[Smt-talk] Fwd: Re: a la mode

Marcel de Velde marcel at justintonation.com
Tue Dec 3 12:05:58 PST 2013

Dear Ildar,

You wrote:
> As for the mathematical toys, such as a stack of fifth--since 
> Pythagoras some inquisitive scientists are trying to harness music 
> with the rules of high school math. If you believe that diatonic is 
> the product of the stack of fifth, take a tuning fork and tune your 
> piano by perfect fifth. Gook luck! In Greece there has been a 
> controversy (canoncs vs harmonics), but practical musicians (including 
> Aristoxenus and his father) did not adopt Pythagorean views. They were 
> too abstract and impractical.

I can assure you that the chain of fifths (and octaves) is not a mere 
"mathematical toy".
The math may be easy, primary school level will do, but the consequences 
profound and fundamental.
Our notation system is based on the chain of fifths, and most of our 
music theory along with it.
The chain of fifths is what makes a diminished fourth a different 
interval from the major third etc, and at the same time gives no "2 
different major thirds"
And the reason this all works so perfectly is that the chain of fifths 
corresponds to how the human brain quantizes the pitch space / how we 
automatically categorize intervals.
Tune the fifths and octaves perfect and you get Pythagorean tuning.
I will not get into a further debate on tuning as it is probably to 
unrelated to the topic of this thread, but I will say that it is not 
merely an old controversy in Greece but that Pythagorean is also still 
practiced today naturally by violin, singers, trombone etc when playing 
solo (both taught and automatically done so). And was the default tuning 
system for polyphonic music in the west until replaced by 1/4 comma 
meantone and other meantone tunings. And the 12-tone equal tempered 
tuning system that we use today for fixed pitch instruments is very 
close to Pythagorean tuning, it is hard to hear the difference with 
Pythagorean as long as one follows the enharmonic notation correctly 
(don't play a diminished sixth as if it is a perfect fifth, or augmented 
second when there should be a minor third etc)

It is very easy to tune the diatonic system based on perfect fifths on a 
piano. Tuning fork is not even needed as it can be done by ear with ease.
Lets start with F tune perfect fifths above it and reduce them by 
perfect octaves when needed and one gets:
F-C-G-D-A-E-B, or C D E F G A B C when going up in steps from C. 
Containing all the strict diatonic scales. A chain of 6 perfect fifths 
linking 7 tones like described.
Tune one fifth extra and you get a chromatic interval, F-F#. And one can 
continue adding fifths till one gets D#-F a diminished third.
Makes good sense to call these the chromatic intervals, from 7 to 11 
fifths. So the chromatic scale covers 12 tones linked by perfect fifths.
If we add one more fifth we get E# - F, which are enharmonically 
related. So a chain of 12 fifths and upwards are the enharmonic intervals.
To me this is the best description there is.
It also allows the following clarification which corresponds to how we 
perceive music.
Something like the harmonic minor scale is by this definition a 
chromatic scale.
However, one can use it melodically in such a way that the melody only 
moves by diatonic intervals within this scale.
So the augmented second is not used as a melodic step. This is the most 
common use of this scale in western music so one can still speak of 
diatonically based music in a way.

Kind regards,

Marcel de Velde
Zwolle, Netherlands
marcel at justintonation.com

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