# woolhouse-unit

[Joe Monzo]

An interval measurement invented by Wesley S. B. Woolhouse, in his Essay on Musical Intervals [1835], and named "woolhouse-unit" by Joe Monzo in this work.

A woolhouse-unit is the logarithmic division of the octave into 730 equal parts. It is calculated as the 730th root of 2, or 2(1/730), with a ratio of approximately 1:1.000949968. It is an irrational number.

The formula for calculating the Woolhouse-unit-value of any ratio r is: woolhouse-units = log10(r) * [730 / log10(2)] or woolhouse-units = log2(r) * 730

Woolhouse created this unit specifically as a small logarithmic measurement to compare interval sizes, because the values for the basic 5-limit intervals which concerned him -- the octave, the two sizes of just intonation "whole-tones", and the diatonic semitone (which he non-standardly called limma) -- are all very close to integer values, making it easy to do computations disregarding fractional or decimal remainders, without accumulating too much error.

[Woolhouse 1835, p 18]

It will be useful to divide the octave into such a number of equal divisions that each interval of the scale may comprise an integral number of them ... such as will render the major and minor-tones and limma whole numbers, since all other intervals result from the various combinations of these elemental ones.

interval      ratio  ~Woolhouse-units

octave         2/1     730
major-tone      9/8     124.0452511
minor-tone     10/9     110.9622582
limma         16/15     67.96986521

A table of integral woolhouse-unit values for various intervals follows:

Woolhouse-units   3,5-monzo     ratio       interval

1        [ 8, 1 >   32805/32768   skhisma
12        [-4,-2 >    2048/2025    diaschisma
13        [ 4,-1 >      81/80      syntonic comma
14        [12, 0 >  531441/524288  pythagorean comma
25        [ 0,-3 >     128/125     enharmonic diesis
43        [-1, 2 >      25/24      small chromatic semitone
56        [ 3, 1 >     135/128     large chromatic semitone
68        [-1,-1 >      16/15      diatonic semitone
111        [-2, 1 >      10/9       small whole-tone
124        [-1,-1 >      16/15      large whole-tone
167        [ 1, 2 >      75/64      augmented-2nd
179        [-3, 0 >      32/27      pythagorean trihemitone
192        [ 1,-1 >       6/5       just minor-3rd
235        [ 0, 1 >       5/4       just major-3rd
248        [ 4, 0 >      81/64      pythagorean ditone
303        [-1, 0 >       4/3       perfect-4th
427        [ 1, 0 >       3/2       perfect-5th
470        [ 0, 2 >      25/16      augmented-5th
495        [ 0,-1 >       8/5       minor-6th
538        [-1, 1 >       5/3       major-6th
563        [-1,-2 >     128/75      diminished-7th
594        [ 2, 2 >     225/128     augmented-6th
606        [-2, 0 >      16/9       pythagorean minor-7th
619        [ 2,-1 >       9/5       just minor-7th

More precise values (to one decimal place) are shown on this 5-limit lattice-diagram:

(Shades of blue and orange designate error plus and minus, respectively, of the actual Woolhouse-unit values from the integral values. See the "Gallery of EDO error lattices" for a full explanation.)

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### woolhouse-units calculator

Ratio may be entered as fraction or floating-point decimal number.
(value must be greater than 1)

For EDOs (equal-temperaments), type: "a/b" (without quotes)
where "a" = EDO degree and "b" = EDO cardinality.
(value must be less than 1)

Enter ratio: = woolhouse-units

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