Review of basic pitch, frequency, interval concepts

• 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
  • |p1 - p2| = |q1 - q2| means that ratio of the p's = ratio of the q's: freq(p1)/freq(p2) = freq(q1)/freq(q2)
  • to go up a whole tone: multiply freq by 9/8
  • to go up a fourth: multiply freq by 4/3
  • to go up a whole tone plus a fourth: multiply by (9/8) x (4/3) = 3/2 (a fifth)
• 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)
• see spreadsheet demonstration...