Is the harmonic series logarithmic?

Hello group. Is the harmonic series, with the pitch doubling
every octave for harmonic numbers 2,4,8...fundamentally logarithmic
(ie, log 2)? Or is this a prejudice of traditional tuning and
harmony?

-Kelly

traktus5 wrote:

> Hello group. Is the harmonic series, with the pitch doubling > every octave for harmonic numbers 2,4,8...fundamentally logarithmic > (ie, log 2)? Or is this a prejudice of traditional tuning and > harmony? The harmonic series knows nothing of log 2. It's roughly a feature of the way the human auditory system perceives pitch. The devil is in the details -- the curve is not exactly logarithmic, and perceptual octaves are not exactly 2:1. There isn't even a one to one relationship between pitch and frequency. Louder tones will be heard as higher in pitch.

In the (ideal) harmonic series, octaves are always 2:1, so in this sense it is fundamentally logarithmic instead of not-quite-logarithmic. There's also a prejudice towards mathematical simplicity. Logarithms are close enough to perceptual reality, and don't require specialist lookup tables on your calculator. Let alone a different lookup table for each listener.

Traditional tuning very often respects octave stretching, particularly with pianos, but even when the theory says otherwise. Harmony is entirely subservient to tuning on this point. The theory usually works as if pitch perception were exactly logarithimic, so you can ignore the fact that is isn't really, unless you're working with electronics.

Graham

--- In tuning@yahoogroups.com, Graham Breed <graham@m...> wrote:
> traktus5 wrote:
>
> > Hello group. Is the harmonic series, with the pitch doubling
> > every octave for harmonic numbers 2,4,8...fundamentally
logarithmic
> > (ie, log 2)? Or is this a prejudice of traditional tuning and
> > harmony?
>
> The harmonic series knows nothing of log 2. It's roughly a feature
of
> the way the human auditory system perceives pitch. The devil is in
the
> details -- the curve is not exactly logarithmic, and perceptual
octaves
> are not exactly 2:1. There isn't even a one to one relationship
between
> pitch and frequency. Louder tones will be heard as higher in pitch.
>
> In the (ideal) harmonic series, octaves are always 2:1, so in this
sense
> it is fundamentally logarithmic instead of not-quite-logarithmic.
> There's also a prejudice towards mathematical simplicity.
Logarithms
> are close enough to perceptual reality, and don't require
specialist
> lookup tables on your calculator. Let alone a different lookup
table
> for each listener.
>
> Traditional tuning very often respects octave stretching,
particularly
> with pianos, but even when the theory says otherwise.

I think we should make a difference between octaves heard melodically
or harmonically.
I don't know anuthing about the melodical perception of 2:1, but
harmonically, an octave is only recongnized as just when there is no
audible beating.
With pure harmonic tones, that is only with 2:1.
Octave stretching is used on the piano because piane tones are not
purely harmonic. The piano tuner stretches the octaves until beating
is at a minimum.
That has nothing to do with the perseption of a melodical octave (2
consecutive tones an octave apart).

> Harmony is
> entirely subservient to tuning on this point. The theory usually
works
> as if pitch perception were exactly logarithimic, so you can ignore
the
> fact that is isn't really, unless you're working with electronics.
>
>
> Graham

Bart Pauwels

Bart Pauwels wrote...

> I think we should make a difference between octaves heard
> melodically or harmonically. I don't know anuthing about
> the melodical perception of 2:1, but harmonically, an
> octave is only recongnized as just when there is no audible
> beating. With pure harmonic tones, that is only with 2:1.
> Octave stretching is used on the piano because piane tones
> are not purely harmonic. The piano tuner stretches the
> octaves until beating is at a minimum. That has nothing to
> do with the perseption of a melodical octave (2 consecutive
> tones an octave apart).

This is a valid distinction, but note that in some cases,
piano tuners intentionally tune octaves wider than beatless
to achieve a certain sound.

-Carl

on 8/10/04 10:31 AM, Carl Lumma <ekin@lumma.org> wrote:

> Bart Pauwels wrote...
>
>> I think we should make a difference between octaves heard
>> melodically or harmonically. I don't know anuthing about
>> the melodical perception of 2:1, but harmonically, an
>> octave is only recongnized as just when there is no audible
>> beating. With pure harmonic tones, that is only with 2:1.
>> Octave stretching is used on the piano because piane tones
>> are not purely harmonic. The piano tuner stretches the
>> octaves until beating is at a minimum. That has nothing to
>> do with the perseption of a melodical octave (2 consecutive
>> tones an octave apart).
>
> This is a valid distinction, but note that in some cases,
> piano tuners intentionally tune octaves wider than beatless
> to achieve a certain sound.

And just to clarify, the octave may not be the target of beatlessness. And
a (single) octave may be "wider than beatless" in order that some
particular-sized multi-octave intervals will be closer to beatless. So in
some sense beatlessness of *some* sized interval might still be a target.

You might be referring to a situation in which the stretch is so wide that
it is beyond the point where *any* odicd interval is beatless. Is that what
you meant?

-Kurt