[Smt-talk] Structure of intervals

Joseph Lubben jlubben at oberlin.edu
Thu Sep 15 21:02:17 PDT 2011


I haven't experimented with a siren myself, but I have seen the phenomenon of sirens crossing the rhythmic threshold described in an acoustics text (I don't recall the author) and by Henry Cowell in his New Musical Resources.  The fine-toothed comb technique was described in Donald Hall's textbook, Musical Acoustics.  I've replicated it myself.  I've also heard it in some electronically generated music, in which the rate of repetition of a percussive attack is increased incrementally until it is transformed into a medium-low-frequency hum.  

The waves of frequencies below 15ish Hz are not "below the auditory limit," but simply below the limit of pitch perception.

Difference tones are analogous to the metric accents that accrue to the shared attack between two cycles of differing cardinality.  For example,  a 3 vs. 2 rhythm (the combination of "frequencies" 3 and 2) will create a higher-level attack at frequency 3-2=1, much as two high pitches a fifth apart will produce a difference tone an octave below the lower pitch.  Interference operates most simply in Krebs' "displacement dissonance" of the the type Dx + x/2, where a strong-weak duple cycle is shifted over by exactly half a phase, so that downbeats are eliminated in a sort of metric flatlining.  It gets more complicated with phase shifts that are not exactly half the length of the cycle.  I should note that I find these analogues interesting, but I recognize that they haven't yet done much to shape my hearing of actual music.


On Sep 15, 2011, at 6:49 AM, Nicolas Meeùs wrote:

> Joseph, are you sure that a siren turning so slowly as to produce a frequency below, say, 15Hz, would produce a "rhythm"? Or, more generally, that infrasounds are rhythms? And that a scraped idiophone, if scraped fast enough, would produce a single sound of low frequency? I do very much doubt both hypotheses. 
>     A rhythm is made up of successive sounds within the auditory range (a rhythm does not exist without sounds), not of waves below the auditory limit. The idea that diminishing sound frequencies would become rhythms seem to me analogous to considering that increasing frequency eventually produces light.
> I fail to see in what sense difference tones (and hardly see in what sense interferences) have rhythmic analogues: can you be more specific about that? 
> Yours
> Nicolas Meeùs
> Université Paris-Sorbonne
> nicolas.meeus at paris-sorbonne.fr
> Le 14/09/2011 04:51, Joseph Lubben a écrit :
>> I too think that a sincere answer should not remain unquestioned.  There are in fact a number of circumstances under which pitch and rhythm can be understood as the same physical phenomena.  Any good oscillator, or siren, or fine-toothed comb, can generate a periodic pattern whose frequency can be altered along a wide spectrum, so that "kicks and blows" below 15ish  cycles per second transform into pitch above that range.  Combinations of kicks vs blows in a ratio of three to two, when sped up, will emerge as a perfect fifth, whose quality will of course depend on the individual timbres of the kicks and the blows.  So I do think one can speak of a "rhythm of sound waves," and whatever its inverse would be.
>> The problems arise when making assertions about the sameness of pitch and rhythm in a musical context, not only because of the complexity of timbres associated with musical textures, but also because of the cognitive divide, probably resulting from the structure of the cochlea, whereby pitch and rhythm are *perceived* as distinct phenomena, and processed by the body/brain in different ways.  Still, some important physical phenomena long associated with pitch combinations, such as difference tones and interference, have interesting rhythmic analogs that seem to arise from the equivalent physical properties that adhere to combinations of rhythmic attacks.
>> Joseph Lubben
>> Associate Professor of Music Theory
>> Oberlin Conservatory
>> 440-775-8239
>> jlubben at oberlin.edu
>> On Sep 13, 2011, at 3:50 PM, Nicolas Meeùs wrote:
>>> Manigirdas,
>>> 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.
>>> Yours,
>>> 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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Joseph Lubben
Associate Professor of Music Theory
Oberlin Conservatory
jlubben at oberlin.edu

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