1: INTONATION, ETC.

Most human musics are based on simple rhythmic and harmonic principles, endlessly elaborated. We pound on things, and discover that pounding in repeated patterns is pleasing. We blow or bow or breathe or rub on other things, and discover that the overlay of repeated sounds is pleasing. And we discover that overlaying sounds whose tones bear a more-or-less simple mathematical retationship to each other, is pleasing.

The simplest such relationship emerges when we bow or pluck a string, then shorten the string halfway and hit it again. OCTAVE! OCTAVE! The second note is twice the frequency of the first.

If we further shorten the string so that it's one-third, one-fourth, one-fifth, one-sixth, one-seventh etc of the original length, we hear new notes that are each a 3x or 4x or 5x or 6x or 7x multiple of the original pitch. This is the OVERTONE SERIES or HARMONIC SEQUENCE. To hear this series, run the following MBASIC/QBASIC program:

  for i= 1 to 20    'build a scale
      t= 50*i       'audible tone
      print t;      'show frequency
      play t,9      'play brief note
  next i            'use next note 

This series of tones is significant because, since each is a perfect multiple of the original [TONIC] note, it resonates with that tonic - play a string tuned to a perfect multiple of a tonic next to an open, unplayed string at the tonic, and the tonic string will vibrate in sympathy with your plucked string. Also, unplayed strings of other perfect multiples will resonate, giving a rich, consonant sound.

In practice, because the harmonic series quickly zooms off into rarefied heights, we build musical scales within an octave by dividing the perfect multiples into usable ratios. So, a simple pentatonic [5-note] scale might have notes at ratios of 5/5 [1, tonic], 6/5, 7/5, 8/5, 9/5 and 10/5 [2, octave]. To hear this scale, run this MBASIC/QBASIC program:

  for i= 5 to 10    'build a scale
      t= (i/5)*100  'audible tone
      print t;      'show frequency
      play t,9      'play brief note
  next i            'use next note 

This isn't the most pleasant-sounding to western ears. A more usable pentatonic scale has the ratios 1, 9/8, 4/3, 3/2, 27/16, 2. Using the above examples, try writing a BASIC program that plays this scale.

Western music evolved an 8-tone octave. The relationships between the notes, based on the harmonic series, are called the Pythagorean scale, with the ratios 1, 9/8, 21/64, 4/3, 3/2, 27/16, 243/128, 2. Another version is the Ptolemaic [Justified Diatonic] scale, with the ratios 1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2. Try programming these scales, and note the differences between them.

These enharmonic or Just-intonation scales are very consonant... as long as music is played in a single key. Changing keys [modulation] reveals often-terrible dissonances, especially when playing chromatic [black-and-white piano-key] notes and not just diatonic [white-piano-key] notes. Bowed strings, human voices, trombones and other such variable-pitch insturments can adapt to the slight differences between the frequencies of Just-intonation notes in different keys, but most wind and brass and fretboard and keyboard instruments can not. Of course, a computerized synthesizer can be programmed to change the pitch relationships when shifting between keys - we'll look at this later.

JS Bach solved the modulation problem by promulgating a new scale system, Even Temperment. THE WELL-TEMPERED CLAVIER is an exposition of the wonders of modulating between keys with such a tuning.

But Even-Tempermant works by slightly detuning ALL notes - the interval between each chromatic [semitone] step, like C to C# or E to F, is a fractional exponent: 2^(1/12), or about 1.059463094... Yow. This works for the production of playable music, but the resonance and richness of pure harmonies are lost. Bother. What to do, what to do???

2: CONSONANCE, ETC.

One way to retain the power of consonance, the resonance of pure harmonies, is to use a scale consisting ONLY of exact multiples of the tonic note. These notes remain extremely consonant up to the 16th overtone, are a bit dissonant in the 17th-20th overtones, and are quite dissonant above that. Once can experiment with fascinating microtonal intervals in the range of the 20th-32nd overtones.

Of course, the standard music staff is insufficient to notate this kind of music. I use logarithmic graph paper, and I'll provide access to a model of this staff Real Soon Now. I promise!

Another way to emphasize consonance, without being run off the high end of an increasingly-dissonant scale, is to use only exact multiples of the tonic [up to and including the 16th overtone], and their even divisions. That is, if the tonic is 50 Hz, exact multiples would include 100, 150, 200, 250, 300, 350, 400 Hz, etc; and the even divisions would include 75 [150/2], 125 [250/2], 175 [350/2] Hz, etc. To hear this scale, run this MBASIC/QBASIC program:

  for i= 1 to 16    'build a scale
      t= 50*i       'audible tone
      print t;      'show frequency
      play t,9      'play brief note
      t= t/2        'down an octave
      print t;      'show frequency
      play t,9      'play brief note
  next i            'use next note 

Of course, these scales only "work" when producing sounds in the same "key" as the tonic or any of its overtones. Modulating from a tonic of 55 [A=440] to a tonic of 64 [C=512] would introduce the same old dissonant problems experienced with any Just scale — UNLESS the entire tonal structure was shifted along with the tonic. That's difficult-to-impossible to do with trumpets, clarinets and acoustic guitars; but it's feasible to program on computer-controlled synthesis systems, whether the driving element is a fretboard, keyhboard, whatever.

