1. Physics 1200 Topic VII Tuning Theory
Very rough draft
Updated Oct 26, 2009

2. Note 2
If some of the sounds don’t play, open your audio control, make sure SW Synth volume is up! [For some reason it often gets turned down]

3. Index 3 Pythagorean Tuning Just Tuning Equal Temperament References
Cycle of 5ths
Pythagorean method of tuning
Analysis of intervals (two types of semitones)
Just Tuning
Triad Chords
Just method of tuning
Analysis of intervals (two types of whole tones)
Equal Temperament
Geometric Progression of Frequency Perception
Definition of “Cents” (arithmetical progression)
Comparison of Scales
References

4. A. Pythagorean Tuning 4 Outline of Section 1. The Cycle of 5ths
(a) Perfect Intervals (P5 and P4)
(b) Cycle of 5ths (or 4ths) hits all 12 notes of chromatic scale
Pythagorean Comma (non-closure of the circle)
2. Pythagorean Tuning Method
(a) Get all 12 notes from cycle of 5ths (or 4ths)
(b) Frequency ratios of the notes
(c) Whole Tones: whole steps between CD, DE, FG, GA, AB are all 9/8
3. Analysis of Intervals
(a) Diatonic Semitone: BC and EF are the same 256/243 ratio
(b) Chromatic Semitone
other semitones are 2187/2048 (e.g. F to F#)
Pythagorean scale hence has 1 whole tone, but 2 different semitones
Ratio of Chromatic to Diatonic semitone=1.0136=Pythagorean Comma
(c) Syntonic Comma
Thirds in Pythagorean scale sound TERRIBLE
Example: CE and FA are ratio of 81/64 instead of harmonic 5/4
Ratio of Phythagorean to Harmonic third is 81/80=1.0125=syntonic comma

5. 1. Cycle of 5ths Or cycle of 4ths (a) The Perfect Intervals:
The Octave P8 is 1:2 ratio
Octave can be divided by the 5th
P5 (e.g. C to G) is 2:3 ratio
P4 (e.g. G to C) is 3:4 ratio or C to F

6. b). Cycle of 5ths hits all notes6
(or cycle of 4ths)
Cycle of 5ths (red)
[P5 is 7 semitones]
Cycle of 4ths (green)
[P4 is 5 semitones]

7. b). Cycle of 5ths (or 4ths) hits all notes7
P5 is 7 semitones
P4 is 5 semitones
This works because 12 is relatively prime to 5 or 7.
Note, no other cycle of a single interval would hit all notes because their interval would divide 12 [except for M7 of 11 semitones]
“Father Christmas gets drunk after every beer”

8. c). Pythagorean Comma 8 End of cycle is 7 octaves higher,
Expect Ratio: C8/C1=27=128
But each 5th raises 3:2 ratio
Results in: (3/2)12=
Overshot by about ¼ semitone
Error is called “Pythagorean Comma” (or diatonic comma)

9. 2. Pythagorean Tuning Construct all notes by P4 or P59
2. Pythagorean Tuning
Construct all notes by P4 or P5
Method shown on the board in class

10. 10
b. Pythagorean Scale
Ratio:

11. c. Pythagorean Whole Steps11
Tones (2 semitones, or whole steps) appear to be harmonic ratio 9:8
All of these are 8 to 9 ratio
C to D
C# to D#
D to E
E to F#
F to G
F# to G#
G to A
G# to A#
A to B
B to C#
The only one that is not is
A# to C

12. 3. Pythagorean Intervals12
(a) Two types of semitones
Diatonic Semitone:
C# to D
D# to E
E to F
B to C
Chromatic Semitone:
C to C#
D to D#
F to F#
G to G#
A to A#

13. 3b. Pythagorean Comma 13 Chromatic Semitone:
Diatonic Semitone: Pythagorean Comma:
This “imperfection” is due to the non-closure of the cycle of fifths.

