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

[Brian Robison]

I'm generally inclined to agree that European classical melodic thinking
tends toward Pythagorean tuning. However, one should take care not to
extend this notion to other repertoires without empirical support.

A couple of years ago, I played along with Blind Willie Johnson's classic
recording "Dark was the night, cold was the ground," using a slide guitar
with the fingerboard marked according to Harry Partch's 43-one system of
11-limit just intonation. Johnson's flattened version of scale degree 3
very clearly and consistently represents a 6/5 relation (i.e., a 5-limit
"minor third") to the tonic pitch, not 32/27.

Yes, this recording represents just one data point, but it's precisely the
sort of information that one ought to accumulate before making any broad
claims about what blues musicians were or weren't (or are or aren't) trying
to do.

All best,

Brian Robison
Assistant Academic Specialist
Department of Music
Northeastern University
Boston MA 02115

b.robison at neu.edu
brian.c.robison at gmail.com





On Wed, Apr 23, 2014 at 7:35 AM, Marcel de Velde
<marcel at justintonation.com>wrote:

>  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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-- 
Brian Robison

www.brianrobison.org
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