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.

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

Sometimes it is good to get back to basics. Sometimes it helps to pull the camera back, throw on the wide angle, and remember some of the really obvious things about musical sound-- such as:

• A tone is a rhythm happening really, really fast.
• A harmonic interval is a polyrhythm happening really, really fast.

But before we get to that, it pays to consider the question "What is a sound?"

Sound is, most literally, the rapid compression and rarefaction (de-compression) of air, which is perceived by tiny hairlike structures in our ears and decoded as sound by our brains. However, all sounds are not tones-- some are impulses and some are noise. A tone (with a definite pitch) results from an oscillation of impulses in a periodic manner. What is an "oscillation of impulses?" Anyone who has ever used an oscillating fan knows that an oscillation is a repeated movement back-and-forth at a constant speed. When air molecules compress and rarefy at a consistent interval in a regular pattern, an oscillation, or tone, is produced.

Because electronic music is all about creating electrical impulses that are analogous to (i.e. "electronically representative of") the air's compression and rarefaction, you can learn a lot about sound and vibration by playing around with a synthesizer. A good analog synthesizer can teach you a lot, particularly if it has multiple oscillators and an intuitive user interface. My favorite tool for this purpose is an old Moog Model D, as its knob-based interface makes sense to me.

On a synthesizer, an oscillator is a circuit designed to generate an AC voltage that swings positive, then negative at regular intervals to electronically represent air that compresses, then rarefies at regular intervals. This voltage can later be converted into actual vibrating air by a transducer, or speaker.

The Moog, like many analog synthesizers, has a few oscillators that can be set to oscillate in one of several ranges, including 5 audible ranges and one labeled "Lo." "Lo" is what is known as an LFO, or "low-frequency oscillator." This means it creates oscillations, or patterns of impulses, that if transduced into sound would be too slow for the ear to recognize as a tone. What do you get when you have a regular pattern of impulses too slow for the ear to recognize as a tone? You get a rhythm!

The slowest rate of oscillation that a human ear will recognize as a tone is approximately 20Hz, or 20 cycles per second. 20 or more impulses in one second, at regular intervals, creates a very low audible tone to most humans. Anything slower, and the typical ear will hear instead a series of successive impulses--A repetitive beat, in other words.

In synthesis, LFOs are typically used in conjunction with other circuits to create vibrato or other modulation effects (among others), and are not typically transduced into sound. However, if one has a synth like the Model D that allows patching of the LFO to the output, listening to what it sounds like can be pretty informative.

When a Model D oscillator is set to "Lo" and routed to the output, depressing a key causes a series of audible, electronically generated clicks, or impulses, that can be heard through the speaker. If keyboard control of the oscillator is turned on, then pressing a higher key causes the impulses to speed up in relation to one another. Depressing a lower key causes a slower series of impulses.

This basically creates a slow-motion movie of exactly what happens when when we use the synthesizer (or any other instrument) to create a tone, except that when we move up to a faster rate of oscillation, we get tones instead of rhythms. The faster-in-succession the impulses, the higher the pitch.

Particularly interesting is to explore the estuary between the audible and sub-sonic ranges of oscillation. For example, selecting the lowest audible range (labeled 32' on the Moog) and depressing the lowest key on the keyboard-- and then selecting "Lo" and depressing the highest key of the keyboard-- will start to give a picture of the transition between "series of impulses" and "tone." Utilizing the tuning and pitch-bend functions can start to give an even more complete picture.

In the following example, I have set the oscillator to a square-wave (which is essentially a binary "off-on" type of impulse, very good for demonstration purposes). I first depress a "low C." You will hear a series of impulses occurring at about 150 beats per minute. Then I move up in octaves, first with a C an octave higher on the keyboard, then another couple of octaves above that until I reach the top key of the keyboard. Each octave up causes the rate of clicks to double (say, from quarter notes to eighth notes, then sixteenths, etc). At this point you can almost begin to hear the impulses wanting to blur together into a tone. To go yet another octave above, we must move now to the 32' setting (signaled by a brief pause), at which point the tone becomes obvious and clear, but very low in pitch. In fact, it is so low in pitch that depending on your speakers, it may appear an octave higher than it actually is. This is because the square waveform contains a lot of harmonics and not all speakers can reproduce very low frequencies with any linearity. Finally, I go one additional octave above for good measure.

The region at which your personal ears begin to perceive tone as opposed to rhythmic pattern can be further narrowed down by utilizing the tuning function on the synth. Using the tuning adjustment and/or the pitch bend wheel, you can find the exact point at which your ear fails to hear the distinct impulses and begins to instead hear a tone.

For another example, I set the oscillator to 32' and turn the tuning and pitch bend controls all the way down. I then set the portamento ("glide") function to its maximum (slowest) setting. This allows me, by depressing the bottom and then top keys of the keyboard, to create a sweep that goes from the sub-sonic range well up into the audible band, and back down again. Notice as the rapid clicks morph into a rising tone. Pretty cool!

In Part II, we will explore harmonic relationships between tones and their relationships to polyrhythmic patterns of impulses.