How did the pitches of the western musical scale came about?

Harmonic SpectrumHarmonics are produced by cutting the fundamental string by ratio

The human ear is sensitive to ratio of frequencies/perfect integer, thus producing intervals

Intervals - distance between 2 notes

A step up or down in pitch which is specified by the ratio of frequencies involvedJust intervals - represented by exact integer ratios

2:1 for an octave up3:2 for a perfect 5th up4:3 for a perfect 4th up5:4 for a major 3rd up6:5 for a minor 3rd up

Example: F - A5 = 440hz A5=? 1/440x2/1 = (440)2 = 880HzExample: E5 =? 1/440x3/2 = (440)3/2 = 660hzExample: D5 =? (440)4/3Example: C#5 =? (440)5/4Example: C5 =? (440)6/5

Consonance - frequencies of tones related by a small integer ration such as the octave (2:1), perfect fifth (3:2), perfect fourth (4:3)Dissonance - Helmholtz proposed that maximum dissonance would arise between 2 tones when the beat rate is roughly 35Hz

Scales and Temperament

Scales - it is a succession of notes in ascending orderTuning - adjustment of pitch to correspond to an accepted normTemperament - a system of tuning in which intervals deviate from acoustically pureIntonation - degree if accuracy with which pitches are producedPythagorean Scale - All notes can be derived by going up or down a 4th or 5th

From C: F = 4/3, G = 3/2, D = 3/2x3/2 or (3/2)^2 = 9/4, A = (3/2)^3 = 27/8, E = (3/2)^4 = 81/61, B = (3/2)^5 = 243/32Arranged in scalar form

C (1), D (9/8), E (81/64), F (4/3), G (3/2), A (27/8), B (243/32)Problems

Semitones created between E-F and B-C are different in size compared to the semitones created by dividing the whole tones, which is above the noticeable difference in pitchDifficulty with modulation of keys

SolvedJust Temperament (Just Tonic) - based on the Major triad

There are 3 major triads in a major scale: Tonic, Subdominant, and DominantFrom C: Tonic: E (5/4), G (3/2)

Dominant: B 3/2 (G)x5/4 = 15/8, D (3/2(G)x3/2) = (9/4)/(2/1(octave lower)) +

9/8Problems

Some 4ths and 5ths are imperfectProduces more major and minor while tones (microtonal)

Equal Temperament - the octave is divided into 12 equal intervals (semitones), producing at the ratio of 2 (octave)^12

A4 (440Hz) to A5 (880Hz) = 440Hz/12 = 36.67HzA4 (440HZ) to A3 (220Hz) = 220Hz/12 = 18.33HzAdvantage

Advantage in modulating to different keyTuning of most instruments today with relatively fixed keys

Cents - are used to express intervals in an equal temperament scale1 semitone = 100 cents1 octave = 1200 cents

SYNTONIC Coma: difference between the just and pythagorean

Trivia 1. Standard pitch of A4 was adopted only during the international conference in London, 19392. Handel's tuning fork, A4 is tuned 422 Hz, which means compositions during those era