reply to Dave Hill and Doren's Mozart question

Dave Hill (I believe that's who Ascend11 is) wrote,

>Although the piano is at present usually tuned to equal temperament, it

>can be tuned flexibly and practically any temperament may be tuned
>on any ordinary piano. It is possible to tune a piano to sound triads
>in just intonation in six major keys and six minor keys. Non-trivial
>music can be performed on a piano in these tunings, although
>with a just intonation tuning and only twelve notes per octave, there
>are considerable limitations. With quarter comma mean tone
>temperament, which has just major thirds and fifths flat by 5.4
>cents - an amount discernible, but not too detrimental to much
>music, music may be performed in eight contiguous major keys
>and eight contiguous minor keys, affording a great deal of
>flexibility.

How exactly are you defining "keys?"

As normally thought of, 12-tone meantone temperament is capable of
playing music in six, not eight, major keys: since each black key must
be either a sharp or flat, one can play in any major key with up to n
flats and up to 5-n sharps, plus C major, for a total of six keys. In
other words, the major key spans seven consecutive notes in a chain of
fifths, while the tuning contains twelve; thus only five transpositions
of a given major key are possible. Minor keys usually require the
leading tone so only three minor keys are really possible on a 12-tone
meantone keyboard.

Just intonantion is not even capable of fully playing one major key
unless a syntonic comma exists between two notes of the tuning. However
if the ii triad is avoided, the problem is circumvented. Still, I cannot
see how more than three of these defective major keys can be accomodated
with 12 notes per octave.

Perhaps you really mean "triads" or "chords" rather than keys?

In any case, I think it quite likely that Mozart thought in something
like 1/6-comma meantone temperament, though in practice his
reinterpretation of augmented sixths as dominant sevenths led to tied
enharmonic equivalents that only make sense in a 12-tone tuning. For
pieces without tied enharmonic equivalents, meantone would probably make
the most appropriate "authentic" performance, and the 1/6 comma variety
makes Mozart's chromatic scales sound smoother (and keeps the leading
tone resolutions smaller) than the more harmonious 1/4 comma variety.

On Mon, 16 Nov 1998, Robin Perry wrote:
> The 200th root of 2 (6 cents) raised to the 15th power is 90 cents.
> The 200th root of 2 raised to the 19th power is 114 cents.
>
> Combinations of these two provide very close approximations to intervals
> up to 10/9.

Robin: try units of the 171th root of 2, and see how well those
intervals are approximated.

--pH http://library.wustl.edu/~manynote
O
/\ "'Jever take'n try to give an ironclad leave to
-\-\-- o yourself from a three-rail billiard shot?"

NOTE: dehyphenate node to remove spamblock. <*>

On Mon, 16 Nov 1998, Paul Hahn wrote:
> On Mon, 16 Nov 1998, Robin Perry wrote:
>> The 200th root of 2 (6 cents) raised to the 15th power is 90 cents.
>> The 200th root of 2 raised to the 19th power is 114 cents.
>>
>> Combinations of these two provide very close approximations to intervals
>> up to 10/9.
>
> Robin: try units of the 171th root of 2, and see how well those
> intervals are approximated.

Reading Robin's message again, I think I may have misunderstood the
thrust of it--the 200th root of 2 doesn't appear to be as important as
I'd originally thought. Rather, Robin seems to have discovered the same
septimal schisma that Margo Schulter just brought up a week or two ago.

--pH http://library.wustl.edu/~manynote
O
/\ "'Jever take'n try to give an ironclad leave to
-\-\-- o yourself from a three-rail billiard shot?"

NOTE: dehyphenate node to remove spamblock. <*>

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End of TUNING Digest 1583
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