Yahoo Tuning Groups Ultimate Backup mills-tuning-list Equal Divisions -- of What?

David wrote,

> Lost ya there. Doesn't "quantization" denote equal divisions? E.g., in
what
> sense are the two spaces in 1:1, 9:8, 5:4 equal, or based on equal
divisions
> of linear frequeny space, whatever that means? Strange as it might seem,
> after years of picturing frequency space logarithmically, I can't get
myself
> to visualize it linearally - it doesn't seem to make any sense that way.
>
The short answer is that 1:1, 9:8, 5:4 means, say, 800, 900, and 1000 Hz
(cycles per second), so that the "equal divisions" are of 100 Hz each.
Couldn't be simpler. Picturing frequency space linearly is typical when
working with Fourier transforms, etc.

However, pitch space is not the same as frequency space. In fact, pitch
space most closely resembles log-frequency space. An example where this is
important is in distinguishing between _true_ or _physical_ white noise and
_psychoacoustic_ white noise. The two are very different. White noise has
equal power in any constant intervals in a linear frequency scale. To quote
Manfred Schroeder,

"Pink noise, also called 1/f noise, has equal power in octave frequency
bands or any constant intervals on a _logarithmic_ frequency scale. This is
a desirable attribute in many applications. For example, pink noise is a
favorite test signal in hearing research and acoustics in general because
it approximates many naturally occuring noises. Pink noise also has the
approximate property of exciting equal-length portions of the basilar
membrane in our inner ears to equal-amplitude vibrations, thus simulating a
constant density of the acoustic nerve endings that report sounds to the
brain. Pink noise is therefore the _psychoacoustic_ equivalent of white
noise."

David seems to be right that picturing frequency space logarithmically
makes more sense. Indeed, it is more natural, given that nature gave us
basilar membranes on which equal lengths correspond to equal portions of
log-frequency space. So it is odd that David asks,

>I'm working on the assumtion that a tuning which correlates to the way we
and
>our world are made would, *by virtue of that fact*, provide a more solid
>foundation for more meaningful and profound compositions. That would seem
to
>be a foregone conclusion, at least through the window of my world-view.
If
>any of you disagrees with that assumption and can cite anything other than
>purely circumstantial evidence against it, please shoot me down. Please!
:-)

with JI in mind as the most natural tuning. David, care to reconsider?

Now for the long answer. In mathematics we have harmonic series as
contrasted with arithmetic series as well as geometric series. The harmonic
series is named after the musical harmonic series, since when looking at
the length of a string, or wavelength of the sound wave, or repetition time
of the waveform, the partials of a perfect string go as

1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, . . .

So that is the harmonic series as known to mathematicians. This certainly
does not appear equally spaced in terms of human quantities like length and
time. So what is? What is the musical equivalent of the mathematical
arithmetic series

1, 2, 3, 4, 5, 6, 7, 8, 9, 10?

The answer: the subharmonic series! Subharmonics are integer multiples of
the string length, wavelength, or repetition time of the "fundamental."
(Note that I am not ascribing any musical significance to this fact).

Finally, consider the geometric series

. .x^-9, x^-8, x^-7, x^-6, x^-5, x^-4, x^-3, x^-2, x^-1, x, x^2, x^3,
x^4, x^5, x^6, x^7, x^8, x^9, . . .

where x>0 and x does not equal 1. Whether the units of measurement are
string length or wavelength or time period, as above, or frequency, as in
the 800:900:1000 Hz example, the result is the same: equal temperament!
Conversely, if you take the log of the above series, you get the integers,
so you can see that an equal temperament is equally spaced in log-frequency
or even log-length or log-time space.

Anyhow, it appears that from a physical point of view, the subharmonic
series is what most people would describe as "equally spaced," while
psychologically, since we hear pitch in log-frequency space, an equal
temperament is the tuning most likely to be heard as equally spaced.

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Equal Divisions -- of What?

🔗PErlich <AAM/AAM/PErlich%Acadian@...>

🔗mr88cet@texas.net (Gary Morrison)