14. 3c. Pythagorean thirds sound bad14
Major 3rds sound Terrible
Desired ratio: 5/4
Actual is slightly bigger 81/64
C to E
F to A
G to B
F# to A#
Error is nearly a quarter tone
Syntonic or Ptolemaic Comma: [ratio of Pythagorean to Just M3]

15. B. Just Tuning 15
Probably first introduced by Gioseffo Zarlino 1558, rediscovered by Helmholtz 1863 (hence known as Helmholtz Scale or Harmonic Scale)
A system based on chords to make the 3rds sound harmonious (as pure harmonic ratios)
1. The Triad Chords
2. Just Tuning Method
3. Just Intervals

16. 1. Triad Chords Major and Minor 3rd ratios From Harmonic Series16
Major and Minor 3rd ratios From Harmonic Series
Major 3rd (e.g. C to E) is 4:5 ratio
Minor 3rd (e.g. E to G) is 5:6 ratio or C to Eb

17. b. Major Triad Chord Major Chords: I. CEG IV. FAC V. GBD Made of M3+m317
Major Chords:
I. CEG
IV. FAC
V. GBD
Made of M3+m3
Harmonic Ratio (4:5:6)
I II III IV V VI VII

18. c. Minor Triad Chord Minor Chords: II. DFA IV. EGB V. ACE18
Minor Chords:
II. DFA
IV. EGB
V. ACE
Made of m3+M3
Harmonic Ratio (10:12:15)
I II III IV V VI VII

19. 2. Just Tuning Method 19
(a) The three major chords cover all 7 white keys
The “I” chord is CEG
The “IV” chord is FAC
The “V” chord is GBD
Tune these chords in (4:5:6)
Tune CEG: C=1, E=5/4, G=6/4=3/2
Tune FAC: C=2, so A=2(5/6)=5/3, and F=2(4/6)=4/3
Tune GBD: G=3/2 so B=(3/2)(5/4)=15/8 and D=(3/2)(3/2)=9/4
(c) Black Keys: could use other chords to get black keys
Diminish the CEG chord to minor triad CEbG to get Eb=6/5.
Diminish the FAC chord to minor triad FAbC to get Ab=8/5.
Diminish the GBD chord to minor triad GBbD to get Bb=9/5.
Augment the ACE chord to major triad AC#E to get C#=25/12.
Augment the DFA chord to major triad DF#A to get F#=45/32.

20. 20
3. Analysis of Just Scale
a. The Just Diatonic Scale
Ratio:

21. 3.b. Analysis of Just Tuning intervals21
(i) Two types of whole tones:
CD, FG, AB are /8
DE, GA are smaller 10/9
(ii) Different Semitones (too many!)
EF and BC are 16/15
Bb to C is 15/14
if make CC# 16/15
Then C#D is different 135/128
If make DbD to be 16/15, then C# is no longer same as Db (enharmonic)

22. 3.c. Inconsistent Intervals22
EGB and ACE come out as perfect (just) 10:12:15 minor triads
DFA is not however a just minor triad
Some of the fifths are not perfect, n.b. DA is off.
So, it really doesn’t work globally. However, vocal groups often really use Just tuning.

23. 3.b. Analysis of Just Tuning intervals23
Two types of whole tones:
CD, FG, AB are 9/8
but DE, GA are 10/9
One type of semitone for white keys
EF and BC are 16/15 (if stick to white keys)
Hence if CC# is 16/15, C#D is different 135/128
Make DbD to be 16/15, then C# is no longer same as Db (enharmonic)
This all disagrees with above method of tuning black keys. Problematic!
(c) Imperfect Intervals (just tuning does not really work)
EGB and ACE come out as perfect (just) 10:12:15 minor triads
DFA is not however a just minor triad
Some of the fifths are not perfect, n.b. DA is off.

24. C. Equal Temperament 24 1. Geometric Progression of Frequency
(a) Frequency Intervals increase with octave
A3 to A4 is 220 Hz (but A4/A3 is 2:1)
A4 to A5 is 440 Hz (but A5/A4 is same ratio 2:1)
Frequency intervals double in each octave, but ratios the same
(b) Geometric average gives the note halfway between two notes
Arithmetic average of A3 and A5 is: or C#5
Geometric average of A3 and A5 is: or A4
Clearly geometric average gives the “halfway” note we hear
(c) Divide Octave geometrically into 12 steps
Semitone Ratio: Octave comes out exactly right: g12= 2
Whole Step Ratio (2 semitones): g2= 1.122
A Major 5th (7 semitones): g7= is slightly less than Pythagorean 3/2, BUT a circle of fifths or fourths will exactly close.

