A little music theory (mostly notation, names, …and temperament)

Physical: It has nothing to do with human beings. Ex: beating Psychophysical, psychological: human anatomy. Ex: fundamental tracking Cultural: society dependent. Ex: appreciation of Beattles songs Nature or nurture

Doubling the frequency feels like the same pitch (pitch periodicity) f and its harmonics: f, 2f, 3f, 4f, … 2f and its harmonics: 2f, 4f, 6f, … This is not a cultural phenomena, it seems to be present in any musical culture.

In Western music the pitch range from f to 2f is split in 12 steps (entirely cultural) f f0f0f0f0 2 f 0 C, C#/Db, D, D#/Eb, E, E#, Fb, F, F#/Gb, G, G#/Ab, A, A#/Bb, B or do, do#/re b, re, re#/mi b, mi, mi#/fa b, fa, fa#, sol, sol#/la b, la, la#/sib, si

CEDFGAB C#D#F#G#A# C... C2C2C2C2 C3C3C3C3 C4C4C4C4

This has changed historically but now it’s standard to take: A 4 = 440 Hz So A 5 = 880 Hz, A 3 = 220 Hz, … For the intermediate notes the whole thing is more contentious (we’ll discuss temperament later)

higher

What about the #’s and b’s ? C# Ab

What about the duration of notes ? halfhalf

Measure time in beats four beats in a measure this will count as one beat

slightly more complex

several instruments

Consonance and dissonance [Let us play some intervals and find what makes them consonant or dissonant]

C C# D D# E F F# G G# A A# B C minor 2 nd major 2 nd minor 3 rd major 3 rd 4 th tritone tritone 5 th minor 6 th major 6 th minor 7 th major 7 th

ratio of frequencies = ratio of small integers consonance Examples: 1/1 unison 2/1 octave = 7 tones 3/2 fifth = 3 ½ tones (actually ) 4/3 fourth = 2 ½ tones (actually ) 5/4 major third = 2 tones (actually )

Consonance/dissonance and the overtone series unison = 0 tones

octave = 7 tones

fifth = 3 ½ tones

fourth = 2 ½ tones

major third = 2 tones

consonance beating roughness consonance roughness …

Temperament Problem: choose the frequencies of the notes (C, C#, D, …) in order to make the consonances very good consonances

Remember: the best consonances are Octaves: 2/1 6 tones = 12 semitones Fifths: 3/2 3 ½ tones = 7 semitones Fourths: 4/3 2 ½ tones = 5 semitones Major thirds: 5/4 2 tones = 4 semitones …

C C# D D# E F F# G G# A A# B C It is impossible to assign frequencies to the notes In such a way as to keep all fifths = 3/2, fourths = 4/3, … exact

C G D A E B F# C# G# D# A# F C not the same

Pythagorean solution Make the octaves and fifths perfect C D E F G A B C 1 9/8 81/64 4/3 3/2 27/16 243/128 2

C D E F G A B C 1 9/8 81/64 4/3 3/2 27/16 243/128 2 one tone = 9/8 ½ tone = 256/243 1 tone = (256/243) 2 = … 1 tone = 9/8 = Pythagorean comma

close, but not the same !

Perfect third : f 2 /f 1 = 5/4=1.25 Perfect third : f 2 /f 1 = 81/64 = 1.265… Can you hear the bad Pythagorean thirds ?

In the Pythagorean temperament some keys are better than others Samuel Barber's Adagio for Strings CAb courtesy of G. Moore

Other temperaments Pythagorean: good fifth (except one), bad thirds Just: some thirds and fifths are good (tonic, dominant and subdominant of some keys) Meantone: better thirds than fifths... Equal temperament: split the difference equally among notes. Nothing is perfect, nothing is too bad

Recap of Music Theory same interval = same ratio of frequencies C 3 C 4 C 3 C 4 half tone tone

Consonances: sensation of calm and repose Frequency ratios name 2/1 octave 2/1 octave 3/2 fifth 3/2 fifth 4/3 forth 4/3 forth 5/4 major third 5/4 major third Dissonances: sensation of tension Frequency ratios name 729/512 tritone 729/512 tritone 243/128 minor second 243/128 minor second

Temperament: an assignment of frequencies to all twelve notes from C to B It is impossible to find a temperament where all the octaves and fifths are perfect Pythagorean: all octaves and all but one fifth are perfect. One fifth is very off (pythagorean comma). Well or equal : split the differences equally. Every semitone = 1.059…

Equal temperament C C# D D# E F F# G G# A A# B C r r2r2r2r2 r 12 =2

Nothing too good, nothing too bad … Fifths: r 7 = instead of 3/2=1.5 Fourths: r 5 = instead of 4/3= Thirds: r3= instead of 5/4=1.25 …