Golden meantone

Golden meantone is based on making the relation between the whole tone and diatonic semitone intervals be the Golden Ratio

$\varphi = \frac 1 2 (\sqrt{5}+1) \approx 1.61803\,39887\ldots\,$

This makes the Golden fifth exactly

$(8 - \varphi) / 11$

octave, or

$(9600 - 1200 \varphi) / 11$

cents, approximately 696.214 cents.

Edo systems in a Fibonacci style recurrence beginning with 7 and 12 are successively better approximations to this ideal.

Construction

Golden meantone is approximated with increasing accuracy by the infinite sequence of temperaments indicated in the table below. In any meantone temperament the five intervals in the column headings form part of a Fibonacci sequence (in the sense that each adjacent pair sums to the interval to its immediate right) and in these equal temperaments the sizes of these intervals (expressed in step units) are consecutive numbers from the integer Fibonacci sequence 0, 1, 1, 2, 3, 5... Both the rows and the columns of the table form Fibonacci sequences, and because the five intervals sums to an octave, the octave cardinalities in the first column are formed by summing the five numbers to their right. As the cardinality increases the interval sequence better approximates a geometric progression.

Temperament Chroma Semitone Tone Minor third Fourth
7edo 0 1 1 2 3
12edo 1 1 2 3 5
19edo 1 2 3 5 8
31edo 2 3 5 8 13
50edo 3 5 8 13 21
81edo 5 8 13 21 34
131edo 8 13 21 34 55
... ... ... ... ... ...

The success of Golden meantone can be understood in terms of the properties of quadratic approximants (q.v.) and the small size of the schisma.

Evaluation

Graham Breed writes: I think of this as the standard melodic meantone because the all these ratios are the same. It has the mellow sound of 1/4 comma, but does still have a character of its own. Some algorithms make this almost exactly the optimum 5-limit tuning. It's fairly good as a 7-limit tuning as well. Almost the optimum (according to me) for diminished sevenths. I toyed with this as a guitar tuning, but rejected it because 4:6:9 chords aren't quite good enough. That is, the poor fifth leads to a sludgy major ninth.