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

CC EEDD FF GG AA BB

C#C# D#D# F#F# G#G# A#A#

CC . . .. . .. . .. . .

CC22 CC33 CC44

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)

higherhigher

What about the #’s and b’s ?What about the #’s and b’s ?

C#C# AbAb

What about the duration of notes ?What about the duration of notes ?

halfhalf halfhalf

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

slightly slightly more more

complexcomplex

several several instrumentsinstruments

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

octave = 7 tonesoctave = 7 tones

fifth = 3 ½ tonesfifth = 3 ½ tones

fourth = 2 ½ tonesfourth = 2 ½ tones

major third = 2 tonesmajor third = 2 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

close, but close, but not the not the same !same !

1.581.58

1.601.60

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

……