Review of last week

1

• Review of last week
• How we hear combination tones depends on anatomy
  and perception.
• When frequency difference between two
  simultaneous notes is a sub harmonic of both,
  then consonance gt dissonance
• Consonant intervals given by simple integer
  ratios, they include
• Octave 2 Perfect fifth 3/2 fourth 4/3
• Major 3rd 5/4 Minor 3rd 6/5 Tone 9/8
• Scales
• Aims
• To demonstrate principles used to construct
  scales and discuss their use as a resource for
  melody, harmony and rhythm
• Learning Outcomes
• A definition for scales, Pythagorean Scale, Just
  Scale, Equal Temperament.

2
Scales

• Discrete set of pitch relationships (notes) to
  yield maximum consonant combinations
• Scales are the resource upon which to build
  melodies, harmonies and rhythms
• Consonant intervals arise when frequency ratios
  are whole number fractions
• e.g. octave 2/1, fifth 3/2, fourth 4/3, third
  5/4, etc
• However, building a scale not as straightforward
  as it may at first seem

3
Psychological perspective

• Notes are clearly identifiable entities
• (even when in harmony or modulated)
• Notes help us remember melodies
• Notes help us share ideas
• For the mind to cope with music it must
• be able to recognise patterns
• draw on familiarities and cultural references
• be challenged by invention

Igor Stravinsky was the master of exploiting
consonance and dissonance. Drawing on familiar
references and offsetting with daring new sounds
4
Real Pianos not quite tuned this way!

• Octave 2/1
• Fifth 3/2
• Fourth 4/3
• Third 5/4
• Minor Third 6/5
• Tone 9/8

Tuning the ideal piano
5
Intervals Not Concurrent
6
Musical Notation

• Diatonic - seven notes, i.e. C,D,E,F,G,A,B
  white notes on piano (Key of C)
• We can represent these notes on lines called
  staves.
• Chromatic twelve notes all notes black and
  white

Piano Keyboard
7
Harmonic Series

• Whole number multiples of fundamental
• HS f , 2f , 3f , 4f , ......... nf
• e.g. from middle C (261Hz)
• f 261Hz , 522Hz , 783Hz , 1044Hz ,
  ................... n x 261 Hz.
• Intervals 2, 3/2, 4/3, 5/4, ...

8
Pythagorean Diatonic Scalebuilt of fifths and
octaves
Go up in fifths
Bring into octave range
Also go down a fifth to get fourth
Sort in ascending order
9
Deriving Pythagorean Chromatic Scale
To get twelve note scale go up and down in fifths
10
Problems with Pythagorean Diatonic and Chromatic

• Major third not the most consonant interval Ideal
  (5/4) 1.250 - Pythagorean (81/64) 1.265.
• Pythagorean Comma (Wolf)
• Chromatic scale - semitone intervals that
  alternate in frequency ratios of 1.053 and 1.068.

11
Triads

• Three note chords
• Major (happy) or Minor (sad)
• Triads have intervals
• perfect fifth (3/2)
• major third (5/4)
• minor third (6/5)

12
Just Diatonic Scale Built on Triads

• Triad at a fifth above, multiplying by 3/2
• i.e. (3/2) (1) (5/4) (3/2) (3/2) (15/8)
  (9/4)
  
• Triad at a fifth below, dividing by 3/2
• i.e. (2/3) (1) (5/4) (3/2) (2/3) (5/6)
  (1)
• Gives three triads 2/3 5/6 1 5/4
  3/2 15/8 9/4
• Bring in octave range (multiply by 2 or 1/2) and
  sort

13
Problems with Just

• There are two whole tones
• 9/8 1.125 called the major tone,
• 10/9 1.1 called the minor tone.
• The semitone has a ratio of 16/15 1.06'.
• The minor third between notes 2 and 4,
  (4/3)/(9/8) 32/27 1.'185' , does not have the
  desired ratio of 6/5 1.2.
• The perfect 5th between notes 2 and 6,
  (10/6)/(9/8) 40/27 1.'481' does not have the
  desired ratio 3/2 1.5.

14
Tempering

• Compromise between requirements
• true tone intonation (3/2, 3/4..etc)
• freedom of modulation (different keys sound same)
• convenience in practical use
• (e.g. keyboards can play along with fretted
  guitars)

15
Equal Temperament All intervals same frequency
ratio
16
Compromise from ideal consonant intervals