Perfect harmony

Title: Perfect harmony: A mathematical analysis of four historical tunings
Author: Michael F. Page.
Abstract: In Western music, a musical interval defined by the frequency ratio of two notes is generally considered consonant when the ratio is composed of small integers. Perfect harmony or an "ideal just scale", which has no exact solution, would require the division of an octave into 12 notes, each of which would be used to create six other consonant intervals. The purpose of this study is to analyze four well-known historical tunings to evaluate how well each one approximates perfect harmony. The analysis consists of a general evaluation in which all consonant intervals are given equal weighting and a specific evaluation for three preludes from Bach’s Well-Tempered Clavier, for which intervals are weighted in proportion to the duration of their occurrence. The four tunings, 5-limit just intonation, quarter-comma meantone temperament, well temperament (Werckmeister III), and equal temperament, are evaluated by measures of centrality, dispersion, distance, and dissonance. When all keys and consonant intervals are equally weighted, equal temperament demonstrates the strongest performance across a variety of measures, although it is not always the best tuning. Given C as the starting note for each tuning, equal temperament and well temperament perform strongly for the three Well-Tempered Clavier preludes examined. © 2004 Acoustical Society of America.

The PDF file can be found from the website http://lit.gfax.ch//tunings/MathAnalysisOf4HistoricalTunings.pdf .

The table below are measures of central tendency and dispersion for absolute deviations from perfect harmony. Two measures of central tendency (the mean and median) and three measures of dispersion (the variance, range, and interquartile range) are shown. Tunings include those mentioned above as well as other selected tunings. Some tunings include links explaining the appropriate tunings. The table is ordered by mean (lowest to highest). 

Name of temperament

Mean

Median

Variance

Range

Interquartile range

Kirnberger I

12.41

19.55

149.00

41.06

21.51

Kirnberger II

12.56

10.79

128.77

41.06

21.51

Beat temperament

12.56

13.38

113.29

40.50

18.71

Kelletat

12.57

12.35

111.90

40.62

18.89

PBP temperament

12.57

12.85

112.73

40.50

18.99

Kellner

12.58

12.12

98.47

38.31

16.82

Kirnberger III unequal

12.60

12.78

108.22

41.06

16.31

Cent temperament

12.61

12.46

107.17

39.35

17.84

Billeter

12.62

10.86

101.98

37.37

17.66

Kirnberger III

12.64

11.14

105.54

41.06

14.94

1568edl (link)

12.65

13.53

85.01

32.20

13.48

Equal temperament

12.71

14.66

77.41

25.42

13.69

Mdevelde (link)

12.79

18.13

147.77

41.06

21.51

Werckmeister III

12.89

9.69

87.83

31.33

10.65

Pythagorean

12.98

19.55

153.26

43.01

21.51

Secor 5/23 TX (link)

12.99

7.84

121.88

38.32

15.14

Chalmers JI (link)

13.34

12.06

161.96

41.06

22.93

Duowell (link)

13.39

5.79

163.56

35.27

18.08

Bifrost (link)

13.47

10.38

141.48

41.06

15.15

Meantone (link)

13.49

7.48

197.71

37.34

7.93

Natural (pure)

13.55

0.00

285.15

41.06

21.51

Malcolm2 (link)

13.58

12.06

178.47

41.06

22.93

Pre-Archytas (link)

13.77

0.00

293.97

41.06

21.51

Meantone

13.77

5.41

277.79

46.52

8.56

Duodene (5-limit JI) (link)

13.79

0.00

306.02

41.06

26.39

John's 31EDO (link)

14.57

5.96

263.83

43.89

29.84

Cauldron (link)

15.14

7.54

258.92

48.77

16.39

Ratwolf (link)

15.24

6.12

345.64

50.14

12.39

Centaur (link)

15.37

7.71

325.70

48.77

27.26

12highschool1 (link)

