[Smt-talk] Question About The First 16 Partials Of The Harmonic Overtone Series

[Marcel de Velde]

Hello Carson,

I may not represent the group view here, but I have been studying tuning 
and just intonation full time for about the past 8 years.
I'll tell you my view on things.

We are from birth hard-wired to "quantize" the interval space by a chain 
of perfect fifths and octaves.
That is, we identify unique intervals according to this "algorithm". 
Music is built upon this.
And our notation corresponds to this. Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#- etc.
We find a functional difference between for instance the intervals C-Gb 
and C-F#.
They are as different as for instance C-Eb and C-E even though Gb and F# 
share the same key on the 12-tone equal tempered piano.
We can tune for instance the C-E according to pure perfect fifths of 
3/2, making C (1/1) - E (81/64), or detune it a little to for instance C 
(1/1) - E (5/4).
The result sounds acoustically a bit different, but we will still 
interpret them both the same, as C-E. There is no functional difference, 
no different interval indicated.
The 5/4 as major third combined with 3/2 perfect fifths will however 
give problems in actual music that cannot be overcome.
As 5/4 cannot be made out of 4 * 3/2 divided by any number of octaves, 
this means that for instance circle progressions and other things do not 
work any more.
I-vi-ii-V-I "breaks" in 5-limit and one must use unnatural comma shifts, 
or "wolf fifths", or allow such progressions to comma drift upon each 
repetition.
Music is full of there sorts of things, and 95% or so of common practice 
music will not function in 5-limit or any higher number of overtones.
This is why things like 1/4 comma meantone were invented which work by 
flattening each perfect fifth by for instance a quarter Syntonic comma 
so 4 flattened perfect fifths reduced by 2/1 octaves give a 5/4 major 
third. This will give the acoustic colouring of the 5/4 that is pleasing 
to so many, while keeping the chain of fifths intact.

Now in your question you try to relate the overtones to the chain of fifths.
Indeed we find that the 11th and 13th partial are almost in the middle 
between 2 simple chain of fifths intervals.
In relation to C (1/1) and reduced to the octave:
F (4/3) perfect fourth at ~498 cents.
F# (729/512) augmented fourth at ~612 cents.
11th partial (11/8) at ~551 cents.
So it's a little bit closer to F.
However, there are also different intervals which are closer.
Gb (1024/512) diminished fifth at ~588 cents.
E# ( 177147/131072) augmented third at ~522 cents.
And one can go on along the chain of fifths to indicate even more remote 
intervals which are still closer in tuning to 11/8, like C-Dx# and C-Abbb.
However I doubt if these are possible to indicate to the brain in a 
chord regardless of tuning. (melodically this may be possible with a 
modulation / change of root)

In the case of the 13th partial reduced to the octave (13/8) at ~841 
cents we have:
Ab (128/81) minor sixth at ~792 cents.
A (27/16) major sixth at ~906 cents.
G# ( 6561/4096) augmented fifth at ~816 cents.
Bbb (32768/19683) diminished seventh at ~882 cents.

However, we have more considerations than simply how close the tuning is 
to an interval.
The music itself can have a stronger influence on which interval we 
perceive than even moderately large tuning differences of 40 or so cents.
What do we hear as the root / fundamental bass, what expectations does 
the melody give, etc.
Also the tuning of the other intervals matter.
We can actually construct progressions where we de-tune intervals in 
such a way that an interval closer to a minor third will still be 
interpreted as a major third.
Furthermore, under many practical circumstances precise tuning 
information is largely lost. We go by the larger tuning differences and 
what the music indicates in other ways.
And lastly, it is even possible to use the partials in such a way that 
we don't perceive them as a separate interval / tone but simply as a 
part of the timbre of another tone.
So I'm sorry, there is no simple answer to your question. Though I hope 
that the above information is still of help in some way.

As for blues.
There is no need to invoke partials to explain blues.
And the flat seventh character of the blues scale comes across as such 
better when we see it tuned just according to the chain of fifths:
1/1 81/64 3/2 16/9. That 81/64 to 16/9 is ~588 cents.
Similarly the minor third is quite low when combined with a major third. 
32/27 to 81/64 ~114 cents.
These kind of characters come across better when tuned just according to 
the chain of fifths.
And it is more due to how it is used musically in blues that gives it 
it's character.

Kind regards,

Marcel de Velde

Zwolle, Netherlands
marcel at justintonation.com



> Hello Everyone,
>
> I'm a new member of SMT.  I am a composer/musician who studied music 
> theory at the University of Washington.  I use the harmonic overtone 
> series in a lot of my compositional work and I have encountered a 
> variety of different partial interpretations for specifically [with C 
> as the fundamental] partials no. 10 and 12 - the pitches F or F# and A 
> or Ab.
>
> C-C-G-C-E-G-Bb-C-D-E-[*F or F#*]-G-[*Ab or A*]-Bb-B-C
>
> I have seen at least three different interpretations.  In Schoenberg's 
> /Theory of Harmony/, Schoenberg references the tones as F and A. In 
> /The Book Of Music edited by Gill Rowley/ the partials are listed as 
> F# and A.  And from internet research I have seen the partials 
> referenced as either F/F# and Ab.  I understand that the reason for 
> the variation is most likely related to the non tempered pitch of 
> those partials and that their pitch may lie somewhere between an F and 
> F# and an Ab and A (taking into account the non tempered frequency of 
> all the partials).  I'm wondering if there is more consensus among the 
> theory group about whether in the above overtone series [if arranged 
> as scale with C as the root] C-D-E-?-G-?-Bb-B-C as to the fourth and 
> sixth degrees?
>
> The reason this is important to me besides the compositional 
> implications of creating scales from overtone structure is a 
> hypothesis/theory I have regarding the jazz/blues scale and it's 
> ability to function with either major or minor diatonic tonalities.  
> When the above scale [which I call the overtone scale] is arranged as 
> the 5th mode:
>
> G-A-Bb-B-C-D-E-F#-G
>
> There is a scale with both minor and major 3rd [a blue note] . . . and 
> if we use the partial variation with F we are ever closer to the flat 
> 7th character of the blues scale:
>
> G-Bb-C-C#-D-F-G
>
> This is my explanation for why the blues scale works with both major 
> and minor modes.   I'd be curious to get some feedback on my idea and 
> more specifically the partials above in question.
>
> Thanks,
>
> Carsonics
>
>
>
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