A little music theoryA little music theory(mostly notation, names, …and (mostly notation, names, …and
temperament)temperament)
Physical:Physical: It has nothing to do with human beings. Ex: It has nothing to do with human beings. Ex: beatingbeating
Psychophysical, psychological:Psychophysical, psychological: human anatomy. Ex: human anatomy. Ex: fundamental trackingfundamental tracking
Cultural: Cultural: society dependent.society dependent. Ex: appreciation of Ex: appreciation of Beattles songsBeattles songs
Nature or nurtureNature or nurture
Doubling the frequency feels like the same pitchDoubling the frequency feels like the same pitch
(pitch periodicity)(pitch periodicity)
f and its harmonics:f and its harmonics: f, 2f, 3f, 4f, … f, 2f, 3f, 4f, …
2f and its harmonics:2f and its harmonics: 2f, 4f, 6f, … 2f, 4f, 6f, …
This is not a cultural phenomena, it seems to be This is not a cultural phenomena, it seems to be present in any musical culture.present in any musical culture.
In Western music the pitch range from f to 2f is In Western music the pitch range from f to 2f is split in 12 stepssplit in 12 steps
(entirely cultural)(entirely cultural)
ff
ff00 2 f2 f00
C, C#/Db, D, D#/Eb, E, E#, Fb, F, F#/Gb, G, G#/Ab, A, C, C#/Db, D, D#/Eb, E, E#, Fb, F, F#/Gb, G, G#/Ab, A, A#/Bb, BA#/Bb, B
oror
do, do#/re b, re, re#/mi b, mi, mi#/fa b, fa, fa#, sol, do, do#/re b, re, re#/mi b, mi, mi#/fa b, fa, fa#, sol, sol#/la b, la, la#/sib, sisol#/la b, la, la#/sib, si
This has changed historically but now This has changed historically but now it’s standard to take:it’s standard to take:
AA44 = 440 Hz = 440 Hz
So ASo A55 = 880 Hz, A = 880 Hz, A33 = 220 Hz, … = 220 Hz, …
For the intermediate notes the whole thing is more For the intermediate notes the whole thing is more contentious (we’ll discuss temperament later)contentious (we’ll discuss temperament later)
Measure time in beatsMeasure time in beats
four beats in a four beats in a measuremeasure
this will count this will count as one beatas one beat
Consonance and dissonanceConsonance and dissonance
[Let us play some intervals and find what makes [Let us play some intervals and find what makes them consonant or dissonant]them consonant or dissonant]
CC
C# D D# E F F# G G# A A# B CC# D D# E F F# G G# A A# B Cminor minor
22ndnd major major
22ndnd minor minor
33rdrd major major
33rdrd 44thth tritonetritone 55thth minor minor
66thth major major
66thth minor minor
77thth major major
77thth
ratio of frequencies = ratio of small ratio of frequencies = ratio of small integers consonanceintegers consonance
Examples:Examples:
1/1 unison1/1 unison
2/1 octave = 7 tones2/1 octave = 7 tones
3/2 fifth = 3 ½ tones (actually 1.4983)3/2 fifth = 3 ½ tones (actually 1.4983)
4/3 fourth = 2 ½ tones (actually 1.22482)4/3 fourth = 2 ½ tones (actually 1.22482)
5/4 major third = 2 tones (actually 1.25991)5/4 major third = 2 tones (actually 1.25991)
Consonance/dissonance and the overtone seriesConsonance/dissonance and the overtone series
unison = 0 tonesunison = 0 tones
consonance beating roughness consonance roughness …consonance beating roughness consonance roughness …
TemperamentTemperament
Problem:Problem: choose the frequencies of the notes choose the frequencies of the notes (C, C#, D, …) in order to make the (C, C#, D, …) in order to make the
consonances very good consonancesconsonances very good consonances
Remember: the best consonances areRemember: the best consonances are
Octaves: 2/1 6 tones = 12 semitonesOctaves: 2/1 6 tones = 12 semitones
Fifths: 3/2 3 ½ tones = 7 semitonesFifths: 3/2 3 ½ tones = 7 semitones
Fourths: 4/3 2 ½ tones = 5 semitonesFourths: 4/3 2 ½ tones = 5 semitones
Major thirds: 5/4 2 tones = 4 semitonesMajor thirds: 5/4 2 tones = 4 semitones
