TUNING digest 979

> The traditional mathematical model of musical tuning and
> acoustics states that these "ghost tones" (for sake of a better
> term) are found at frequencies which are small integer ratios
> to the fundamental or open string frequency.
> This is the assumption upon which Just Intonation is based.

In physics I think they're called modes of oscillation. The fundamental mode
is the one with lowest frequency. Maybe the others are just called
non-fundamental modes.

A while back someone described a delightful experiment with prepared
guitar, verifying that these non-fundamental modes need not have frequencies
which are small integer ratios of the fundamental. Under ideal conditions -
uniform strings, and I hear one would need massless frictionless pivots to
hold the ends of the string - ideal conditions would yield a spectrum of
frequencies all integer multiples of the fundamental.

But this is not the only place that integers come up. I just read an amazing
thing, I think in IEEE SPECTRUM, an engineering magazine. There's this new
kind of loudspeaker that's been invented. It's based on the fact that air is
a non-linear acoustic medium. Non-linear means that air isn't a perfect
spring, as you squeeze air it doesn't resist in exact proportion to how much
you squeeze. So the new loudspeaker takes advantage of this. It's composed
of two ultrasonic loudspeakers that each emit sound up around 200 KHz. Dogs
etc. can't hear way up there either. For all I know these may be the best
way to eliminate cockroaches! Anyway, the two ultrasonic speakers actually
play at somewhat different frequencies - for example, suppose one played at
200 KHz and the other at 201 KHz. What happens, when two oscillating forces
hit a nonlinear medium, is that new oscillations get induced at the sum and
difference frequencies. So new oscillations appear at 401 KHz and 1 KHz. Of
course 401 KHz is inaudible. But the 1 KHz... hey, where'd that sound come
from!

The IEEE Spectrum mentioned as I recall that Carver is involved in producing
a product based on this new invention.

Anyway, if you start with two frequencies and start taking sums and
differences:

a, b, a+b, a-b, a+2b, 2a-b, 2a+2b, 2a, 2b, 3a+b, etc.

somehow amazingly integers pop up. Now I'm not particularly enamoured of the
creationist view of the world, but I'll follow Haverstick at least to the
point that it looks like the integers are just somehow the way things are.

The ear itself is undoubtedly non-linear. I imagine that's why we can hear
"beat frequencies".

Anyway, my whole point is that just intonation need not be founded on the
frequencies of modes of oscillation of a string. Personally I don't think you
can find secure foundations for any conceptual system, though even insecure
foundations can be quite useful. The best foundation for just intonation,
seems to me, is just the way various pitch combinations sound to us
humans. Of course, our perception is also shaped by the conceptual systems
that have been used to construct our environment. But anyway if you look for
why humans might like simple ratios of frequencies, there are more answers
around than just strings.

Jim Kukula

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Bill Alves wrote:

> However, if you're using a tuning system that does not replicate any
> particular interval, how do you specify pitches? Do you just pick one pitch
> as the starting point and number them? That seems rather awkward.

You might do it in Hz, or, if triggered via MIDI, simply as note numbers,
or, if on paper or screen, as vertical positions, etc. Color is a
non-repeating parameter, yet painters seem able to handle it. In a way,
repeating systems could be considered awkward in that you have paradoxically
simultaneous difference and sameness which must be kept track of.

For my someday pan-pitch sequencer, I intend to represent pitch vertically
allowing continuous vertical placement and having no guidelines, or
user-definable guidelines, relying instead on audible feedback as notes
are placed in the score and moved to tune. I suspect that hand/eye/ear
coordination would adapt to this, and the software would become a sort
of score-becomes-instrument. I see no reason to keep the old ideas of
note names, etc. when composing electronically using the entire pitch
spectrum as a resource.

Matt Nathan

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Bill Alves raised a very meaningful question about 88CET or other
tunings that don't boil down to some number of steps per so-and-so
interval:

> if you're using a tuning system that does not replicate any particular
> interval, how do you specify pitches? Do you just pick one pitch as the
> starting point and number them? That seems rather awkward.

Well, Bill posed that question in a very general way, but certainly not
an inappropriate way. I'll suggest though that a better way to approach it
is to assume that you'll devise some sort of organization for the tuning.
Without some sort of organizational principle, it's chaos. You have no
framework in which to compose.

Some would suggest that establishing a framework or underlying
organization is too restrictive for the freedom needed to compose
innovative new music. I personally find the alternative too difficult a
nut to crack - there are too many possibilities for creativity to flourish.
I find that I have to break the problem down to something simpler before I
can get anything done.

So, generally speaking, I recommend organizing the notes not into spans
of octaves but into spans of something else. For example, in the case of
88CET or Carlos' Alpha (78CET), you have good approximations to a 3:2 P5.
So, you in either of those tunings, you can organize their pitches into
patterns that repeat in fifths.

Were I working (so far I haven't yet) with my viola in 88CET, I would
almost certainly build music from the scale pattern 2 1 2 2 1 (i.e.,
whole-step, half-step, whole-step...) repeating in fifths. That's very
convenient on the viola or violin since their open strings are fifths
apart. (It's a little trickier on a 'cello since you run out of fingers.)

But for most of my work, I instead organize 88CET pitches by a "cycle"
of perfect fifths wrapping within a subminor seventh (7:4 approximation).
That works out to a seven-step-per-cycle scale with the pattern 2 1 2 2 1 2
1, and I give them traditional letter names, in this case A B C D E F G A.
Any note and the next higher or lower tone of the same note name is a
subminor seventh apart rather than an octave apart. There's an
inconvenience that G-A is a half-step, but I learned my way around it
pretty quickly. I simply omit the G#/Ab key from the system.

After you do that, you have a complete system of pseudokeys and
pseudokey signatures. Fifths look like what we normally think of as
sixths, so sharps add in the order of C# A# F# D# B#..., and flats add in
the order of Eb, Gb, Bb, Db, Fb... So some of the pseudokeys work out to:

Gb: Gb A Bb C D Eb F Gb
Eb: Eb F Gb A B C D Eb
C: C D Eb F G A B C
A: A B C D E F G A
F: F G A B C# D E F
D: D E F G A# B C# D
B: B C# D E F# G A# B

From that point you learn things analogously with how you'd learn things
in an octave-based tuning (pretty much rote but with auditory reinforcement
and memory), for example:
Intervals:
- Subminor third (~7:6) above the root is the third step in the scale,
- Supramajor third (~9:7) is the fourth step in the scale,
- The perfect fifth is the sixth step in the scale,
- The major tenth is the fifth step in the second cycle of the scale,
etc.
Chords:
- A supramajor third-stack triad is scale steps 1, 4, and 7.
- A harmonic-series fragment (4:6:7:9:10:11:15) approximation can be
found
at scale steps
First cycle: 1, 6,
Second cycle: 1, 4, 5, #6,
Third cycle: #3.

It becomes its own new world of its own with all new nomenclature,
possibly new notation, new harmony rules, and such. But once you get to
know it, it makes as much sense as the traditional system. In some cases
it's a similar kind of sense, and others it's a very different kind of
logic.

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On Fri, 7 Feb 1997 15:22:01 -0800 Bill Alves said:
>However, if you're using a tuning system that does not replicate any
>particular interval, how do you specify pitches? Do you just pick one pitch
>as the starting point and number them? That seems rather awkward.
>
I dispense with "pitch" altogether, and just work in frequency space
using ratio designations (a la Partch, Harrison, & Johnston/Tenney
matrices). The frequency of current reference is 1:1, and then subsequent
pitch(es) reflect from that point via ratios (> 1 ascend; < 1 descend).
-- Brian B.

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