaments where x is very small.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> Wendell defined his beat ratio in terms of the major and minor
thords.
> If f is the fifth and t is the major third, then the brat is
> (6t-5f)/(4t-5). It makes for an equivalent but more straightforward
> theory to define it in terms of the major third and fifth instead.
Let
> us set b5 = (4t-5)/(2f-3). Then we have the relationships
>
> brat = (3b5 - 5)/2b5
> b5 = 5/(2-3*brat)

I get b5 = 5/(2*brat-3), does that matter?

>
> If the tuning of the approximate 3 is 3+x, then the tuning of the
> approximate 5 will be 5+b5x. Let us define the higher limit beat
> ratios so that for prime p, the tuning of the approximate p is p +
> bp*x. So long as we are not dealing with a pure fifths tuning,
where x
> is zero, the list of beat ratios (b5 b7 ... bp) gives a complete
> picture. In a pure fifths system, we would need to refer everything
to
> something else, such as the third.
>
> We may derive the composite beat ratios from the ratios for primes;
> for instance, if the "3" is 3+x, the "9" will be 9+2x+x^2, with a
> ratio of 2+x, close to 2. The "15" will be 3b5+5+b5x, and so forth.
>
> We may also derive an approximate formula for x, which immediately
> gives us the tuning for a given set of beat ratios and a given
comma.
>
> If a comma c has monzo |c2 c3 ... cp>, then
>
> c2ln(2) + c3*ln(3+x) + c5*ln(5+b5*x) + ... + cp*ln(p+bp*x) = 0
>
> for a value x correctly chosen to temper out c. This can be
rewritten
>
> ln(c) + c3*ln(1+x/3) + c5ln(1+b5*x/5) + ... + cp*ln(1+bp*x/p) = 0
>
> Linearizing the logarithm, we get the approximate formula
>
> x ~= -ln(c)/(c3/3 + b5*c5/5 +...+ bp*cp/p)
>
> This gives an excellent approximation to the tuning, particularly
for
> more accurate temperaments where x is very small.
>

>Wendell defined his beat ratio in terms of the major and minor thords.
>If f is the fifth and t is the major third, then the brat is
>(6t-5f)/(4t-5). It makes for an equivalent but more straightforward
>theory to define it in terms of the major third and fifth instead. Let
>us set b5 = (4t-5)/(2f-3). Then we have the relationships
>
>brat = (3b5 - 5)/2b5
>b5 = 5/(2-3*brat)
>
>If the tuning of the approximate 3 is 3+x, then the tuning of the
>approximate 5 will be 5+b5x. Let us define the higher limit beat
>ratios so that for prime p, the tuning of the approximate p is p +
>bp*x. So long as we are not dealing with a pure fifths tuning, where x
>is zero, the list of beat ratios (b5 b7 ... bp) gives a complete
>picture. In a pure fifths system, we would need to refer everything to
>something else, such as the third.
>
>We may derive the composite beat ratios from the ratios for primes;
>for instance, if the "3" is 3+x, the "9" will be 9+2x+x^2, with a
>ratio of 2+x, close to 2. The "15" will be 3b5+5+b5x, and so forth.
>
>We may also derive an approximate formula for x, which immediately
>gives us the tuning for a given set of beat ratios and a given comma.
>
>If a comma c has monzo |c2 c3 ... cp>, then
>
>c2ln(2) + c3*ln(3+x) + c5*ln(5+b5*x) + ... + cp*ln(p+bp*x) = 0
>
>for a value x correctly chosen to temper out c. This can be rewritten
>
>ln(c) + c3*ln(1+x/3) + c5ln(1+b5*x/5) + ... + cp*ln(1+bp*x/p) = 0
>
>Linearizing the logarithm, we get the approximate formula
>
>x ~= -ln(c)/(c3/3 + b5*c5/5 +...+ bp*cp/p)
>
>This gives an excellent approximation to the tuning, particularly for
>more accurate temperaments where x is very small.

Wow dude.

-Carl