Of course, one could always just base their melodies on tones other than the tonic, and play modal scales. That's how musicians dealt with these problems before the era of even-temperament - but it still doesn't solve the dissonance of modulation. Solutions? Live with it, or bend it, or ignore it - there're not many other choices.

But what happens when we turn the harmonic sequence around? Instead of dealing with an OVERTONE series, how about an UNDERTONE series? Start with a high tonic, say 2000 Hz, and build a scale of exact divisions rather than of exact multiples. To hear this scale, run this program:

   for i= 1 to 20    'build a scale
      t= 2000/i     'audible tone
      print t;      'show frequency
      play t,9      'play brief note
  next i            'use next note 

Notating this scale would require a reverse-logarithm-graph staff. Yow.

And just as above where we extended the overtone series by including even divisions of the notes in the series, we can extend the undertone series by including even multiples of the series. Such a scale would contain the tones 2000, 1000, 666.66, 1333.33 [666.66*2], 500, 400, 800 [400*2], 333.33 Hz, etc. Try writing a program to play this scale.

Now try building a mixed scale: up to the 20th overtone of 100 Hz, and down to the 20th undertone of 2000 Hz. Include the even divisions of the overtones, and the even multiples of the undertones. Write a program to produce and play this scale. Is it consonant? Does it swing?

Of course, consonance may be highly overrated. We'll deal with that next.

3: DIDDLING SAMPLES

The diatonic chromatic scale, as implemented on a well-tuned piano or MIDI device, is an artificial construct with only the slightest connexion with consonance, resonance, pure harmonies. It is necessary for modulating between musical keys, and can lead us to further realms of controlled dissonance. (NOTE: Instruments like a piano or gamelan, where multiple elements [strings, chimes, etc] are used to produce an individual note, often have those elements mistuned slightly, so the resultant "beating" note has a throbbing or shimmering effect. Dissonance has its uses, eh?)

We can construct new scales based on e [2.71828...], the basis of natural logarithms; or on pi [3.14159...], the ratio of circular measurements; or on any mathematical function. And we can build scales based on ease of computation, scales that are simple and very fast for computers to manipulate., especially when handling sound samples. And we can create straightforward notation systems for these scales.

Consider how digital sound samples are constructed and manipulated. The amplitude of the sampled sound is measured every 1/nth of a second with a certain resolution. The sampling rate determines the highest frequency that can be sampled; this cutoff frequency is roughly half the sampling rate. Each sample is stored in 8 or 12 or 16 or 20 or 24 bits, with more bits per sample giving finer resolution, more accurate recording and reproduction.

So, a sampled sound consists of a string of bits, representing a complex mix of waveforms over a period of time. The CD sound standard is 44.1 MHz per channel, 16 bits per sample. Thus, CD-quality sound requires storage space of about 172 kilobytes per second.

The apparent pitch of a sound to the human ear is a psychological effect, not determined just by a tone's frequency. Different mixes of overtones, producing the timbre or flavor of a sound, can give very different tonal experiences to a listener.

But the overall apparent pitch of a sound CAN be changed in a controlled manner by digital manipulation. Let's say we have a single-channel [mono] recording of a 2 KHz flute tone, sampled at 11 KHz, 8-bit sampling, with the piece lasting 10 seconds. If we remove every other 8-bit byte and save the processed sample, it will now sound like a 4 KHz flute tone playing for 5 seconds. If we repeat every byte once, the result sounds like a 1 KHz flute tone playing for 20 seconds. It is EXACTLY like doubling or halving the speed of a tape deck - the faster the tape the higher the pitch, and the slower the tape the lower the pitch. Removal or replication of bytes in a digital sample have the same effects.

[Of course, there's no free lunch. Using these tape-speed tricks to change the pitch of a sound also changes the pitch of the sonic environment, unless the sound was recorded in an acoutistically-dead environment. So all the echo/reverberation, background sounds etc. are shifted too, so the result is quite unnatural - I think this is called "the chipmunk effect.]

We're not limited to just doubling or halving a sample's pitch and duration. By deleting or replicating a ratio of bytes, we change the pitch/duration by that ratio. And by deleting/replicating GROUPS of bytes, we can change the duration while leaving the apparant pitch unchanged — within certain limits of fidelity, of course. Muck around with a sample enough and it starts sounding dirty, and there's no good cure for that, even working at the highest resolution possible.

Higher resolutions [16-bit to 24-bit] sound better than sampling at lower resolutions [8-bit or 12-bit]; a higher sampling rate [44.1 or 48 KHz] also sounds better than a lower rate [8 or 11 KHz]; but for the sake of simplicity in the discussion, I'll deal with 8-bit 11 KHz samples.

How can we use this sampling stuff in building new harmonies, new scales?

[to be continued...]