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RE: TUNING digest 998

🔗Manuel.Op.de.Coul@ezh.nl (Manuel Op de Coul)

2/27/1997 8:29:37 AM
From: PAULE

Amazing how my posts are continually misunderstood. I wrote,

>> In particular, would some mathematician not have interpreted
>>Aristoxenus' ideas in terms of the twelfth root of two?

Why does that lead people to argue that Aristoxenus did not advocate equal
temperament? Of course he did not. The idea of irrational numbers was too
repugnant to the ancient Greeks. Aristoxenus just described what he heard,
and a different mathematical mindset may have led to different
interpretations of his descriptions. As these descriptions were
self-contradictory, it seems useless to try to figure out what Aristoxenus
"actually meant" in mathematical terms; however, this didn't prevent ancient
Greeks or 16th century theorists from doing just that.

>> I do agree that in most cases the diatonic scale existed without a
>> standardized tuning. I believe that the creation of similar tetrachords
>> was the only guiding principle in most cases, since I do not believe that
>> 5-limit or higher ratios are relevant to purely melodic music. The 3-limit
>> (and even 2-limit) approximations have a wide range of tolerance when
>> harmony is not an issue, or when inharmonic instruments make the value of
>> just intonation questionable.

>What is a 2-limit approximation of a diatonic scale. The "2-limit" gives us
>nothing but octaves. Or did you mean 2^1/x, in which case the "limit"
>terminology is out of place.

No, I meant all 2-limit approximations, i.e., octaves. Many cultures use
distinctly out-of-tune octaves, yet still recognize an equivalence between
pitches so separated.

Actually, I prefer to use the "limit" terminology in Partch's original
sense, for all odd numbers and odd numbers only. However, when discussing
tuning systems rather than consonance/dissonance issues, the prime usage can
occasionally be useful.

I don't know how the word "mystical" got attached to Ptolemy, if I made the
attachment, I apologize, I should have substituted a better word.

>> Matt, you could be given a Mozart string piece which would have to
descend
>> by, say, seven commas from beginning to end, if it is to have all chords
>> in just intonation and no comma shifts in sustained tones (this is
>> actually a typical scenario). Even though the beginning and ending keys
>> may be notated exactly, you would say that the piece ends in a distincly
>> different key than it began. Would your analysis reflect the musical
>> reality better than the traditional analysis? I think not! What if the
>> piece was a keyboard piece?
>> Forget it!

>Leopold Mozart, being representative of a 17th/18th-century string playing
>tradition, specified 1/6th-comma meantone as the tuning which expert
players
>should employ. His son, it seems, endorsed this (see the Chesnut article
>"Mozart's Teaching of Intonation" in JAMS, c.1980 -- sorry about the vague
>date), and with it the corollory of tuning flats higher than sharps that
>mean-tone tunings share with just intonation. Since string players are
>limited by the tuning of their open strings, they cannot incorporate any
>irreversable comma shifts (as opposed to some local fluctuation), but the
>use of any meantone tuning removes this problem, by means of their
different
>characteristic compromises.

Thank you for the info, Jonathan, meantone was my choice as well.

-Paul Erlich

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