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

[Victor grauer]

In response to Carson Farley's inquiry regarding the possible relation
between elements of the overtone series and the ability of the "blue
note" to function in both major and minor:

I have always
had a problem with scale- or tuning-based interpretations of musical resources
and processes, a trap many music theorists tend to fall into. Why? Because they
are non-dialectical, and for me music is through and through based on
dialectical processes and not simply
the following of certain procedures and rules.
 
The attempt to account for Stravinsky's harmonic
practice by pointing to his use of the octatonic scale is a good example. The
problem here is not simply that attempts at analysis along such lines tend to
stretch the definition of "octatonic" to the breaking point, but primarily
that a non-dialectical approach can tell us only a very little about what is
really going on in such works. For me, the older view, based on polarization, remains
the most promising approach, as it reflects what I hear as the highly
dialectical nature of his tonal language.

Returning to Farley’s query, the emergence of the “blue
note” is in my view best explained according to a historical process that is,
at heart, dialectical (also noted by the ethnomusicologist Gerhard
Kubik, in his book, Africa and the Blues).
What we find in some of the earliest examples of “country blues,” is in many
cases almost indistinguishable from certain practices commonly found among
Bantu peoples in Africa. And, as in so many (though certainly not all) African
melodies, we tend to find melodies based, essentially, on pentatonic scales. 


The
guitar accompaniment is often only a single major chord, which functions almost
like a drone. As blues musicians gradually adapt to the standards of the
popular music of the time, the two other primary “common practice” chords, the major
subdominant and major dominant, are added, but the pentatonic melodies often
remain. 

For example, a melody accompanied by triads drawn from C major, may be
based on the pentatonic C, Eb, F, G, Bb. In such cases, Eb and Bb can be understood simply as “blue
notes” in the context of C major. And if we are content to analyze simply in terms of a scale, it would be C, D, Eb, E, F, G, Bb, B. However, such an “analysis”
would tell us little or nothing about how this music was actually constructed,
or in fact heard by the musicians who produced the style in the first place. 


The historical explanation, on the other hand, makes it clear that in the blues we are
dealing with a dialectical process in which two very different musical cultures
encounter and enrich one another. As I see it, the notion of a “blue note” is a
sort of copout, as is the usual explanation of the tonic with added minor seventh, so common in the blues, simply as V of IV. If you hear the music in such
terms you are missing the dialectical “frisson” produced by the clash of two
different tonal realms, emanating from two different musical cultures.


On Wednesday, April 23, 2014 7:07 AM, Nicolas Meeùs <nicolas.meeus at scarlet.be> wrote:
 
One must realize that harmonic partials (or partials in general) are "notes" (or pitches) only in a metaphoric way. I am surprised to read, in a research that I am running just now about the German concept of Klang, that the opinion (of Helmholtz, Riemann, Schenker, Schoenberg, etc.) was that harmonics could normally not be heard – an opinion that would hardly be shared today.
>
>One may suppose that 'spectral' music, a music that seeks to
      replicate the spectrum of harmonic sounds in distinct pitch
      classes, expects the hearer to (consciously or not) reconstruct,
      or recognize the spectrum. A certain level of approximation must
      be allowed, not only because the music is tempered, but also
      because the pitch classes cannot all be in the correct octave
      (partial 10 and 12 are more than three octaves above their
      fundamental). This approximation probably consist in a tolerance
      of our hearing, as already noted by Euler in the 18th century.
>
>It is a simple matter to calculate the exact position in cents of
      each of the harmonic partials. The formula is log(x)*1200/log(2), where x is the number of the partial. The formula can be written as such in an Excel spreadsheet (if you want to permanently make this a function available in your spreadsheet, read http://www.plm.paris-sorbonne.fr/Convertisseur-de-Cents). Partial 10 is 3986 cents, 12 is 4302 cents above their fundamental. Remove 3600 cents to account for the three octaves, their value in cent is 386 (a pure major third) and 702 (a pure fifth) above the fundamental. This evidences that what you call partial 10 and partial 12 are in fact partials 11 and 13 – note that the fundamental itself is its own first partial.
>    For 11 and 13, the values are 4151 and 4441, i.e. 451 and 841
      after correction for the three octaves. For C as fundamental
      (partial 1), this is 51 cents above F, and 59 cents below A
      respectively. Keep in mind that a tempered semitone is by
      definition 100 cents (this is the definition of the cent, not of
      the semitone). Partial 11 is thus very slightly (1 cent) more than halfway between F and F#, partial 13 is more decidedly on the side of Ab.
>
>After that, how you make use of these, as approximations of what
      pitch classes, is your choice as composer or as theorician. Note
      in addition that the instruments for which you write probably are
      slightly inharmonic, but in a hardly predictable way, which leaves
      you with even more freedom to analyze your scales. Partial 11
      could probably be used as approximation of both F and F#, and
      partial 13, even although closer to Ab, probably would form an
      acceptable approximation to A.
>
>Nicolas Meeùs
>Professeur émérite
>Université Paris-Sorbonne
>nicolas.meeus at scarlet.be
>
>
> 
>Le 23/04/2014 06:12, CARSON FARLEY a écrit :
>
> 
>>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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