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<font face="Calibri">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 <i>Klang</i>, that the opinion
(of Helmholtz, Riemann, Schenker, Schoenberg, etc.) was that
harmonics could normally <u>not</u> be heard – an opinion that
would hardly be shared today.<br>
<br>
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.<br>
<br>
It is a simple matter to calculate the exact position in cents of
each of the harmonic partials. The formula is log(<i>x</i>)*1200/log(2),
where <i>x</i> 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 <a class="moz-txt-link-freetext" href="http://www.plm.paris-sorbonne.fr/Convertisseur-de-Cents">http://www.plm.paris-sorbonne.fr/Convertisseur-de-Cents</a>).
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.<br>
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 </font><font
face="Calibri">(1 cent) </font><font face="Calibri">more than
halfway between F and F#, partial 13 is more decidedly on the side
of Ab.<br>
<br>
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.<br>
<br>
Nicolas Meeùs<br>
Professeur émérite<br>
Université Paris-Sorbonne<br>
<a class="moz-txt-link-abbreviated" href="mailto:nicolas.meeus@scarlet.be">nicolas.meeus@scarlet.be</a><br>
<br>
<br>
</font>
<div class="moz-cite-prefix">Le 23/04/2014 06:12, CARSON FARLEY a
écrit :<br>
</div>
<blockquote
cite="mid:38441367.1093940.1398226361871.JavaMail.root@embarqmail.com"
type="cite">
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font-size: 12pt; color: #000000">
<p style="background-color: rgb(255, 255, 255);">Hello Everyone,</p>
<p style="background-color: rgb(255, 255, 255);"> </p>
<p style="background-color: rgb(255, 255, 255);">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. </p>
<p style="background-color: rgb(255, 255, 255);"> <br>
</p>
<p style="background-color: rgb(255, 255, 255);">C-C-G-C-E-G-Bb-C-D-E-[<b>F
or F#</b>]-G-[<b>Ab or A</b>]-Bb-B-C</p>
<p style="background-color: rgb(255, 255, 255);"> </p>
<p style="background-color: rgb(255, 255, 255);">I have seen at
least three different interpretations. In Schoenberg's <i>Theory
of Harmony</i>, Schoenberg references the tones as F and A.
In <i>The Book Of Music edited by Gill Rowley</i> 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? </p>
<p style="background-color: rgb(255, 255, 255);"> </p>
<p style="background-color: rgb(255, 255, 255);">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:</p>
<p style="background-color: rgb(255, 255, 255);"> </p>
<p style="background-color: rgb(255, 255, 255);">G-A-Bb-B-C-D-E-F#-G </p>
<p style="background-color: rgb(255, 255, 255);"> </p>
<p style="background-color: rgb(255, 255, 255);">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:</p>
<p style="background-color: rgb(255, 255, 255);"> </p>
<p style="background-color: rgb(255, 255, 255);">G-Bb-C-C#-D-F-G</p>
<p style="background-color: rgb(255, 255, 255);"> </p>
<p style="background-color: rgb(255, 255, 255);">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. </p>
<p style="background-color: rgb(255, 255, 255);"> </p>
<p style="background-color: rgb(255, 255, 255);">Thanks,</p>
<p style="background-color: rgb(255, 255, 255);">Carsonics</p>
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