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
