Re: [tuning] Complex number temperaments

It is tempting to apply various mathematical processes to music, either in theory or in practice. For example, theme and variations may correspond to permutations and combinations. The concept of randomness appears as aleatory music, such as in some of the works of the late John Cage, and some computer generated compositions.

Some would say that the rigid application of math such as the above is a poor substitute for genuine inventiveness, inspiration and creativity.

In thinking about musical scales, one should first consider the nature of sound and hearing before deciding what mathematical processes might be suitable to apply. For example, our perception of the second most basic consonance (after identity), usually referred to as the "octave" or "2/1" relationship, is logarithmic, rather than linear. So it is based on the multiplication and division of frequencies (pitches) by two, rather than addition and subtraction of any particular number of cycles per second (Hertz, abbreviated Hz). So powers and roots can be involved, even with multiple "octaves", but not necessarily roots of negative numbers.

Even though we describe music in terms of rhythm, melody, harmony, tone color and structure and dynamics and many other aesthetic considerations; its basis, sound, is essentially one dimensional. It consists in variations in air pressure over time. Were that not so, recording on disks or tapes (whether analog or digital) would not be possible. It is due to the limitations of our sense organs that we perceive slower variations as rhythm, medium speed variations (about 20 to 8000 Hz) as pitch and faster variations (about 40 to 20,000 Hz) as tone color, with the interactions between simultaneous tones as harmony.

On the other hand, painting could be described as a two dimensional art form and sculpture as three dimensional. Mobiles could be described as four dimensional as they reintroduce the time element. (Not everyone agrees that time is the fourth dimension.) Videotape is based on one dimensional scanning at a rate slightly faster than our persistence of vision with a separate track for audio (or two for stereo).

Music, then, described as the physical phenomenon of sound, requires only two axes mathematically, with pressure variations above and below the "x" or time axis. Imaginary numbers require an additional axis. You don't need it. However, from the physiological standpoint of perception, it could be difficult to determine how many axes or dimensions one might want or need. But simply for a way to construct a scale, tempered or not, it would seem an unnecessary complication. Would it yield an imaginary scale on which one could play imaginary music?

Although... purely mathematically generated music may not be a good
substitute or true creativity - it shure is nice to listen to for
inspiration. Just like natural sounds in the world around us. :)

have y'all listened to Phil Thompson's Organised Chaos?
http://www.organised-chaos.com

(I have wanted to combine the fractal music with JI, but I haven't the
software to do either at the moment)

Music, Magick, Love, and Sex
Alexander Azi Vajravai
the
ChaosMonkey
my website: http://www.vajravai.com
my mp3.com: http://www.mp3.com/chaosmonkey
the ChaosMonkey Collective:
http://www.chaosmonkey.com

John F. Sprague wrote,

>In thinking about musical scales, one should first consider the nature of
>sound and hearing before deciding what mathematical processes might be
>suitable to apply.

I agree wholeheartedly. I've used a great deal of involved mathematics in my
posts to this list, but each instance was founded in some sort of model of
how we hear. Just playing with numbers or formulas for their own sake is no
way to make music.

>It is due to the limitations of our sense organs that we perceive slower
>variations as rhythm, medium speed variations (about 20 to 8000 Hz) as
pitch

That is a common misconception, but untrue. For example, we will hear a 30
Hz variation as a rhythm if it is a variation in _amplitude_, and as a pitch
if it is a variation in _air pressure_.

>That is a common misconception, but untrue. For example, we will hear a 30
>Hz variation as a rhythm if it is a variation in _amplitude_, and as a pitch
>if it is a variation in _air pressure_.

Your ears must be very different from mine, Paul, if I understand what you
mean. When I try 30 hz amplitude modulation, I definitely cannot hear the
individual changes ("rhythm") at such a fast rate. Instead, I hear the
carrier together with the sum and difference tones that AM produces. (Just
to be sure, I tried it out moments ago in Csound.)

Bill

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
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Bill, what are you modulating? And, more importantly, do you hear a 30 hz
tone?

>Bill, what are you modulating? And, more importantly, do you hear a 30 hz
>tone?

I was modulating a middle C sine wave. I tried different modulating
waveforms: sine, triangle, square, and sawtooth. I did not hear a 30 hz
tone, though some of the difference tones produced by the square and
sawtooth waves were rather low and raspy. My point is that I did not hear a
"rhythm," but a single tone. I can try it with other carrier frequencies
and waveforms, but I don't think it would change that perception.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Bill Alves wrote,

>I was modulating a middle C sine wave. I tried different modulating
>waveforms: sine, triangle, square, and sawtooth. I did not hear a 30 hz
>tone, though some of the difference tones produced by the square and
>sawtooth waves were rather low and raspy. My point is that I did not hear a
>"rhythm," but a single tone. I can try it with other carrier frequencies
>and waveforms, but I don't think it would change that perception.

I would try a higher carrier pitch, but it is difficult to hear a 30 Hz
rhythm (listen to lots of Tony Williams and then try again:)). _My_ point is
that you didn't hear a 30 Hz pitch, even though _something_ was changing at
30 Hz.

How about 20 Hz? Can you hear a 20 Hz tone? A 20 Hz rhythm? Would you agree
that it is possible to hear one and not the other in a given circumstance?
What I was challenging was the popular misconception that they are one and
the same thing.

>I would try a higher carrier pitch, but it is difficult to hear a 30 Hz
>rhythm (listen to lots of Tony Williams and then try again:)). _My_ point is
>that you didn't hear a 30 Hz pitch, even though _something_ was changing at
>30 Hz.
>
>How about 20 Hz? Can you hear a 20 Hz tone? A 20 Hz rhythm? Would you agree
>that it is possible to hear one and not the other in a given circumstance?
>What I was challenging was the popular misconception that they are one and
>the same thing.

Perhaps they are not one in the same thing, but in my experience, the
change-over from perceiving individual changes in low-frequency modulation
to a single "tone" is around 20 hz, that is, right around the same spot
that we start hearing pitch. Pinning down the exact point is difficult,
because there is a somewhat nebulous area around the change-over, people's
perceptions are slightly different, and speakers have a very tough time
reproducing such frequencies reliably. (Stockhausen liked to play with this
nebulous area in works such as Kontakt.) In my class, I often play a
modulator tone slowly sweeping up in frequency and poll students as to
where "rhythm" ends and "tone" begins for modulation. For most people, it
seems to be at about the same point that "pitch" begins. If you have an
example that demonstrates the independence of these perceptions for a given
circumstance, I would certainly like to hear it.

Bill

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

>If you have an
>example that demonstrates the independence of these perceptions for a given
>circumstance, I would certainly like to hear it.

I thought you agreed that as the frequency of the modulation becomes to high
to be perceived as a rhythm, one still does not hear a tone at that
frequency. Similarly, if one decreased the frequency of a loud sine wave
below 20 Hz, one simply heard nothing -- one does not hear a rhythm at that
frequency.