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

Marcel de Velde marcel at justintonation.com
Tue Dec 3 17:00:33 PST 2013

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 
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 tune.

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!).
> Best,
> Ildar Khannanov
> Peabody Institute
> solfeggio7 at yahoo.com <mailto: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.
> http://www.youtube.com/watch?v=pr7PTxGnI1I
> And here an example of a sampled piano tuned to Pythagorean (so 
> perhaps try again on your own piano as well ;-)
> http://www.youtube.com/watch?v=JgLz5qYdM88
> 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 <mailto: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 <mailto: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 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 <mailto:marcel at justintonation.com>

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