[plt-scheme] music theory (was Natural numbers)

From: Jos Koot (jos.koot at telefonica.net)
Date: Thu Mar 12 08:10:02 EDT 2009

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.

----- 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)

> 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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