25. Comparison: C to G (fifth)25
Comparison: C to G (fifth)
The scales really sound differently for some intervals.
Pythagorean
Just intonation
Mean-tone temperament
Equal intonation

26. 26
Comparison: C to E (3rd )
The scales really sound differently for some intervals.
Pythagorean
Just intonation
Mean-tone temperament
Equal intonation

27. Comparison: C# to F (3rd )27
Comparison: C# to F (3rd )
This is really terrible in some scales:
Pythagorean
Just intonation
Mean-tone temperament
Equal intonation

28. Comparison: C to A (6th ) Again Compare Pythagorean Just intonation28
Comparison: C to A (6th )
Again Compare
Pythagorean
Just intonation
Mean-tone temperament
Equal intonation

29. Comparison: C# to A# (6th )29
Comparison: C# to A# (6th )
This interval is quite bad in some
Pythagorean
Just intonation
Mean-tone temperament
Equal intonation

30. C.2 Arithmetic Progression of Cents30
C.2 Arithmetic Progression of Cents
(a) Definition of Cents
Each semitone is exactly 100 cents
All Octaves are exactly 1200 Cents (12 semitones)
Does not change with octave. C1E1 is 400 cents, so is C3E3.

31. C.2.b Mathematical Definition of Cents31
C.2.b Mathematical Definition of Cents
Difference in cents between two frequencies:
In terms of base 10 log:
Inverse Relation:

32. C.2.c Intervals in Cents 32 50 cents Quartertone
100 cents Semitone (m2)
200 M2
300 m3 (not quite a “just” minor third of 6/5 which would be 316 cents)
400 M3 (not quite a “just” third of 5/4 which would be 386 cents)
500 M4 (not really a “perfect” 4th which would be 498 cents)
600 Tritone (minor 5th or augmented 4th )
700 M5 (not really a “perfect” 5th which would be 702 cents)
800 m6
900 M6
1000 m7
1100 M7
1200 P8, octave

33. C.3 Comparison of Scales 33 (a) Pythagorean See table on next slide
Two different semitones (90 and 114) vs equal temperament’s 100
Fifths are 702, which are 2 cents longer than equal temperament, which is why the cycle of 5ths does not close. Hence one fifth is “sacrificed” to have a massive error (678 cents) and will sound absolutely terrible
Fourths: same problem, they are 2 cents short, hence will fail to close, so one interval will be too big (522 cents).
Major 3rds: two different types, some which are only 2 cents off from the perfect “just” harmonic ratio of 5:4 (386), the rest are far too big at 408 cents and will sound absolutely terrible, which is why Greeks didn’t use thirds

34. C.3.a Pythagorean 34 Octave Note Cents semitone Fourth Fifth Major 3rd
2400
498
702
384 2 B
2310 90
408 Bb
2196
114 A
2106 G#
2016
522 G
1902 F#
1812 F
1698 E
1608 Eb
1494
678 D
1404 C#
1314
1200 1
1110
996
906
816
612
294
204

35. C.3 Comparison of Scales 35 (b) Just Temperament
See table on next slide. Even for just the white keys, not all of the fifths are 702 cents (n.b. DA is terrible 680).
Many of the Major 3rds are a sweat 386 cents, but this is far short of the equal temperament 400 cents, so some of the 3rds will come out very big (e.g. 428) and will sound absolutely terrible. Since the whole point of the just tuning was to make the 3rds sound good, the system fails
Regardless, performers will often “bend” the notes so that thirds are closer to just tuning because they sound better. This is easy to do with voice, but a problem if you are going to use instruments.

36. C.3.b Just 36 Octave Note Ratio Cents Semitone M5th M4th M3rd
WholeTone 3 C
2400
702
498
386
182 2 B
2288
112
204 Bb
2218 70
520
428 A
2084
134
680 G#
2014
744
224 G
1902 F#
1790 F
1698 92 E
1586 Eb
1516
246 D
9/4
1404 C#
1270
456
1200 1
15/8
1088
1018
884
814
3/2
590
5/4
316
9/8

37. 37
C.3 Comparison of Scales

38. 38
Comparison of Scales

39. 39
How to tune Piano
Tuning a perfect 5th is easy, you eliminate the “beats”. But this would give you a 3:2 ratio (1.5 exactly), whereas the “equal temperament” interval is 27/12=1.4983
So you tune the G slightly flat so that you hear a beat frequency of 0.89 Hertz.

40. 40
References