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

From: Stephen Bloch (sbloch at adelphi.edu)
Date: Wed Mar 11 22:59:39 EDT 2009

> On Wed, Mar 11, 2009 at 8:31 PM, Jos Koot <jos.koot at telefonica.net>  
> wrote:
>> (For who it concerns: I have put 'plus' between asterixes, because  
>> when
>> dealing with frequencies, you must multiply ratios of frequencies,  
>> not add
>> them :)

Robby replied:

> I'm not sure of the exact timing of these things myself either, but I
> believe the multiplication wrinkle is around Bach's time (the
> well-tempered claiver) which is significantly later than 4ths and
> 5ths, which seem to go way back.

> On Mar 11, 2009, at 9:51 PM, Jos Koot replied:
>> That could very well be the case. I am not sure about that and the  
>> only book I have that treats the matter well does not discuss  
>> history. The books that treat history of music (as far in my  
>> possession) are written by people afraid of numbers. Anyway, at  
>> the time the tempered scale was introduced, it must have been  
>> known that there are no natural number n and m such that (expt 3/2  
>> n) = (expt 2 m) (where 3/2 is a quint interval and 2 an octave  
>> interval).This is a very fortunate fact, because it allows to put  
>> tension into harmony  and melody.

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  

Stephen Bloch
sbloch at adelphi.edu

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