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

Ildar Khannanov solfeggio7 at yahoo.com
Tue Dec 3 16:39:53 PST 2013

Dear Marcel,

if you managed to tune piano to Pythagorean fifth without destroying other intervals you deserve a Nobel Prize in music theory. 

It is commonly known that 12 perfect fifths do not fit into 7 octaves (the overshooting is Pythagorean comma). Let us say, you have tuned your middle octave as follows: C-G-D-A-E-B (in a zigzag fashion). Your B will be higher than the equally tuned B exactly 5/12 of Pythagorean comma. 

You will have to detune your fifths (shrink them) and they will stop being Pythagorean perfect fifths. For ever. You can do two things: leave four fifth Pythagorean and sacrifice one of the fifth (say, supertonic to submediant) by detuning it 5/12 of the P. comma. That fifth will start howling like a wolf! And this fifth will not be able to serve as the basis for the tonic triad, if you would like to modulate to the key of supertonic. And in general, you will be able to play only in this key (on the condition of avoiding the supertonic triad). Or, you can detune each fifth to two beatings per second, and detune each fourth to one beating per second. All your fifth and fourth will be miserably out of tune, but you will receive a system of equal temperament. And a badly tuned instrument. 

Of course, hundreds of theorists before us had a dream of perfect tuning because, if the fifth is perfect, the two tones create resonance. Violinists like it, singers cannot live without it. The equally tuned piano, on the other hand, marks the end of Western music (at least, such is the opinion of Phillip Gosset). The piano makers put three strings for a note, in order to compensate for the dull and uninteresting sound of equally tempered intervals. Still, circonium is no match for diamonds (ask any woman!).


Ildar Khannanov
Peabody Institute
solfeggio7 at yahoo.com

On Tuesday, December 3, 2013 6:14 PM, Marcel de Velde <marcel at justintonation.com> wrote:
Dear Ildar,

That is very odd.
Indeed the fifths are perfectly tuned when the beating stops and
        the interval sounds perfectly stable.
And 4/3 fourth down should work as well in theory, though
        probably a bit harder to get perfect by ear.
It should work the way you describe. I'm not an expert in piano
        tuning at all but I did tune a piano this way myself once and it
        worked perfectly.
Though I was surprised by the great amount of detail needed, the
        smallest movement of the tuning wrench had an enormous effect on
        the tuning, I spent about a full day tuning the piano.
I tuned first the middle octave by adding fifths and reducing
        them by an octave when needed. Then I tuned the other octaves
        based on this middle octave.
I may not give a perfect result as the overtones of the piano
        become slightly inharmonic especially for the lower keys which
        gives a slight streching relative to the lower keys I belief.
        But audibly the piano I tuned this way was tuned to Pythagorean
        correct enough (and I know Pythagorean tuning very well).
I've read reports of others who have tuned this way as well with
        good results.
I can't tell what went wrong in your case, but please don't
        blame Pythagoras ;-)
Perhaps someone more experienced with piano tuning can explain
        what may have gone wrong.

As for the 5/4 major third vs the 81/64 Pythagorean major third.
This is the stuff of great debates.
The 5/4 major third gives what most people (including me) find a
        very pleasurable sound. Same goes for 7/4 for a minor seventh.
They are similar to the 3/2 major third in the way that it has
        the lowest harmonic overtone relation and therefore gives a very
        "synchronous buzzing" sound.
But is it in tune? I personally, after about 7 years (I lost
        count) of research, think it is not. At least not in the strict
        sense of "in tune".
I think the 5/4 is a coloration of the major third. This is not
        always wrong, and as I said it is a very pleasant sounding
        coloration. And I see good reason to prefer it to a perfectly
        "in tune" 81/64 major third in many situations.
But "in tune", or "just intoned" had a deeper meaning than
        simply a pleasant sound. It is to tune according to how we
        internally "understand" the interval.
And we internally categorize pitches according to a chain of
        fifths and octaves.
I can give many demonstrations of this but that would make this
        an email the size of a book perhaps haha.
One very simple example is to tune a circle progression. For
        instance I-vi-ii-V-I or I-IV-ii-V-I.
A very simple natural diatonic progression. To use 3/2 for
        perfect fifths, 5/4 for major thirds and 6/5 for minor thirds
        will not correspond at all to how we interpret this progression.
As all the notes are linked by a perfect fifth somewhere in this
        progression, and since 4 perfect fifths make an 81/64 and not a
        5/4, there is no mathematical solution that will allow one to
        tune this progression without some very unnatural modifications.
One will have to either allow a "wolf fifth" which sound
        horrible and makes no sense. Unacceptable in this progression.
Or one will have to allow a "comma shift", meaning a (possibly
        held) note will all of a sudden become a different note a comma
        higher. Sounds horrible and makes no sense either. Unacceptable.
Or one will have to allow the whole progression to "comma
        drift", which means it ends up a (Syntonic) comma lower every
        time it repeats. Sounds strange, makes no sense, and for many
        other progressions this does not work.
Another example: try tuning a big chord like F-A-C-E-G-B-D-F to
        5-limit "just intonation". Won't work without wolves.
5-limit or higher limit "just intonation" was nice to theorize
        about, but it has no practical place in music as for 90%+ of
        common practice music it is not mathematically possible, and it
        clearly does not correspond to how we perceive music.
Many people think pure "just intonation" is impossible. I think
        it is Pythagorean.
One can however tune the above things with the pleasant
        coloration of the 5/4 major third if one gives up the pure 3/2
        fifth, then one gets 1/4 comma meantone.
But 1/4 comma meantone performs very badly the more chromatic
        the music gets. It is pleasant for harpsichord playing
        renaissance music, both because of the music and because of the
        harsh harsichord timbre which can use the sweetening effect of
        5/4 very well. But for for instance late romantic or modern
        music it is much worse performing.
Here an example of an enharmonic progression in 1/4 comma
        meantone vs Pythagorean. Notice how the Ab goes down to G# in
        1/4 comma meantone and the Ab goes up to G# in Pythagorean.
And here an example of a sampled piano tuned to Pythagorean (so
        perhaps try again on your own piano as well ;-)

As for the chromatic tetrachord singing. Listen to Arabic /
        Turkish / Persian maqam music. They do make these distinctions.
In some Turkish music theory it is even described as such. For
        instance the Rast tetrachord is for instance D E Gb G and they
        sing it! (though not reliably so, there's great variety in

Kind regards,

Marcel de Velde
Zwolle, Netherlands
marcel at justintonation.com

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 mailto: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
>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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