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

[Steve Haflich]

On Tue, Apr 22, 2014 at 9:12 PM, CARSON FARLEY <ccfarley at embarqmail.com>wrote:

> 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
>
Whatever you want to call these things -- partials or overtones or
harmonics -- it is clearer and notationally more useful to number them
starting with the fundamental as 1.  (Your references number the
fundamental as 0.)  Under that convention the frequency of any partial is N
times the fundamental.  Partial N where N is a power of two, and only those
partials, are the same note under octave equivalence.  (More usefully, the
2^i partial will be i octaves above the fundamental.)  If two partials are
related as partial N and partial (2^i)*N, they will be the same note under
octave equivalence.  (E.g., partials 3, 6, 12, 24 etc. will all be the
perfect 5th scale degree.)

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
>
Here is where you start to stand on inform ground!  No overtone is tempered
except the ones that are octaves.  Assuming everyone understands that a
pitch interval is a ratio (the Greeks understood millennia ago) then the
ratios of an overtone to the fundamental is an integer.  Since pitch is
logarithmic, it is convenient to measure intervals in cents, where an
equal-tempered semitone is 100 cents, and an octave is 1200 cents. _None_
of the other partials agree exactly with equal tempered pitches.  Some of
them are quite far off equal temperament, or perhaps it would be
historically and operationally clearer to say that equal temperament
doesn't agree with the partials...

Here is a table of partials 1..16 in cents, reduced to intervals within the
octave by reducing the cent values mod 1200.

 1:       0.00
 2:       0.00
 3:     701.96
 4:       0.00
 5:     386.31
 6:     701.96
 7:     968.83
 8:       0.00
 9:     203.91
10:     386.31
11:     551.32
12:     701.96
13:     840.53
14:     968.83
15:    1088.27
16:       0.00

Note that even the 5ths are sharp by about 2 cents (near the limit of human
discernability under successive presentation.  The "just" M3 is very flat
compared to equal temperament, which is why those neat electronic tuners
have hash marks somewhere on their flat-sharp indicators to show performers
where to adjust M and m degrees -- the latter is adjusted upwards -- lest
triads sound badly out of tune.

> and that their pitch may lie somewhere between an F and F#
>
The 11th harmonic in the table above (551.32 cents) is so close to being
exactly half way between F and F# that you would be better off to think of
it as a quarter töne -- F half sharp.  Similarly for the 13th harmonic.

> 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?
>

What does consensus matter?  The name you or Schoenberg or anyone else
assign to these partials has nothing to do with mathematics or physics or
psychology. It is more an intentional decision that one wants to think
about the 11th harmonic as an F or an F#.  That's an analytical or
compositional choice, not a mathematical or physical property.

(In my night job, I'm a horn player -- horn as in Corno -- and in the
pre-valve literature we get to think of that 11th harmonic as the 4th or
4#th degree, depending.  Right hand position, embouchure, and wishful
thinking (mind over matter) allow either to be achieved.  Or else them
modern doohickies commonly called "valves" help resolve the ambiguity.)

> 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...
>
I don't want to comment on the ëxplanation" of the blues scale.  That's a
bottomless subject.  But this detail:

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:
>
> Do you intend these notes to be the equal-tempered versions of the named
notes -- what the player gets when he presses the named buttons on his
instrument -- or the accurately achieved overtone tunings of these notes?
 They are often quite different scales!  Either might make sense, and lead
to different compositions, but I suggest you need to decide.  The names you
assign to scale degrees aren't directly related to their performance,
except that those names drag along a lot of historical and pedagogical
flotsam from music theory.  You can't fight that, but you need to be aware
of it.

If I understand correctly, that Bb derived from the overtone series (969
cents) will be about 1/3 semitone flatter than equal temperament.  Hard to
play on lots of kinds of axes.


> 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.
>
"Ëxplanations" in music theory are treacherous.  I've found that the word
"because" is quite rare in theoretical discourse.  When it it or its
analogs are used, the connection between observation and its explanation
rarely survive critical (scientific?) scrutiny.

=====

Steve Haflich
Berkeley CA
academically unaffiliated
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