According to Wikipedia, the size of the perfect fifth in 1/4 comma meantone is "flattened by one quarter of a syntonic comma, with respect to its just intonation used in Pythagorean tuning (frequency ratio 3:2)." How exactly is this done mathematically?

A pythagorean fifth is the ratio 3/2.

A fifth in 1/4 comma meantone is 5^1/4 (1.495).

The syntonic comma is 81/80.

When I try to calculate the size of the fifth I end up with 1.246875 instead of the expected 1.495. Here's how I'm getting there: subtracting (81/80 * 1/4) (a quarter of the syntonic comma) from a pure fifth (3/2). What am I doing wrong?


Mathematically, everything about pitch is logarithmic. "Adding a perfect fifth" really means "multiply the pitch by 3/2. So "subtracting a syntonic comma" means multiplying by the reciprocal of the comma; and "a quarter of" means the fourth root of. So try working out:

  • 3/2 x 4th-root(81/80)

...and see if this is more like the correct answer. (I haven't checked, but 4th root is at least easy on a calculator). HTH

  • This was really helpful- thank you! The math you wrote out was almost correct but you forgot to use the reciprocal of the the 4th-root of 81/80 as you suggested. Here's how you would write the calculation in the Lisp programming language (chosen because it makes the order of operations clear): (* 3/2 (/ 1 (expt 81/80 1/4))) – patterkyle Jan 16 '16 at 15:26
  • Good; I almost deliberately left it to you to get it right. Note that the other answer (by @Dave) also includes the magic word "logarithmic". This is the key to it, and the basis for the logarithmic "cent" system -- a cent is a multiplicative constant equal to 2^-1200. – Brian Chandler Jan 16 '16 at 15:41

In terms of frequency ratios "flattening" is not "subtracting" at least not mathematical subtraction. In the way that you are expressing it the mean tone fifth would be (3/2)/[ (81/80)**(1/4)]=1.495... The reduction is achieved by division.

Often you want to think about things in terms of cents: a logarithmic measure of pitch. By working with these the multiplication and division (like the division in the expression above) turn into addition and subtraction; raising to a power (like the **(1/4) above) turns into multiplication. An equal tempered semitone is 100 cents; a second is 200; minor third 300, and so on up to an octave which is 1200. Mathematically, the cents value for a given ratio is cents = 1200*log2( ratio).

In terms of cents, a just intonated fifth is 702 cents; the sytonic comma is 21 cents; and the meantone fifth ends up at 702-(1/4)*21 = 697. Using cents makes the idea of subtracting intervals from one another make sense (ha!).

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