Graham Breed wrote:
>> This link says Huygens discovered the close relation****p between
>> meantone and E31:
>> http://www.xs4all.nl/~huygensf/english/huygens.html
>
> It says he demonstrated it. Not that he discovered it, in the sense
> that nobody knew it before. And the page itself mentions precedents.
Good point. In the context I merely wanted to point out how early the
idea was. The Archicembalo a little earlier did not use E31, though
being quite close if the values given are to assumed:
http://en.wikipedia.org/wiki/Archicembalo
(Apparently adding a few extra keys to get a few more pure fifths.)
With the key map I made for Scala, it is only essential that the pitch
system is diatonic. So playing in meantone is the same as in E31, though
I do not know if Scala can produce it.
>> But ETs is just a convenient way to produce tunings. The main point is
>> that the music is diatonic: any choice of minor and major seconds will
>> produce an intonation.
>
> For suitable definitions of "diatonic".
I mean a pitch system generated by a minor and a major second. The Just
intonation that uses two major seconds of different sizes sound weird in
my ears, and they need to be tempered out in order to work musically.
So I think only a true diatonic pitch system will work well.
> Some of us prefer to say
> "meantone" in a general sense.
Most of these concepts have a general and a narrow meaning. Here, I
meant the one where the major second is sqrt(5/4), said to be common in
the Renaissance.
> The defining feature is that four
> perfect fifths add up to a major third, give or take some octaves.
What I call a diatonic scale in the strict sense requires the major
third to be made up by two major seconds. This excludes the traditional
(narrow sense) Just intonation, then.
But notion of diatonic pitch system works with scales degrees other than
seven per octave. This may be needed if more partials should be
included. (Cf. Paul Erlich's 22-tone system.)
> There were plenty of just intonation scales called "diatonic" throughout
> history.
This term can also be used ina general sense, i.e., any scale using
rational intervals. But I used it narrowly, the one using the intervals
5/4 and 3/2.
>> Only some modern music use the enharmonic equivalence of E12 in an
>> essential way. But for music produced in E12, as in a MIDI file, the
>> problem is to extract the underlying diatonic structure before
retuning.
>
> There's more to it than that. Chromatic chords still have to be tuned
> and sometimes a chord may work best with a tuning outside meantone
theory.
I think the OP situation is the tuning into another diatonic pitch
system with no other pitches outside it available. So no tone outside
are available. So one is faced essentially with translating enharmonic
E12 names into diatonic. And this may not work if the music is chromatic
E12 is an essential way.
Hans
|
