Physics of Music Lecture 10

1
Physics of MusicLecture 10

• Tuning and Temperament
• Berg Stork, Chapter 9
• Prof. Charles Hyde-Wright
• Old Dominion University

2
Tuning

• Concert A 440 Hz
• Historic pitch Large variations in custom with
  period and location. Gradual historical tendancy
  for pitch to rise.
• Baroque period typically a major third (1.25)
  lower.
• Octaves Frequency Interval 21

3
Scales, Temperament

• 12 half steps in an octave.
• C-Major Scale (Diatonic)
• CDEFGABC Steps 1, 1, ½ , 1, 1, 1 , ½
• G-Major Scale
• GABCDEF,G Steps 1,1,1/2, 1, 1, 1, ½
• Every major scale is constructed from the same
  pattern 1 1 ½ 1 1 1 ½ built on any one of the
  twelve notes of the chromatic scalecalled the
  tonic

4
Temperament

• Pythagorean Equal Tempered Tuning
• Pythagorean
• Tune to circle of fifths
• Musical fifth 5th note in diatonic scale
• C-Major Major Chord CEG
• Third, Fifth Ideal frequency intervals are
• 5/4, 3/2
• Harmonics of fundamental
• 1, 2, 3, 4, 5, 6, 7, 8 (Frequency ratios to
  fundamental)
• 1, 1,3/2, 1,5/4, 3/2, 7/4, 1 Ratios within an
  octave

5
Temperament

• Ideally, tune all fifths to 32 ratio
• Tune all thirds to a 54 ratio.
• Tune all octaves to 21 ratio.
• Impossible.
• Some compromise is necessary

6
Pythagorean Temperament

• Tune all but one fifth to 32 ratio
• Tune all octaves to 21 ratio
• Perfect fifths C G D A E B F D-flat
• Perfect fifths A-flat E-flat B-flat F C
• (3/2)12 129.75
• 27 128.
• Comma of Pythagorus
• Mistuning of 129.75/128 1.0136 23 cents
• Tune D-flat A-flat (3/2)/1.0136
• 23 of one half-step flat.

7
Equal Temperament

• Bach Well Tempered Clavier
• Distribute the mistuning equally in all
  intervals.
• 12 half steps are each 12th root of 21
• Half step frequency ratio 1.059521/12
• For tuning discrimination, divide ½ step into 100
  cents.
• Each cent is a frequency ratio 21/1200 1.00058

8
Major ChordIdeal Frequency ratios 11.251.50

• Equal Tempered
• Every chord is the same as CEG
• 1 1.2599 1.4983
• Equal tempered improves the third with only a
  tiny flatness to fifth

• Pythagorean
• CEG, GBD, DFA, ACE,
• 1 1.2656 1.5000
• EGB, BD F, G-flat B-flat
• 1 1.2486 1.500

9
When is A not equal to B-flatEnharmonic notes

• Some (Baroque??) keyboards have multiple
  keyboards (and strings) allowing for different
  tunings.
• F-Major scale F G A B-flat C D E F
• Construct one keyboard tuned to the scales
• F,C G, D, A, E, B (see BS Fig. 9-1)
• B-Major scale B C D E F G A B
• Construct second keyboard tuned to scales
• B G-flat D-flat A-flat E-flat B-flat.
• A and B-flat will differ by ratio 1.0136
• comma of pythagorus.

10
Can you hear the differencein different tunings?

• Cool Edit File EqualPythagComma.wav
• Sequence of interval of fifth AE
• 11.5 Perfect, or Pythagorean
• 11.498 Equal Tempered
• Superimpose 1.498 and 1.5 (times 440 Hz)
• 11.5/1.013 Comma of Pythagorus (shifted from
  D-flatA-flat interval to AE).
• Superimpose complex tone (A 440 Hz) with Comma of
  Pythagoras
• Differences in Thirds AC are larger