……
C C# D D# E F F# G G# A A# B CC C# D D# E F F# G G# A A# B C
It is impossible to assign frequencies to the notesIt is impossible to assign frequencies to the notes
In such a way as to keep all fifths = 3/2, In such a way as to keep all fifths = 3/2, fourths = 4/3, … exactfourths = 4/3, … exact
C G D A E B F# C# G# D# A# F CC G D A E B F# C# G# D# A# F C
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
77 octaves 2
12
7
3129.746
2
2 128
not the samenot the same
Pythagorean solutionPythagorean solution
Make the octaves and fifths Make the octaves and fifths perfectperfect
C D E F G A B CC D E F G A B C
1 9/8 81/64 4/3 3/2 27/16 243/128 21 9/8 81/64 4/3 3/2 27/16 243/128 2
27 316 2
2
3 32 22
C D E F G A B CC D E F G A B C
1 9/8 81/64 4/3 3/2 27/16 243/128 21 9/8 81/64 4/3 3/2 27/16 243/128 2
one tone = 9/8one tone = 9/8 ½ tone = 256/243½ tone = 256/243
1 tone = (256/243)1 tone = (256/243)2 2 = 1.1098…= 1.1098…
1 tone = 9/8 = 1.1251 tone = 9/8 = 1.125
Pythagorean Pythagorean commacomma
Perfect third : fPerfect third : f22/f/f11 = 5/4=1.25 = 5/4=1.25
Perfect third : fPerfect third : f22/f/f11 = 81/64 = 1.265… = 81/64 = 1.265…
Can you hear the bad Pythagorean thirds ?Can you hear the bad Pythagorean thirds ?
In the Pythagorean temperament some keys are better In the Pythagorean temperament some keys are better than othersthan others
Samuel Barber's Samuel Barber's Adagio for StringsAdagio for Strings
CC AbAb
courtesy of courtesy of G. MooreG. Moore
Other temperamentsOther temperaments
Pythagorean: good fifth (except one), bad thirdsPythagorean: good fifth (except one), bad thirds
Just: some thirds and fifths are good (tonic, dominant Just: some thirds and fifths are good (tonic, dominant and subdominant of some keys)and subdominant of some keys)
Meantone: better thirds than fifthsMeantone: better thirds than fifths
. . .. . .
Equal temperament: split the difference equally among Equal temperament: split the difference equally among notes. Nothing is perfect, nothing is too badnotes. Nothing is perfect, nothing is too bad
Recap of Music TheoryRecap of Music Theory
same interval = same same interval = same ratioratio of frequencies of frequencies
CC33 C C44
half tonehalf tone tonetone
Consonances: sensation of calm and reposeConsonances: sensation of calm and repose
Frequency ratios nameFrequency ratios name
2/1 octave2/1 octave 3/2 fifth3/2 fifth 4/3 forth4/3 forth 5/4 major third5/4 major third
Dissonances: sensation of tensionDissonances: sensation of tension
Frequency ratios nameFrequency ratios name
729/512 tritone729/512 tritone
243/128 minor second243/128 minor second
Temperament: an assignment of Temperament: an assignment of frequencies to all twelve notes from C to Bfrequencies to all twelve notes from C to B
It is impossible to find a temperament where all the It is impossible to find a temperament where all the octaves and fifths are perfectoctaves and fifths are perfect
Pythagorean: Pythagorean: all octaves and all but one fifth are perfect. all octaves and all but one fifth are perfect. One fifth is very off (pythagorean comma).One fifth is very off (pythagorean comma).
Well or equal :Well or equal : split the differences equally. Every split the differences equally. Every semitone = 1.059…semitone = 1.059…
Equal temperamentEqual temperament
C C# D D# E F F# G G# A A# B CC C# D D# E F F# G G# A A# B C
rr
12 122 2 1.05946...r r
rr22
rr1212=2=2
Nothing too good, nothing too bad …Nothing too good, nothing too bad …
Fifths: rFifths: r7 7 = 1.498 instead of 3/2=1.5= 1.498 instead of 3/2=1.5
Fourths: rFourths: r55 = 1.3348 instead of 4/3=1.3333 = 1.3348 instead of 4/3=1.3333
Thirds: r3=1.25992 instead of 5/4=1.25Thirds: r3=1.25992 instead of 5/4=1.25
……