Yahoo Tuning Groups Ultimate Backup tuning RE: [tuning] Re: 53-tet, meantones, etc. -- for Monz, Paul, and G raham

Margo wrote,

>A very simple way of demonstrating this is to note that in 53-tet
>treated as a 5-limit approximation, a ~5:4 major third will be 17
>steps. Now 17 is an odd number, and cannot be divided into two
>"mean-tones" of equal integer size which together produce this
>third.

[...]

>If we want a 53-tet-like tuning that _does_ support meantone in the
>usual sense, then maybe 106-tet would be an ideal solution. Here our
>near-5:4 major third would be 34 steps, so that a mean-tone of 17
>steps (the mean of 9/53 octave and 8/53 octave) would indeed answer to
>your requirements. This mean-tone is ~192.45 cents, very slightly
>smaller than the whole-tone in 1/4-comma meantone (~193.16 cents)
>because our 53-tet or 106-tet "schisma third," like its Pythagorean
>counterpart, is slightly narrower than 5:4.

Margo, you can use a similar argument to the first paragraph above to show
that 106-tET is _not_ a meantone tuning. If the major second is 17 steps,
the major ninth is 106+17 = 123 steps. The major ninth can be divided into
two perfect fifths. But since 123 is an odd number, the meantone tuning you
intend does not have a perfect fifth.

With another doubling, however, the problem is solved -- 212-tET contains
both 53-tET and meantone tunings. This could have been apparent from the
start by noting that one needs to flatten the fifth by 1/4-comma in a
meantone, and that in 53-tET, which has virtually pure fifths, one degree is
approximately equal to a comma, so by dividing each degree into four parts
(obtaining 53*4=212-tET), one effects a meantone tuning.

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Along the lines of 212-tET . . .

In my paper I mention 76-tET as a kind of "lowest common denominator" for
all the tonal systems discussed therein. Using its 44-degree (694.7¢)
fifths, it supports 5-limit diatonicism as in 19-tET (since 76=19*4), and
the 7-limit 14-tone scales I found in 26-tET and that Dave Keenan found were
better in 38-tET (since 76=38*2). Using its 45-degree (710.5¢) fifths, it
supports the 10-tone scales that are the main subject of the paper (since 76
contains the half-octave interval) and would also support a rather extreme
xeno-gothic tonality. Either of these fifth sizes could be used for 3-limit
pentatonicism, though with strongly different melodic characters; and the
44-degree fifth could be used for octatonicism with its eight 5-limit
consonant triads, by interlacing two 4-tET scales 44 degrees apart . . .