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

Ildar Khannanov solfeggio7 at yahoo.com
Tue Dec 3 19:01:22 PST 2013

Dear Marcel and the List,

this seemed to be an interesting topic (returning to and revisiting the old one, of course). I suggest creating a panel at the upcoming EUROMAC 2014 in Leven. I know (from the recent Congress of the Russian society for theory of music) the names of at least two more scholars, working in this area, Pyotr Tchernobrivets from St. Petersburg and Franck Edrzejewsky from Paris. We could organize a truly international panel On Tuning in Composition Today. Here is the wiki discussion room, prepared by the organizing committee of the EUROMAC 2014: http://groups.google.com/group/euromac-2014/topics 

Of course, the topic can be extended to the manipulation of partials and harmonics in a synthesized sound.

Best wishes,

Ildar Khannanov
Peabody Conservatory
Member of the Organizational Committee
solfeggio7 at yahoo.com

On Tuesday, December 3, 2013 8:01 PM, Marcel de Velde <marcel at justintonation.com> wrote:
Dear Ildar,

Aah, now I get what you mean! I thought we were talking about
        merely tuning the diatonic scales in Pythagorean on the piano.
But no, of course, you are right. One can only tune 12 notes per
        octave on a standard piano (though pianos have been built that
        have 24 tones).
So a piano tuned to Pythagorean is very limiting in what one can
        play in it!
For this reason I work with virtual pianos, sampled pianos on my
        computer. I can tune them to any enharmonic and am no longer
        limited to 12 tones per octave.
The Fauré piece I linked to below has more than 12 tones per
        octave and all of them are tuned to perfect Pythagorean. (try to
        spot the places where an A# is followed by a Bb)

What is often misunderstood is that Pythagorean is not a
        temperament for 12-tones.
And for instance the "wolf fifth" is not a fifth at all.
If you tune your piano to Pythagorean, even for only 12 tones
        per octave, then there is not an out of tune interval there.
But if one tries to use the diminished sixth, which is tuned
        perfect, as a perfect fifth then this diminished sixth will
        sound like a horribly out of tune perfect fifth of course.
There is no enharmonic equivalence in Pythagorean tuning. So for
        instance an Ab is tuned differently than a G#, and one had
        better know some music theory to use it correctly / play in

The optimal tuning for a piano if one wants to be able to play
        everything on it is 12-tone equal temperament like we use
        standard today.
I'll be the last person to propose a change for that.

Kind regards,

Marcel de Velde
Zwolle, Netherlands
marcel at justintonation.com

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 mailto: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
>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
>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
>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
>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
>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
>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.
>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
>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
>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 practice)
>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
>>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
>>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
>>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
>>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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