[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 
     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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