Musical sequence

[Marc LeBrun]

 >=Jon Wild
 > ...for many, many more details than you'd care to know about.

Here's far too many of those details, which I feel compelled to offer as a 
public service.

Tuning systems are fun, but they seem often to become a sort of obsession, 
divergent from sense.

If you want to pursue this subject, you should know that the musical 
realities are subtler than one might at first assume:


 >=David Wilson:
 > What if we abandon the notion of octave, but keep the idea of equally 
spaced tones,
 > so that our scale can be described by...a ratio r between adjacent tones 
on the scale.

There are doubtless other good examples, but one bit of computer-generated 
music I'm familiar with that does this is John Chowning's 1978 "Stria".

This piece uses the golden ratio, not only for defining the scale tones 
used in the composition, but also in the spectra of the synthesized sounds 
themselves.

When these sounds are played at the specified scale tones the harmonics 
tend to line up and reinforce eachother, producing a pleasant, though 
exotic, kind of consonance.


Nonlinearities in physical vibrating systems--not only in the instruments 
but also in the ear--cause all real sounds to have complicated spectra.

The "fundamental" frequencies of a composition, as defined by the formal 
scale, are only one small aspect of the perceived musical result.


 > Is there a measure of goodness of r that would rate r = 2^(1/12) high?

Yes, it's motivated by the happy coincidence that the value of 2^(7/12) = 
1.4983... is close to 3/2.  But before you start making lots of continued 
fraction expansions of weird numbers, see below...


 > Presumably this rating would be based on the closeness of tone ratios to 
simple rationals which represent pleasing harmonies.

Yes, that's true to a first approximation, but, as the "Stria" example 
indicates, there's a lot more to the story.

As others have mentioned, in musical contexts 2^(7/12) is indeed a 
reasonably close approximation to the so-called "perfect fifth".

But note how numerically coarse this "good" approximation actually is!

Moreover, it has no particular bearing on 2^(k/12) for k#7.  As mentioned 
elsewhere, 2^(4/12) = cbrt(2) = 1.2599... is an OK approximation to 5/4, 
but there's nothing particularly nice about 2^(1/12), and 2^(6/12) = 
sqrt(2) was called "diablo in muisica".

On top of all this, dissonance is a non-monotonic function of frequency 
separation.  The nonlinear intermodulations referred to above produce 
difference components.  For very close pitches they are subaudible.  As the 
pitches diverge you get increasing roughness, dissonance and beating, which 
then tapers off again at larger separations.

This intermodulation applies to *all* the spectral components of both tones.

That is the reason why close approximations to the "simple ratios" tend to 
produce consonance--it's the reinforcement and blending,  not because the 
ear somehow mysteriously does number theory!

For example playing two tones with fundamentals in 3:5 ratio means the 5th 
overtone of the former and the 3rd overtone of the latter will fall 
together inside the "critical band".

But this says nothing about, say, the 7th overtone of one against the 4th 
overtone of the other, etc.

Or chords of three or more tones.

Now it's true, very roughly, that the signal power in higher harmonics 
tends to fall off.   You could imagine weighting the strength of each 
cross-interaction, to produce some kind of global measure.

But there are many very significant deviations from monotonic fall-off--for 
example in initial transients (which the ear is critically sensitive to for 
"source identification"), the formants that distinguish sung vowels, etc etc.

Further, the overtones of real vibrating systems are at best only integral 
to a first approximation--and then only for linear vibrators--surfaces like 
drums and bells have all kinds of crazy modes.

Even in linear sources the physics often tends to stretch the actual 
overtones upward, which in turn causes humans to prefer "bright" 
tunings.  Tests on the conductor Pierre Boulez, for example, demonstrated 
(much to his surprise) a definite preference for sharp octaves--so even 
that most basic 2:1 ratio is a fudge.  Historically, I believe the music 
world legislated the A=440 Hz standard in part because of the stress from 
continual gradual "pitch inflation".


 > Given this rating, what would be the best r?

Any such "best" you might want to define might produce an interesting 
sequence for the OEIS, but I hope I've demonstrated its relevance to actual 
real music is at most tenuous.

Have fun, but don't take it too seriously.  Beware, tuning systems tend 
encourage certain species of mathematical crankiness.  There is a glut of 
microtonal crud out there whose creators believe *has* to be great, reality 
notwithstanding, because it embodies their pet theories.

But honestly, the best such music seems to succeed in spite of, or aside 
from, the exotic tunings.

Perhaps I'm too reactionary.

Just to be contrary, a few years ago, as an experiment, I had a guitar 
refretted with 14 equal-tempered tones per octave (instead of the usual 
12).  This peculiar number was chosen expressly to maximize musical 
apostasy (and practical joke potential).  Surprisingly, it was not too 
difficult to noodle on it in a reasonably melodic way.  I believe this was 
partly due to players' tendency to unconsciously "bend" notes a little, 
even on normal instruments, to deal with slight mistunings, untrue 
frettings, etc.  Experiments with exact synthetic tones tended to sound 
much worse.

I concluded that other aspects of musicality vastly overwhelm tuning 
technicalities.