Historical temperaments

>From postings....
>I wonder why meantone tunings were abandoned.
>Could it have been due to the limitations imposed on composers and
>performers, when using twelve fixed pitches per octave?

Almost certainly that was the most basic reason.


>...but rather that the well temperaments
>were an abandoned cul-de-sac within a generally meantone era,

In terms of required resources that makes sense, but not in terms
of
actual usage to the best of my knowledge. Correct me if I'm wrong,
but I
know of no evidence that meantone was never actually used in any
appreciable abundance in the Classical Era or later.

- - - - - - - - - - - - - - - - - - - etc.

Greetings List
I would be interested to learn about the use of meantone in any Era.

Whatever tuning schemes were used, Equal Temperament "proved" better.
At any rate, that is what pianos are tuned to,and most all keyboards
today. When ET was first used for tuning, (and I assume it was on
pianos other wise it would have been on harpsichrods, or even earlier
on organs) that is an interesting question. (My guess is after 1850)
We know that ET is based on the 12th root of 2.
When that formula came into being and by whom I would like to know.
The other question is could Equal Temperament exist without knowledge
of the 12th root of 2 formula?? I think so, but that is for another
post.
Regarding the 12th root of 2, it doesn't mean much if you don't know
the frequency of the note. When did it become possible to measure
the frequency of a musical pitch? The frequency of pitch also allows
determination of harmonics. Harmonics is what produces the beats (or
lack of) by which any instrument is tuned, especially when tuning by
intervals such as the fifth, fourth, third, octave and so on. But one
doens't need to know frequencies or math formulas to hear beats
resulting from tuned intervals.
The schemes for tuning intervals on the keyboard is interesting,
however after tuning pure fifths it soon becomes apparent that one
fifth will be "wild" (and consequently its fourth). (and only one
fifth which is interesting) Because of this one lone "wolf" we have
the conecpt of temperament. Keep in mind this is a phenomona of
keyboards of 12 semitones to the octaves. Orchesterial and choral
performances don't seem to be encumbered by temperament.

Richard Moody piano tuner technician

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>I wrote,
>
>> Marion, I am using geometrical pictures to express my thoughts and your
>> LCM idea, as Graham pointed out, is equivalent to a particular
>> geometrical picture as well, a rectangular one where 2 is a distinct
>> factor that has its own axis, and the log of the LCM is just the
>> distance (according to the city-block metric) between the two most
>> distant points in the chord.
>
>I was being a little muddy here. I should have written, the log of the LCM is
>just the distance between the two most distant points in the smallest
>(hyper-)rectangular region that contains the chord.
>
>> In other words, the LCM is a measure of the
>> longest extent of the ROHS of a chord.
>
>I think that was correct, using Graham's terminology.
>
>> Since I feel that a minor triad
>> is more consonant than a major seventh chord, I don't like this method
>> and would prefer instead to use the triangular lattice. What replaces
>> the LCM in that case? If we use the isosceles triangles eluded to above,
>> the answer is just the Partch limit!
>
>That is, the log of the Partch limit will indeed be the distance (according
>to a city-block-type metric) between the two most distant notes of a chord in
>the isosceles triangular lattice. There is no need to construct anything like
>a ROHS. If a certain note appears twice in the lattice, as will happen in the
>9-limit, use the closer one to evaluate the distance from any other note. Of
>course, the distance between two instances of the same note is zero. Keeping
>these rules in mind, the fact that the two recently discussed 6th chords are
>merely 9-limit will be clear.
>
>Marion and Graham, have either of you not yet received TD 1122? If so, I will
>repost my posts from there, so we can continue our discussions.

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>>>i.e., a major seventh is just as dissonant/complex as a major
>>>sixth.

Although I certainly respect that as an opinion, I'm amazed that anybody
would feel that way.

If by dissonant you mean the commonplace (but technically not correct)
meaning of "discordant", then I'd say that there's no comparison; to my
ears, a M6 sounds every bit as "sweet" as a major third, whereas a M7
sounds very jarring. Of course that's all at least somewhat dependent upon
timbre.

If by dissonant you mean the more traditional (but less common) meaning
of "less resovlved within a keyframe", the I'd say that the contrast is
even more extreme, since any seventh chord in traditional harmony is
regarded as a dissonance.

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>>>i.e., a major seventh is just as dissonant/complex as a major
>>>sixth.

Gary Morrison wrote

>Although I certainly respect that as an opinion, I'm amazed that anybody
>would feel that way.

I don't know if anybody actually does feel that way, though I'm glad to
have Gary on my side in this. I was simply pointing out that if the
rectangular matrix (Euler-Fokker, Tenney, Rapoport) view is applied to
questions of consonance and dissonance, and octave equivalence is
assumed, the above comes out as a consequence. I know that certain
applications of Euler-Fokker theory have assumed octave equivalence, for
example using the genera as scales for compositions (I recently pointed
out that in a triangular latice, there are many structures as compact or
more compact than the E-F genera). I also know that James Tenney
examines the lattice along with a city block metric to give a measure of
harmonic complexity, but I don't know whether he assumes octave
equivalence.

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