[plt-scheme] music theory (was Natural numbers)
Good Story. We are off topic I am afraid, but nevertheless the following:
I am not sure that bach used the nowadays equally tempered scale for Das
Whol Temperierte Klavier (I have been told differently once by a music
teacher) But, as you point out, it certainly was not the mean-tone tuning.
Mark though that a good singer, player of string or wind instruments (other
than organ),orchestra, chorus or ensemble do not use the equally tempered
scale all the time. In particular the terts of a final chord in major should
be taken somewhat higher than in equally tempered scale in order to obtain
good harmony.
Jos
----- Original Message -----
From: "Stephen Bloch" <sbloch at adelphi.edu>
To: "PLT Scheme ML" <plt-scheme at list.cs.brown.edu>
Sent: Thursday, March 12, 2009 3:59 AM
Subject: Re: [plt-scheme] music theory (was Natural numbers)
snip
>
>
> I'm not sure what you mean by "the multiplication wrinkle," but music
> theorists have known that an octave corresponded to doubling or halving
> of something, and a fifth corresponded to a factor of 1.5, at least since
> Pythagoras. If you recognize that an octave corresponds to doubling or
> halving, then a two-octave interval must correspond to a factor of 4, a
> three-octave interval to a factor of 8, and octave and a fifth to a
> factor of 3, and so on; again, I'm pretty sure this has been known since
> Pythagoras. Of course, Pythagoras was measuring string length or
> wind-column length; the connection to frequency may only go back to
> Helmholtz in the 19th century (I'm not sure).
>
> Anyway, this is enough to know that, as Jos points out, there is no
> positive-integer number of perfect fifths which exactly matches any
> positive-integer number of octaves, and therefore that Pythagorean tuning
> (tuning all the fifths to be exact 3:2 ratios, which is actually pretty
> easy to do aurally) can't possibly give you octave- invariant definitions
> for all the notes of the scale. In particular, (3/2)^12 is very close
> to, but not exactly, 2^7. One solution is to insert a "Pythagorean
> comma," a fudge factor that makes one of the fifths quite far off from a
> 3:2 ratio, at some point in the scale; ordinarily you put it someplace
> harmonically remote, like from F# to C#, someplace you'll never need if
> you're playing in keys like C, F, and G.
>
> Pythagorean tuning was largely replaced in the 15th century (IIRC) by
> "mean-tone" tuning, which also tries to get major thirds to be 5:4
> ratios, and minor thirds to be 6:5 ratios. This turns out to be
> inconsistent BOTH with perfect fifths AND with octave invariance, so they
> inserted a couple of other "commas" to fudge things into place.
>
> What happened in Bach's time was the replacement of both Pythagorean and
> "mean-tone" tuning by "equal-tempered" tuning, in which each semitone is
> exactly a frequency ratio of the 12th root of 2. Octaves still
> correspond to a 2:1 ratio, but NOTHING else is exactly any integer ratio.
> The result is that you can transpose to any key you want and the piece
> will sound pretty much the same, i.e. equally out- of-tune (from the
> perspective of somebody accustomed to "perfect" intervals). Bach
> demonstrated the system by writing a cycle of 96 short keyboard pieces,
> two preludes and two fugues in each of the 12 major and 12 minor keys.
> He seems to have been successful, because almost all Western music
> written in the past 350 years has been in the equal-tempered system.
>
> And now back to your usually-scheduled discussion of programming
> languages....
>
> Stephen Bloch
> sbloch at adelphi.edu
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