the time relationship between pitch and rhythm (and harmony and polyrhythm): Part II

If you haven't yet read Part I, I recommend glancing at it here before reading this installment.

In the previous episode, we talked about the relationship between rhythm and single tones, and how a series of impulses at regular intervals, when speeded up, would generate a tone.

But what about multiple tones? What is harmony? If we have simultaneously-occurring tones that sound consonant, why do they sound consonant? What is the mathematical relationship of their oscillating impulses? Finally, what would happen if we slowed these tones down to low-frequency oscillations of audible impulses-- what would the rhythms sound like?

Interestingly, there can be rhythmic harmony of varying density just as there can be tonal harmony of varying density. The essence of harmony and its density lies in low-prime mathematical ratios.

The first prime ratio of significance is the 2:1 ratio. In tonal harmony, this creates a simple perfect octave. For example, a tone of 880 Hz juxtaposed with a tone of 440 Hz creates a 2:1 ratio of vibrations resulting in the perfect consonance of two A naturals one octave apart. In rhythmic harmony, it creates a simple division of the pulse. For example, if a series of impulses at 120 beats per minute were identified as quarter notes, a second series at 240 bpm would be perceived as eighth notes-- a 2:1 ratio.

Where it begins to get more interesting--and harmonic--is with the next prime ratio of significance: the 3:2 ratio. Two tones with a vibrational ratio of 3:2 will produce a just-tuned perfect fifth. For example, one tone at 220Hz and another at 330Hz will produce an open fifth of A natural and E natural above. When slowed down and heard rhythmically, the 3:2 ratio gives us the very common "three-against-two" polyrhythm--typified by a triplet in duple time. The actual consonant ratio is best perceived when the "3" pulse and "2" pulse are occurring simultaneously, of course.

The basis of the Euro/Western conception of tonal harmony reaches its overtone-based limit with the next prime ratio of significance: the 5:4 ratio. Two tones with a vibrational ratio of 5:4 will produce a just-tuned major third, which incidentally is about 14 cents narrower than an equal-tempered major third. Two tones at 220Hz and 275Hz will produce A natural and C# above. When slowed down and heard rhythmically, it creates the slightly more elusive "five against four" polyrhythm, typified by squeezing 5 beats into the space normally occupied by four. Again, the rhythmically consonant ratio--much more complex this time--is most clearly perceived when the "5" pulse and "4" pulse occur simultaneously.

While certain non-western/European music, including much music of Asia, Africa, and the Americas (including the blues!) utilize higher ratios such as 7:6 and even beyond, the European 5-limit concept of tonal harmony from which modern equal temperament was derived is based strictly on 3:2 and 5:4 ratios reckoned in different directions to create the tones which have been approximated on the equal-tempered keyboard. For more reading on the subject, I highly recommend the book Harmonic Experience by W.A. Mathieu. For this reason, I'll focus here on the most common 3:2 and 5:4 ratios.

In the following examples, I have tuned two oscillators of the trusty Moog Model D to as close an approximation as is practical of just-tuned consonances. When tuning for just consonances, it is desirable to hear the higher tone 'lock' with the corresponding partial frequency (overtone) of the lower tone. The partial frequencies of the overtone series are generated themselves by low-prime ratios, and are the basis of harmony. For this reason, just consonances often possess a rich, otherwordly quality compared to the imperfect consonances created by the series of compromises that is equal-tempered tuning.

An important note: the Model D's one flaw for our purposes (though it is hardly a flaw in its character as a musical instrument!) is the fact that its three oscillators are imperfectly clocked not electronically synched with one another. What this means in real-world terms is that there might be some tuning drift which can result in phasing between the two oscillators. Interestingly, what manifests as a sweeping, oscillating comb-filtering effect when applied to harmonically related tones (not unlike a 'phaser' effect) manifests itself in rhythm as one oscillator "turning the beat around" on the other at regular intervals, or dropping a beat here and there. I have tried to work around this by selecting short enough audio examples that illustrate the polyrhythm without confusing dropped beats.

Example 1 illustrates 3:2. First, using a sawtooth wave on the 8' range, the oscillators are tuned nearly as possible to a just perfect fifth. I depress an "A" note, and you hear "A" and "E" sounding simultaneously. Then I switch to "Lo" range, and you will hear the 3-against-2 polyrhythm, if you listen carefully. When the two oscillators click nearly-simultaneously, that is "beat 1." Then you will hear the distinct "ONE (and) TWO AND THREE (and)" signature of the 3:2 polyrhythm.

Example 2 is a sweep of 3:2 from just below the audible range (fast enough to be rhythmically indistinguishable but still not tonal) well into the audible band (and back). For this, the keyboard is set up similarly to Example 2 from Part I.

Example 3 illustrates 5:4. Set up similarly to Example 1, we tune this time to a just major third. First on the 8' range, when I depress "A," you will hear A and C#. Then on "Lo" range, you will hear 5:4. Again, the 'flam' generated by the two oscillators sounding almost-simultaneously is "beat 1." 5:4 can be an elusive polyrhythm to hear in this context, but if you are familiar with its sound, you will be able to distinguish the particular character of this rhythmic consonance in this example.

Finally, Example 4 is a sweep of 5:4 from just below the audible range well into the audible band (and back).

So that's scratching the surface of that concept. Of course there is a lot more there if you want to dig for it. Electronic composers might enjoy creating pieces based on the ratios, where ratio-based rhythmic consonances speed up into tonal consonances, or slow down from consonances into rhythms. Some might want to explore the complex rhythms inherent in dissonances using electronic means of slowing down recorded intervals. Most importantly, it is just enlightening to know what makes the sounds we make relate to one another the way they do.