Forwarded: Spud duBoise's remarks on Kornerup's golden meantone

I wrote,

>>>>Is there any rhyme or reason to the ratios you're
>>>>associating with the circle of fifths?

Spud wrote,

>The original choice of just intervals was not intended to be arbitrary. It
was based on the observation that the preferred cultural development of
scales goes from 5-tone to 7-tone to 12-tone (each displacing but not
eliminating the former). The most efficient division of the scale by each of
these numbers has been reported in ethnomusicological literature: 5-TET in
Uganda; 7-TET in Thailand; 12-TET in Europe (starting with luthiers).
However, equal temperaments have limited capacity for expansion as
meantones, because additional pitches coincide exactly with those already
there.
>
>The addition of tones to most meantone systems will eventually hit
stretches where the added pitches fall too close to those in existence to be
practical in use. The only method I have found that avoids this inefficiency
is division by the golden ratio. Applying the golden ratio to 5-tone,
7-tone, or 12-tone produces Golden Meantone
(<http://www.rev.net/people/aloe/music/golden.html>), which is close to
1/4-comma, although not identical. Because Golden Meantone works
consistently for low numbers, it is extended to higher numbers. That
produces close correspondence to the Fibonacci sequence {2, 5, 7, 12, 19,
31, 50, 81, 131, . . .}. So Golden was used as the base for choosing just
intervals from those of the smallest limit practicable. However, I have
started to make exceptions to the limit rule, with some misgivings. I may
eliminate those exceptions, because I don't see justification for them.
>
>Conceptually, new notes don't seem to generate from a void. They are more
likely to be daughters of pitches in the previous generation, e.g., minor
second descended from natural (major) second. This parallels another
acoustically-based digital theory, that of phonology, in which more complex
systems of vowels can be explained by the imposition of additional variables
to simple systems. The vowels in a standard system of five (the most popular
in the world, occurring in such languages as Spanish, Russian, Japanese,
Swahili, and Hawaiian) can be differentiated by the use of three variables.
Splitting the mid vowels "e" and "o" into two states of tension produces a
seven-vowel system, which Italians recognize only in speech, Portuguese
acknowledge with accents, and ancient Greeks designated with different
letters. Alternatively, allowing front (or as theory describes them,
non-back) vowels to be rounded (a quality otherwise reserved for back
vowels) results in the seven-vowel system of German articulation (although
the rounding determines the identity of the vowel, with an umlaut nullifying
the location in the back of the mouth. (Germans use a subset of a
theoretical 12-vowel gamut that also includes Finnish and Romanian, plus a
long-short durational distinction that can also be explained in terms of 14
of 24 vowels). As with most behavioral sciences, the abilities to use these
distinctions and to explain them do not necessarily coincide. I wonder
whether there has been research to see if there is any relationship between
the overtones that identify vowels and the frequencies of notes in chords
that accompany the syllables in which they are sung.
>
>To pursue a tangent, the undesirability of "inefficient" spacing of pitches
seems to be aesthetic, not merely hypothetical. People seem to prefer scales
with two or more sizes of adjacent intervals, because uniformity is bland.
On the other hand, if the spacing is too severe, such as quarter-tones in a
pentatonic scale (<http://www.rev.net/people/aloe/music/enharmonic.html>),
the scale is not comfortable. Quarter-tones might be acceptable in a
double-digit scale. It's too early to tell, because the heptatonic scale has
yet to be eclipsed in popularity by the dodecatonic. (Although equivalent
changes in taste have previously taken centuries, they are subject to
acceleration by technology, as Ed Foote indicates.)

>>Adding tones after the first three always divides intervals in the
>>golden ratio, which you're right is roughly 5:3, but is exactly
>>(sqrt(5)+1)/2 . . .
>
>Yes. I wanted to give the reader some idea where the ratio fell, but it is
not as clear as intended. I'd better take another look at it.