Review of basic pitch, frequency, interval concepts

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  • distinguish: psychoacoustic domain vs. acoustic domain
  • pitched sound = locally periodic sound wave signal (approximately)
  • period is measured in seconds; frequency is measured in cycles per second (Hertz)
  • period and frequency are reciprocals of one another (period of a tenth of a second means: 10 cycles per second)
  • pitch and frequency move up and down together
  • but equal pitch intervals correspond to equal frequency ratios (illustrate all this using Audacity). Get used to these computations:
    • Given 4 pitches p1, p2, q1, q2, the equality of their intervals (p1 - p2) = (q1 - q2) means that ratio of the frequencies of the p's = ratio of the q's: freq(p1)/freq(p2) = freq(q1)/freq(q2)
    • ascending and descending ratios are reciprocals (they cancel each other out)
    • to go up an octave: multiply freq by 2
    • to go down an octave: divide freq by 2 (or multiply by 1/2)
    • to go up a Pythagorean whole tone: multiply freq by 9/8
    • to go down a whole tone: divide freq by 9/8 (or multiply by 8/9)
    • to go up a fourth: multiply freq by 4/3
    • to go down a fourth: divide freq by 4/3 (or multiply by 3/4)
    • to go up a whole tone plus a fourth: multiply by (9/8) x (4/3) = 3/2 (a fifth)
    • to go up a fifth and down a whole tone: (3/2) * (8/9) = 4/3 (a fourth)
  • tempered scale: divide octave into equal parts
    • 12 equal parts: s x s x s x s x s x s x s x s x s = 2, i.e. s^12 = 2. What is s? 2^(1/12)
    • 24 equal parts: q = 2^(1/24)
    • 1200 equal parts (cents): c = 2^(1/1200)
  • to measure intervals in units that add we need to measure freq ratio R in terms of some small unit ratio r.
    • Given frequency ratio R we ask: how many times do we multiply r to reach it?
    • r x r x r x r x ....x r = R
    • answer: log(R) base r
  • thus to measure an interval R in terms of
    • octaves: use log(R) base 2
    • 9/8 (Pythagorean whole tones): use log(R) base (9/8)
    • semitones: use log(R) base 2^(1/12) = 12 * log (R) base 2
    • cents: use log(R) base 2^(1/1200) = 1200 * log (R) base 2
  • Octave equivalence and reducing intervals to a single octave
    • Octave stretches from ratio 1 to 2 (e.g. 440 cps to 880 cps)
    • if the ratio is >2, then divide it by 2, and repeat until it's between 1 and 2
    • if the ratio is <1, then multiply it by 2, and repeat until it's between 1 and 2
    • ex: "up two fifths" = 9/4, but 9/4 > 2, so reduce it to 9/8
    • ex: "down a fifth" = 2/3, but 2/3 < 1, so reduce it to 4/3
  • now consider the "circle of fifths" obtained by raising pitch by 3/2 over and over: start at A4 = 400 Hz, say, then take ascending pitch series:
    • 440 * 3/2
    • 440 * 3/2 * 3/2
    • 440 * 3/2 * 3/2 * 3/2
    • 440 * 3/2 * 3/2 * 3/2 * 3/2 ...
    • 440 * (3/2)^n
  • finally reduce these pitches to a single octave by dividing by 2 (repeatedly if necessary)
  • the resulting pitches will never repeat, since each one has a unique frequency: 440 * 3^n/2^m
  • see spreadsheet demonstration...
  • Pythagorean music theory, the circle of fifths, naming notes and intervals. Pythagoreans believed "all is number", loved integer ratios. Two key intervals:
    • The octave (2/1) (defines equivalence)
    • The fifth (3/2) (generates new pitches)
      • by generating an infinite series (3^n/2^n)...
      • ... reduced to a single octave (3^n/2^m)
  • Pythagorean major scale, formulated another way: starting from a pitch, go up 5 fifths, and down 1 fifth:
    • @v1, @0, @^1, @^2, @^3, @^4, @^5
    • @0 is the tonic this time
    • @^1, @^2, @^3, @^4, @^5: going up
    • @v1: going down
    • Arranged in order we get:
      • @0, @^2, @^4, @v1, @v1, @v3, @v5 as the functional scale tones.
      • ...because moving two fifths upwards generates a step-wise tone: T, separated by 2 fifths.
      • we're left with smaller gaps between series: from @^4 to @v1, and from @^5 to @0. These are limmas (L), separated by 5 fifths.
      • If we go up to @^12 we get a pitch slightly higher than the starting point, by a Pythagorean comma (C).
  • Pythagorean scale
    • The ascending fifth (3/2)
    • The ascending fourth (4/3) = descending fifth (2/3) raised an octave (x 2)
    • The ascending tone (9/8) = up two fifths (3/2) x (3/2) = 9/4, and then down an octave (C to D)
    • The ascending sixth: up three fifths (3/2)^3 = 27/8, then down an octave = 27/16 (C to A)
    • The ascending third: up four fifths (3/2)^4 = 81/16, then down two octaves = 81/64 (C to E)
    • The ascending seventh: up five fifths, down two octaves = 3^5/2^7 = 243/128 (C to B)
    • The descending limma: ascending seventh but dropped an octave = 3^5/2^8 = 243/256 (C to B below it)
    • The ascending limma: reciprocal of descending limma = 256/243, around 90.2 cents
    • The comma (Pythagorean comma)(3^12:2^19) = up 12 fifths, down seven octaves, around 23.5 cents
  • Tempered major scale
    • Theorists noted that 12 fifths comes close to a unison (off by a Pythagorean comma), and divides the octave into 12 nearly equal pieces
    • Decided to make them exactly equal: the octave equals 12 tempered semitones, the "error" of the comma being spread among all 12
  • Just scale
    • The number 5 is introduced to avoid the high integers used in sixths, thirds, and sevenths. Thirds come first, and the other two intervals are generating from the third:
    • Third: 5/4
    • Sixth: (5/4)*(4/3) = 5/3 (a fourth above a third)
    • Seventh: (5/4)*(3/2) = 15/8 (a fifth above a third)
  • Pythagorean, Just, and Tempered major scales (see spreadsheet, hear Audacity files)
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