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

Ildar Khannanov solfeggio7 at yahoo.com
Tue Dec 3 14:27:24 PST 2013

Dear Marcel,

I have tried just that. I decided to save on tuning and do it myself. I bought a tuning fork (some 100USD), took one evening off and started tuning. When are the two strings tuned perfectly at a fifth? When the beating stops. You will feel how two notes blend with each other, creating a beautiful sound, almost like a single note sounding. It made me so happy. So, I tuned the notes in the first octave, going by fifth up and fourth down (they are the same, just inversions, am I right?). However, when I listened to an octave--it did not sound tuned, at all. The more notes I tuned by Pythagorean perfect fifth, the more the notes were running away from me, like rabbits. I got very angry at Pythagoras! So much for the diatonic scale tuned by fifth! And, check the major thirds then--they will be also out of tune, definitely not expected 5/4 ratio. And if you stack the fifth from the Do in the bottom of the keyboard until you have reached another Do, the two
 Do's will be 1/5 of a semitone apart. 

The resume: Pythagorean slogans are like Communist Party slogans: they may sound good but digress significantly from reality. Just as the joke of Archytas the Pythagorean, which everyone seemed to buy: the three types of tetrachord. Was he a practical musician? Where did he get these three from? No answer. Please, find me a person who can sing effortlessly a chromatic tetrachord after the enharmonic tetrachord. From Mi down to Ti. Try to hit that C double flat after the D sharp.

Best wishes,

Ildar Khannanov
Peabody Conservatory
solfeggio7 at yahoo.com

On Tuesday, December 3, 2013 4:33 PM, Marcel de Velde <marcel at justintonation.com> wrote:
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
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
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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