[Smt-talk] Question About The First 16 Partials Of The Harmonic Overtone Series
Nicolas Meeùs
nicolas.meeus at scarlet.be
Wed Apr 23 01:50:34 PDT 2014
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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