Golden meantone

Golden meantone (or "golden diatonic" temperament-agnostically) is based on making the relation between the whole tone and diatonic semitone intervals be the Golden Ratio

[math]\displaystyle{ \varphi = \frac 1 2 (\sqrt{5}+1) \approx 1.61803\,39887\ldots\, }[/math]

This makes the Golden fifth exactly

[math]\displaystyle{ (8 - \varphi) / 11 }[/math]

or

[math]\displaystyle{ (3\varphi + 1) / (5\varphi + 2) }[/math]

octave, equivalently

[math]\displaystyle{ (9600 - 1200 \varphi) / 11 }[/math]

cents, approximately 696.214 cents.

This can be approached by successively taking soft child MOSes of pentic (2L 3s, 5L 2s, 7L 5s, 12L 7s, 19L 12s, etc). Each time, the generator range "narrows in" on the golden diatonic generator.

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

The process behind constructing golden meantone can be generalized to other MOSes.

Construction

Golden meantone is approximated with increasing accuracy by the infinite sequence of temperaments indicated in the table below. In any meantone temperament the 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. As the Octave is the Sum of two Fourth intervals and a Tone this can be rearranged as the Sum of the first five intervals in this table, the sequence of EDOs is a Fibonacci like sequence where terms are the sum of 5 consecutive Fibonacci numbers.

As the cardinality increases the interval sequence better approximates a geometric progression. 81edo marks the point at which the series ceases to display audible differences and approximates all of theses intervals within 1 cent. For 131edo and further, the best 5th can no longer be used as the generating interval for a golden meantone tuning.

Temperament	Chroma	Semitone	Tone	Minor third	Fourth	Minor sixth
7edo	0	1	1	2	3	5
12edo	1	1	2	3	5	8
19edo	1	2	3	5	8	13
31edo	2	3	5	8	13	21
50edo	3	5	8	13	21	34
81edo	5	8	13	21	34	55
131edo	8	13	21	34	55	89
...	...	...	...	...	...	...

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

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[-odd]-limit tuning. It's fairly good as a 7[-odd]-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.

—Graham Breed