PH 105

1
PH 105
Musical Scales
Dr. David Renneke Lecture 14
2
Musical Intervals and Scales

• Function generator -
• Any frequency (within 0.1 Hz)
• Musical scale
• An array of selected, discrete frequencies
• Note
• A certain frequency on the scale

3
Intervals

• Definition An interval is two notes sounded
  together or in succession.
• The ear recognizes intervals by ratios rather
  than by differences in frequencies.
• Interval representation ratio R fupper /
  flower
• Examples. octave R 300 Hz / 150 Hz 2
• R 440 Hz / 220
  Hz 2
• perfect 5th R 600 Hz / 400 Hz
  3/2 1.5

4

• Consonant (harmonious) intervals -
• Frequency ratios are small whole numbers (sound
  smoother)
• Dissonant intervals -
• Freq. ratios are not small whole numbers (sound
  rough, harsh)
• These are somewhat subjective classifications.
• Interval names are based on the relationship to
  C
• Note C D E F G A B C
• Demo Keyboard

8th (octave)
7th
4th
6th
5th
3rd
2nd
5

• Four perfect intervals unison, 4th, 5th, octave
• Four major intervals 2nd, 3rd, 6th, 7th
• Major triads three notes sounded together with
  frequencies in ratio 456
• Example f1 200 Hz, f2 250 Hz, f3 300 Hz
• So the ratio is 200 250 300 which is
  the same as 456 (when you divide
  through by 50)
• In general, the divisor is f1 / 4 .
• On the musical scale the major triads
  are F A C C E G G B D
• Demo Keyboard

6
Pythagorean Scale

• (Latin word scala meaning ladder or staircase)
• It is based on two consonant intervals fourth
  4/3, fifth 3/2
• Let f frequency of the note C
• C D E F G A B C
• Notes Interval Ratio Frequency Note
• a. C F up fourth 4/3 4/3 f F
• b. C G up fifth 3/2 3/2 f G
• c. G C up fourth 4/3 x 3/2 2 2 f C
• d. G D down fourth 3/2 4/3 9/8 9/8 f D
• e. D A up fifth 9/8 x 3/2 27/16 27/16 f
  A
• f. A E down fourth 27/16 4/3 81/64 81/64
  f E
• g. E B up fifth 81/64 x 3/2 243/128 243/128
  f B

a
b
c
d
e
g
f
7

• Summary of the Pythagorean Scale
• Note C D E F G A B C D
• Freq. (Hz) 260.7 293.3 330 347.7 391.1 440 495 5
  21.5 586.7
• Ratio to C 1 9/8 81/64 4/3 3/2 27/16 243/128 2
  
• Calculations fC 440 27/16 260.74 fD
  260.74 x 9/8 293.33 etc.
• Ratio to C 1 1.125 1.266 1.333 1.5 1.688 1.898
  2 (decimal values corresponding to the
  fractions above )
• Ratio to the --- 9/8 9/8 256/243 9/8 9/8 9/8 256
  /243
• preceding freq.
• Calculation RF / RE 4/3 81/64 256/243
  1.053 (This is called a semitone.)

8

• Just Diatonic Scale (more consonant intervals)
• Note C D E F G A B C D
• Freq. (Hz) 264 297 330 352 396 440 495 528 594
• Ratio to C 1 9/8 5/4 4/3 3/2 5/3 15/8 2
• Calculations fC 440 5/3 264 fD 264 x
  9/8 297 etc.
• Question What is the frequency of A to the
  left of C (not shown) ?
• Answer 440 2 220 Hz (down one octave)
• Ratio to C 1 1.125 1.250 1.333 1.5 1.667 1.875
  2 (decimal values corresponding to the
  fractions above )
• Ratio to the --- 9/8 10/9 16/15 9/8 10/9 9/8 16
  /15
• preceding freq.
• Triad 1 4 5 6 (C E F - divide by
  264/4 66)
• Triad 2 (G B D - divide by 99) 4 5 6
• Triad 3 4 5 6

9
True BASIC programs for sound

• Program phone
• for ring 1 to 3
• for x 1 to 30 ! repeat 30 times for each
  ring
• sound 600, 0.03 ! freq. 600 Hz, duration
  0.03 s
• sound 1500, 0.03
• next x
• next ring
• end
• Program scales
• ! Frequencies for the 8 notes of the
  Pythagorean scale.
• data 8, 260.7, 293.3, 330, 347.7, 391, 440,
  495, 521.5
• read notes
• for x 1 to notes
• read freq
• sound freq, 0.5 ! play each note ½
  second
• next x
• end
• Demo 1. phone, 2. scales (Pythagorean), 3.
  scales (Just Diatonic),
• 4. scales (alternate Pythagorean and Just
  Diatonic)

10
Returning to the Just Diatonic Scale

• There is a problem Gsharp ? Aflat
• Remember the three ratios
• major semitone 16/15 1.067
• major whole tone 9/8 1.125
• minor whole tone 10/9 1.111
• The last two numbers are close but not identical.
• Compromise Let (two semitones 1 major whole
  tone) and eliminate the minor whole tone. That
  is, add five notes to the octave between C D,
  D E, F G, G A, A B.
• Result The octave is divided into 12 equal
  intervals.
• We have just arrived at the definition of the
  ---

11
(Equal) Tempered Scale (this is our current scale)

• Definition Let a ratio of adjacent notes
• We must have a x a x a x a x a x a x a x a x a
  x a x a x a 2
• or a12 2
• Solving for a a 21/12 1.059463
• Note a2 1.122 (in between 1.125 and 1.111)
• Look at the extensive table on p. 183.
• Demo 5. scales (Tempered white keys 8
  notes)
• 6. scales (alternate Just, Tempered 16
  notes)
• 7. scales (Tempered all keys 13 notes)

12

• Lets use the value of a to check out a few of
  the frequencies in the table on p. 183.
• Note freq. (Hz)
• C4 261.63
• C4 277.18 261.63 x a 261.63 x
  1.05946 277.18
• D4 293.66 277.18 x a 277.18 x
  1.05946 293.66
• etc.
• Special unit for an interval - cent
• Definition Each interval corresponds to 100
  cents. So, an octave is 1200 cents.
• Recall ratio R fupper / flower
• Then (no. of cents) n 1200 (log R / log 2)
  3986 log R
• Examples perfect 5th R 3/2, n 3986 log
  (1.5) 702 cents
• perfect 4th R 4/3, n 3986 log
  (1.333) 498 cents
• Look at Table 9.3 on p. 184 (textbook) for other
  values.

13
Audio Demonstrations via CD(see the handout)

• Track 34 (137)
• 18. Logarithmic and Linear Frequency
  Scales Compares the 7-step diatonic and the
  12-step chromatic scales.
• Track 32 (059)
• 16. Stretched and Compressed Scales
• Track 52 (120)
• 27. Circularity in Pitch Judgment
• This illustrates an auditory illusion called the
  Shepard Scale which is ever-ascending. It also
  plays the continuous descending scale of
  Jean-Claude Risset.