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

Marcel de Velde marcel at justintonation.com
Tue Dec 3 15:14:26 PST 2013

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 
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 
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 
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 
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 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 <mailto: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 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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