[music-dsp] Re: pitch shift

[Richard Dobson]

The harmonic calc should be 550 = 440 * 5/4 (or, 440/2 * 5/2); the major
third as the fifth harmonic (two octaves plus a third). The major third
is the worst interval in equal temperament - seriously wider than the
'pure' meantone interval.

To cut a horrendously complicated story short, the non-equal
temperaments tried to square the circle - get as many perfect fifths as
possible, in the 12-note scale (can't be done - 12 perfect fifths is
wider than seven perfect octaves, or (3/2)^12 > 2^7), and, even more
difficult, get as many pure thirds as possible. Typically, one fifth was
sacrificed as the 'wolf' fifth , e.g. Ab to D#, so that the rest could
be purer. To make the wolf fifth less scary, a few other fifths could be
narrowed, etc, etc etc.

Current scholarship considers that the "Well-Tempered Klavier" was
written not for equal temperament, but for a good ("well") unequal
temperament; possibly Werckmeister III. Yes, there are 24 preludes and
fugues, one in each major and minor key.

Richard Dobson



"Sergio R. Caprile" wrote:
> 
> >> The equal tempered scale advances on a 2^1/12 rate. given A=440, C# is
> >> 440*(2^1/12)^4=554.365 . Calculating C# by harmonics, it is 440*5/2=550.
> >
> >Sergio, there's some wrong with this equation...
> >And what's the theory behind the ideal tempered scale? (something about
> >harmonics I guess?)
> 
> I've been using this theory for a long time, don't remember where it came
> from. May be I should have written the exponent in parenthesis: k=2^(1/12),
> that is, "the twelfth root of two".
> If you want to advance an octave:
> new_note=old_note*k^shift
> 1octave(12 semitones)=> [2^(1/12)]^12=2, it holds
> When I tested it the calculations where coincident with the note tables (of
> an equal tempered scale). If you apply a log2 then you get log2(k)=1/12 and
> all the notes are equally distributed on a line on a logarithmic scale.
> 
> Before the equal tempered scale (Bach times ?), instruments were tuned for
> each particular scale, I mean, if the musical piece was in Am, and then you
> were to play a tune in Dm, you had to retune your instrument. I'm not sure,
> have to ask my "real" musician friends, but I guess they used harmonics
> (natural sounds ?).
> Also, C# and Db were actually different notes, as they came out from
> different roots (being in different diatonic scales). AFAIK, Bach showed (in
> "Well tempered clavier") that it was possible to play with a uniform scale.
> I've been told that "Well tempered clavier" includes a piece in every key,
> to show that all the keys can sound OK with an instrument in an equal
> tempered scale; however, I didn't check.
> 

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