Food for thought - a puzzle

I have been playing with the tone set generated by using all small whole
number ratios in the 9 limit with their inversions. I chose 9 because the
11s didn't sound consonant to me and generated just too many notes for my
taste. Here's the challenge: There are some really large intervals near
the edges of the scale. What 2 ratios and their inversions would you use to
fill in the gaps and why?

Here's the scale from lowest to highest:

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


7 2 9 5 3 7 4 9 5 ? ? 2
10 3 14 8 5 12 7 16 9 ? ? 1


35 20
I chose 36 and 21 because they were common in the set of resultant intervals
produced by this scale. If you have a reason to choose more than 2
intervals that's fine too.

I look forward to a variety of answers! :-)

-Glen Peterson


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(1) James Kukula wrote:

'' Nice uniform strings will generate integer overtones.''

Perhaps a string player on this list would like to comment: has anyone ever
encountered a ''nice uniform string''?

Neither drawn metal wire nor nylon has produce the kind of uniformity
suggested here, and the renewed and growing preference for gut strings
suggests that that kind of uniformity has not been a desired
characteristic. (I am certain that there is a huge body of literature on
this subject from the Catgut Acoustical Society and elsewhere: perhaps a
list subscriber with expertise in this area can give us some pointers).

In his all natural harmonics string quartet, Chronos Kristalla (1990,
Material Press), La Monte Young found it necessary to tune the harmonics
themselves, not the fundamentals in order to get the level of consonance he
requires. In fact, when the open strings of his quartet are then played
with one another, beating is abundant although the intervals between their
natural harmonics are essentially beatless. (All of this is said ignoring
the role of bow technique).

Additional complications of _wind_ instrument spectra that I neglected to
remark upon earlier include (1) in musical contexts what the instruments
play changes and is rearticulated so rapidly that there is little or no
chance to establish stable systems comparable to Kukula's example of ocean
waves forming crests, and (2) the stability of the air pressure from the
player's lungs is far from perfect.

(2) I hope my observations are not misunderstood as dismissing the integer
overtone series. Far from it: The series remains the best basic model for
the spectra of most instruments, and is the essential model (both positive
and negative) for voice leading and orchestration. But a degree of greater
precision with regard to the complexities of real musical instruments
should help to reveal the extent and limits of the model as well as suggest
new technical possibilities for music making.

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> Perhaps a string player on this list would like to comment: has anyone
ever
> encountered a ''nice uniform string''?

Dan's point is certainly valuable, and very amusingly put, but I think
personally that there's only so much validity to it.

I spent several years working on a program that decomposes musical tones
into envelopes of harmonics or near-harmonics. Based upon that experience,
I really think it's fair to say that all of the common string and wind
instruments have partials at frequencies sufficiently close to harmonics
that, for the vast majority of musical purposes, they can be considered
practically harmonic.

You will find a fair number of subharmonics in those tones, by the way.


Dan's point about very short notes is definitely an important caveat to
that. But those have to be very short notes indeed.

Of course bells, timpani, xylophones, and of course snare drums, for
example, are an entirely different question altogether. The piano is
definitely on the hairy edge, but still falls on the harmonic side.

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