[Smt-talk] Structure of intervals

Nicolas Meeùs nicolas.meeus at paris-sorbonne.fr
Tue Sep 13 12:50:07 PDT 2011


You might not receive many answers, I am afraid, because your argument 
really does not resist close examination. As I hate leaving a sincere 
question unanswered, here is my opinion:

There is no such thing as a "rhythm of sound waves". A wave is not 
delimited by kicks or blows and its duration (its period) can be 
measured between any specific point in the first wave to the analogous 
one in the second. The idea that sound vibrations were successions of 
blows, i.e. of individual points, is very ancient, and you are in the 
very good company of antique philosophers in thinking so. I believe, 
however, that the idea had to be abandoned with the development of 
physics in the 17th century, even if a mathematician like Euler probably 
still believed in it in the middle of the 18th century. Sound vibrations 
result from continuous motions.

The addition of notes sounding at an interval results in the addition of 
curves, involving not only the fundamental of the notes, but also their 
partials. These highly complex resulting curves can be analyzed in their 
individual (sinusoidal) components through Fourier analysis, a 
formidable achievement of late-18th-century mathematical theory.

You are right that these individual resulting complex curves may 
individualize intervals and chords. The problem is that they also 
strongly depend on individual timbres (i.e. harmonic content of the 
individual constituent notes). They certainly do account, in general 
terms, for the specific sonorities of given consonant intervals or 
chords, nobody denies that. But (a) they certainly correspond in no way 
to rhythms: the conditions or the respective perceptions of complex 
waves and of rhythms are incommensurate; (b) they hardly could define 
general physical categories, because of the important role of timbre 
(harmonic content) in their construction: consonances and dissonances 
are way too dependent on timbre to elicit "different emotions" that 
could be categorized on purely physical grounds.

It remains, though, that fifths definitely are consonances, and major 
7ths dissonances, regardless of their timbre. This is because they form 
entities not only in a physical context, but also in a semiotic one -- 
i.e. in the context of a musical "language". Circumstances might be 
created where, from a physical point of view, some fifths may sound more 
dissonant than major 7ths; much as known words may sound very bad in the 
mouth of some of us (me, for instance, when I try to speak English). But 
even badly pronounced, words do retain their emotional content.

To sum up: you obviously would be happy to be able to link musical 
emotion to physical properties. These two realms (emotion and physics) 
however do not overlap, so that your hope will not be fulfilled.
     There is a consolation to this, though. If musical emotions really 
were linked to specific physical properties, then very little space 
would be left for alternative musics -- musics of the folks, musics of 
the world -- and the claim of some of us, that our Occidental music is 
the best (they don't usually specify _which_ Occidental music) would 
seem true. Fortunately, the emotion in music _does not_ depend on 
physics, it can take many forms, and allows space for many different tastes.
     We would like our music, or music at large, to be "natural", i.e. 
founded on (physical) nature. Yet, as Walter Wiora (an outstanding 
musicologist) once claimed, any attempt at proving the "naturalness" of 
some music (there have been several such attempts in the history of 
music) usually aimed at proving the unnaturalness of some other (usually 
more recent) music.


Nicolas Meeùs
Université Paris-Sorbonne

Le 12/09/2011 06:52, Manigirdas at cs.com a écrit :
> Musical intervals with the notes sounding together are actually very rapid
> rhythns of sound waves. For example, the 5th can be represented as follows.
> The distance between two marks represents the duration of each sound wave of
> the fundamental partial of a note. The times involved are on the order of
> 1/200 of a second:
> |     |     |     |   (durations of sound waves of the upper note g)
> |        |        |   (durations of sound waves of the lower note c)
> ---------------------
> |     |  |  |      |   (combined rhythm of the waves of the two notes)
> We can see that the sound waves of these two notes are related in the
> rhythm of 2 against 3.
> Similarly, the 4th has the rhythm of 3 against 4:
> |     |     |     |     |   (f)
> |       |       |       |  (c)
> ---------------------------
> |     | |   |   | |     |   (combined)
> The octave has the rhythm of 1 against 2.
> |       |       |  (c an octave higher)
> |               |  (c)
> -------------------
> |       |       |  (combined)
> And so on.
> Chords are also rhythms. For example the waves of a major triad in second
> inversion look like this:
> |           |           |           |           |           |    (top note
> e)
> |              |              |              |              |    (middle
> note c)
> |                   |                   |                   |    (bass note
> g)
> ---------------------------------------------------------------
> |           |  |    |   |     |     |    |    |  |            |   (combined)
> This is the rhythm of 3 against 4 against 5.
> Of course, the sound waves will not necessarily be neatly in phase as in
> the diagrams. If they are shifted relative to each other, the rhythms formed
> will be somewhat more complex.
> These rhythms are too rapid to be discerned by the ear as rhythms, but they
> are there nevertheless. They repeat continuously while the notes are
> sounding.
> Since each interval and chord has an unique rhythm, it may be speculated
> that each rhythm contributes to a different emotion.
> Manigirdas Motekaitis
> Piano teacher
> 714 W. 30th St.
> Chicago, IL 60616-3005
> (312) 804-4324</HTML>
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