15.38

7.71

325.97

48.77

27.26

Carlos harmonic Mean = 27.78 // Median = 27.05 // Variance = 437.10 // Range = 67.90 // Interquartile range = 36.34 // Weighted means = {12.04, 36.61, 25.09}
Grail Mean = 27.89 // Median = 11.00 // Variance = 1097.70 // Range = 103.03 // Interquartile range = 27.55 // Weighted means = {21.45, 23.12, 56.72}
Wilsonistic Mean = 29.53 // Median = 31.19 // Variance = 389.77 // Range = 70.10 // Interquartile range = 38.91 // Weighted means = {22.01, 32.18, 26.62}
Bicycle Mean = 31.61 // Median = 29.23 // Variance = 772.38 // Range = 106.81 // Interquartile range = 48.77 // Weighted means = {10.84, 32.06, 30.07}
Genovese 12 Mean = 33.52 // Median = 31.19 // Variance = 735.81 // Range = 97.36 // Interquartile range = 48.77 // Weighted means = {11.41, 36.00, 38.23}
Portsmouth Mean = 34.65 // Median = 35.05 // Variance = 534.63 // Range = 87.68 // Interquartile range = 23.93 // Weighted means = {35.62, 24.06, 38.92}
Omaha temperament Mean = 35.39 // Median = 34.89 // Variance = 1034.61 // Range = 113.44 // Interquartile range = 40.69 // Weighted means = {22.12, 35.57, 37.54}
Stelhex Mean = 37.24 // Median = 27.26 // Variance = 1640.90 // Range = 133.63 // Interquartile range = 64.89 // Weighted means = {36.44, 20.25, 52.28}
Omaha Mean = 37.45 // Median = 38.91 // Variance = 1131.13 // Range = 118.87 // Interquartile range = 42.33 // Weighted means = {24.00, 37.57, 40.21}
No fives Mean = 38.45 // Median = 48.77 // Variance = 634.43 // Range = 105.25 // Interquartile range = 27.26 // Weighted means = {32.98, 32.84, 44.56}
Serafini 11 Mean = 40.91 // Median = 48.77 // Variance = 848.22 // Range = 94.33 // Interquartile range = 37.14 // Weighted means = {18.16, 40.47, 38.85}
Blues JI Mean = 41.09 // Median = 48.77 // Variance = 1393.63 // Range = 140.95 // Interquartile range = 70.08 // Weighted means = {39.64, 35.42, 45.06}
Öljare Mean = 41.67 // Median = 48.77 // Variance = 1459.07 // Range = 155.14 // Interquartile range = 64.89 // Weighted means = {55.71, 49.98, 30.58}
Glumma Mean = 41.99 // Median = 42.23 // Variance = 1277.86 // Range = 133.63 // Interquartile range = 70.67 // Weighted means = {18.97, 63.57, 20.86}
Stelhex 2 Mean = 42.13 // Median = 21.51 // Variance = 2128.45 // Range = 182.40 // Interquartile range = 70.67 // Weighted means = {15.97, 59.83, 16.10}
Otones 12/24 Mean = 46.66 // Median = 43.83 // Variance = 872.94 // Range = 111.73 // Interquartile range = 47.01 // Weighted means = {56.99, 44.05, 31.07}
Locomotive Mean = 46.02 // Median = 31.77 // Variance = 999.53 // Range = 99.67 // Interquartile range = 33.06 // Weighted means = {38.32, 26.80, 56.49}
Superpyth Mean = 46.79 // Median = 48.97 // Variance = 836.20 // Range = 115.02 // Interquartile range = 48.97 // Weighted means = {38.01, 67.74, 40.92}
Sevish Mean = 47.29 // Median = 31.77 // Variance = 1854.55 // Range = 165.00 // Interquartile range = 45.56 // Weighted means = {26.15, 42.40, 39.65}
Terrain Mean = 47.88 // Median = 49.49 // Variance = 769.87 // Range = 84.79 // Interquartile range = 49.17 // Weighted means = {51.61, 48.89, 49.70}
Bihexany Mean = 50.19 // Median = 48.77 // Variance = 857.55 // Range = 102.04 // Interquartile range = 41.37 // Weighted means = {52.16, 49.38, 41.41}
Stelhex 5 Mean = 50.35 // Median = 48.77 // Variance = 1325.72 // Range = 176.65 // Interquartile range = 56.66 // Weighted means = {59.63, 49.29,37.20}
Harrison Revelation Mean = 58.39 // Median = 38.99 // Variance = 2794.97 // Range = 140.95 // Interquartile range = 103.31 // Weighted means = {24.85, 81.08, 34.57}
Rectoo Mean = 63.15 // Median = 70.67 // Variance = 2357.63 // Range = 182.40 // Interquartile range = 85.26 // Weighted means = {45.36, 51.01, 73.72}
Young/LaMonte Mean = 63.40 // Median = 76.03 // Variance = 2239.17 // Range = 140.95 // Interquartile range = 77.08 // Weighted means = {58.26, 83.29, 59.36}
Barton Mean = 67.11 // Median = 51.02 // Variance = 2441.20 // Range = 199.41 // Interquartile range = 79.96 // Weighted means = {68.31, 70.46, 61.89}
Breedball 3 Mean = 67.30 // Median = 48.77 // Variance = 2514.83 // Range = 196.20 // Interquartile range = 76.76 // Weighted means = {52.80, 61.83, 73.95}
Max2 Mean = 71.23 // Median = 66.85 // Variance = 4076.02 // Range = 182.40 // Interquartile range = 119.44 // Weighted means = {37.47, 68.25, 75.48}
Max1 Mean = 71.23 // Median = 66.85 // Variance = 4076.02 // Range = 182.40 // Interquartile range = 119.44 // Weighted means = {37.47, 69.82, 75.48}
Max5 Mean = 86.05 // Median = 84.47 // Variance = 3842.72 // Range = 231.17 // Interquartile range = 90.74 // Weighted means = {63.85, 118.88, 73.22}
Max6 Mean = 86.05 // Median = 84.47 // Variance = 3842.72 // Range = 231.17 // Interquartile range = 90.74 // Weighted means = {76.44, 69.89, 89.10}
Sonbirkez sorted Mean = 111.68 // Median = 111.73 // Variance = 3871.00 // Range = 226.86 // Interquartile range = 74.45 // Weighted means = {101.46, 101.63, 119.04}
Max3 Mean = 111.91 // Median = 98.10 // Variance = 3875.60 // Range = 231.17 // Interquartile range = 78.68 // Weighted means = {81.23, 129.17, 117.11}
Max4 Mean = 111.91 // Median = 98.10 // Variance = 3875.60 // Range = 231.17 // Interquartile range = 78.68 // Weighted means = {92.25, 102.53, 115.58}