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Physics 371 March 14, 2002
• Scales (end)
names of intervals
transposition
the natural scale
the tempered scale
meantone tuning
The Natural Scale
bugle, bach trompet, natural horn etc.
1
C
2
C
3
G
4
C
5
E
example: find tones of horn in G
6
G
7
(Bb)
8
C
9
D
10
E
the problem of transposition
how many keys do I need to retune?
what if I don’t retune?
what about adding more keys to the keyboard?
Help in visualizing scales:
• equal musical intervals - equal frequency ratio
• 44on a “multiplicative” number line (=log scale) equal ratios are equidistant
• advantage: in graphs below equal intervals have same length
1
1.5
2
octave
C
D
4
3
octave
fifth
E F
8
octave
G
1
A
B C
2
JUST
x Triad
x
6
x
x
12
The Tempered Scale
Octave divided into 12 equal intervals
Semitone ratio x: x12 = 2
x = 1.05946….
How to calculate tempered frequency ratios:
Example on blackboard:
fifth - 7 semitones
major third - 4 semitones
3/27/2001
Tempered Scale - good or bad?
all 12 halftone intervals are identical (100 cent)
solves problem of transposition
how good (“exact”) are the tempered intervals?
the fifth is good (called “perfect” by musicians)
700 cent but should be 702 cent
ratio = 1.498
major third is sharp! 400 cent instead of 386 cent
1.26
instead of
1.25
example: 480+605 Hz
480+600 Hz
minor third is flat!
300 cent
315 cent
fourth is good
498 cent instead of 500 cent
Meantone tuning - another interesting compromise
1. Use just major third, divided into two equal intervals
2. This defines a wholetone interval - how many cents?
(answer: 193.2 cents)
3. five whole tones use up 966 cents of the octave
4. Problems remain - see homework # 7
What do musicians not bound to a fixed scale
(e.g unfretted strings) actually play?
Measurements on violin soloists (Green 1937)
Minor third
Major third
Fifth
mean 296 cents (range 289-301)
406
400-417 (just: 386!!)
708
692-731 (just: 702!!)
Maybe one likes what one is used to……
Evidence: originally, tempered scale sounded bad
now just or meantone sounds out of tune
Other cultures - other scales…...
Example:
measurements on Burmese Xylophone show that
wholetones are smaller and half tones larger than
Western scales, tending toward seven almost equal
steps.
Similar scales are used in tuning of the Burmese harp.
Demo: scale with seven equal intervals within octave.
Tuning matters mostly in polyphonic music Only when playing cords is consonance the main issue
Became wide spread only in late Renaissance (after 1600)