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

A little music theory (mostly notation, names, …and temperament). Nature or nurture. 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.

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

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

  2. Nature or nurture 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

  3. 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.

  4. In Western music the pitch range from f to 2f is split in 12 steps (entirely cultural) f f0 2 f0 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

  5. C# D# F# G# A# . . . . . . C D E F G A B C C2 C3 C4

  6. This has changed historically but now it’s standard to take: A4 = 440 Hz So A5 = 880 Hz, A3 = 220 Hz, … For the intermediate notes the whole thing is more contentious (we’ll discuss temperament later)

  7. higher

  8. What about the #’s and b’s ? Ab C#

  9. What about the duration of notes ? half half

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

  11. slightly more complex

  12. several instruments

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

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

  15. ratio of frequencies = ratio of small integers consonance Examples: 1/1 unison 2/1 octave = 7 tones 3/2 fifth = 3 ½ tones (actually 1.4983) 4/3 fourth = 2 ½ tones (actually 1.22482) 5/4 major third = 2 tones (actually 1.25991)

  16. Consonance/dissonance and the overtone series unison = 0 tones

  17. octave = 7 tones

  18. fifth = 3 ½ tones

  19. fourth = 2 ½ tones

  20. major third = 2 tones

  21. consonance beating roughness consonance roughness …

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

  23. 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 …

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

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

  26. 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

  27. one tone = 9/8 ½ tone = 256/243 C D E F G A B C 1 9/8 81/64 4/3 3/2 27/16 243/128 2 Pythagorean comma 1 tone = (256/243)2 = 1.1098… 1 tone = 9/8 = 1.125

  28. 1.58 1.60 close, but not the same !

  29. Can you hear the bad Pythagorean thirds ? Perfect third : f2/f1 = 5/4=1.25 Perfect third : f2/f1 = 81/64 = 1.265…

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

  31. 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

  32. Recap of Music Theory half tone tone C3 C4 same interval = same ratio of frequencies

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

  34. 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…

  35. Equal temperament C C# D D# E F F# G G# A A# B C r r2 r12=2

  36. Nothing too good, nothing too bad … Fifths: r7 = 1.498 instead of 3/2=1.5 Fourths: r5 = 1.3348 instead of 4/3=1.3333 Thirds: r3=1.25992 instead of 5/4=1.25 …

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