List of pitch intervals

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Comparison between equal-tempered (red) and Pythagorean (blue) intervals showing the relationship between frequency ratio and the intervals' values, in cents. Note that one octave equals 1200 cents.

Below is a list of musical intervals. Some terminology used in the list:

  • In music, the prime limit (henceforth referred to simply as the limit) is a number measuring the harmony of an interval. The lower the number, the more consonant the interval is considered to be. It is defined as the largest prime number occurring in the factorizations of the numerator and denominator of the frequency ratio. The limit of the just perfect fourth (4 : 3) is 3, but the just minor tone (10 : 9) has a limit of 5, because 9 can be factorized into 3·3, and 10 into 2·5. There exists another type of limit, the odd limit, which differs slightly from the prime limit, but is not used here.
  • Equal-tempered refers to 12-tone equal temperament with intervals corresponding to 100 cent multiples (e.g., 100, 200, 300, etc.).
  • Pythagorean means 3-limit just intonation—a ratio of numbers with prime factors no higher than three.
  • Just means 5-limit just intonation—a ratio of numbers with prime factors no higher than five.
  • Similarly, septimal, undecimal, tridecimal, and septendecimal mean, respectively, 7, 11, 13, and 17-limit just intonation.
  • By definition every tone in a 3-limit unit can also be part of a 5-limit tuning and so on. By sorting the limit column all tones of that limit can be brought together (tip: sort backwards by clicking the button twice).
  • Meantone refers to meantone temperament, the most common of which is quarter-comma meantone. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in a meantone, each fifth is narrowed by the same small amount (a portion of the syntonic comma such as 1/4).
  • Since the table is sortable, you can also sort the table by frequency ratio, by cents or alphabetically.

List[edit]


Code Legend
E 12-tone equal temperament.
Q 24-tone equal temperament, or Arab tone system.
2 2-limit tones (only fundamental and octaves).
2 3 3-limit just intonation, or Pythagorean.
2 3 5 5-limit (not 3-limit) just intonation, or just.
2 3 5 7 7-limit (not 5-limit) just intonation, or septimal.
2 3 5 7 11 11-limit (not 7-limit) just intonation, or undecimal.
2 3 5 7 11 13 13-limit (not 11-limit) just intonation, or tridecimal.
2 3 5 7 11 13 17 17-limit (not 13-limit) just intonation, or septendecimal.
2 3 5 7 11 13 17 19 19-limit (not 17-limit) just intonation, or novendecimal.
M Meantone temperament.
U A unit of measurement.
S Superparticular ratio.
List of musical intervals
Cents Note (from C) Freq. Ratio Factors Interval Name E Q 2 3 5 7 11 13 17 19 M U S
0.00
C[1] 1 : 1 1 : 1 About this sound playUnison[2] or monophony,[3] perfect prime[2] E Q 2 3 5 7 11 13 17 19 M
0.40
4375 : 4374 54·7 : 2·37 About this sound playRagisma[2][4] 7 11 13 17 19 S
0.72
E7777triple flat+ 2401 : 2400 74 : 25·3·52 About this sound playBreedsma[2][4] 7 11 13 17 19 S
1.00
21/1200 : 1 About this sound playCent U
1.20
21/1000 : 1 About this sound playMillioctave U
1.95
B++ 32805 : 32768 38·5 : 215 About this sound playSchisma[2] 5 7 11 13 17 19
3.99
101/1000 : 1 About this sound playSavart or eptaméride U
7.71
B7 upside-down 225 : 224 32·52 : 25·7 About this sound playSeptimal kleisma,[2][4] marvel comma 7 11 13 17 19 S
8.11
Bdouble sharp- 15625 : 15552 56 : 26·35 About this sound playKleisma or semicomma majeur[2][4] 5 7 11 13 17 19
10.06
Adouble sharpdouble sharp++ 2109375 : 2097152 33·57 : 221 About this sound playSemicomma,[2][4] Fokker's comma[2] 5 7 11 13 17 19
12.50
21/96 : 1 About this sound playSixteenth-tone U
13.07
1728:1715 26·33 : 5·73 About this sound playOrwell comma[2][5] 7 11 13 17 19
13.79
Ddouble flat7 upside-down 126 : 125 2·32·7 : 53 About this sound playSmall septimal semicomma,[4] small septimal comma,[2] starling comma 7 11 13 17 19 S
14.37
121 : 120 112 : 23·3·5 About this sound playUndecimal seconds comma[2] 11 13 17 19 S
16.67
21/72 : 1 About this sound play1 step in 72 equal temperament U
19.55
Ddouble flat--[1] 2048 : 2025 211 : 34·52 About this sound playDiaschisma,[2][4] minor comma 5 7 11 13 17 19
21.51
C+[1] 81 : 80 34 : 24·5 About this sound playSyntonic comma,[2][4] major comma, komma, chromatic diesis, or comma of Didymus[2][4][6][7] 5 7 11 13 17 19 S
22.64
21/53 : 1 About this sound playHoldrian comma, Holder's comma, 1 step in 53 equal temperament U
23.46
B+++ 531441 : 524288 312 : 219 About this sound playPythagorean comma,[2][4][6][7] ditonic comma[2][4] 3 5 7 11 13 17 19
25.00
21/48 : 1 About this sound playEighth-tone U
27.26
C7 upside-down- 64 : 63 26 : 32·7 About this sound playSeptimal comma,[2][4][7] Archytas' comma[2] 7 11 13 17 19 S
29.27
21/41 : 1 About this sound play1 step in 41 equal temperament U
31.19
D7 56 : 55 23·7 : 5·11 About this sound playPtolemy's enharmonic:[8] difference between (11 : 8) and (7 : 5) tritone 11 13 17 19 S
33.33
21/36 : 1 About this sound playSixth-tone U
34.98
B7 upside-down7 upside-down- 50 : 49 2·52 : 72 About this sound playSeptimal sixth-tone or jubilisma, Erlich's decatonic comma or tritonic diesis[2][4] 7 11 13 17 19 S
35.70
D77 49 : 48 72 : 24·3 About this sound playSeptimal diesis, slendro diesis or septimal 1/6-tone[2] 7 11 13 17 19 S
38.71
21/31 : 1 About this sound play1 step in 31 equal temperament U
40.00
21/30 : 1 About this sound playFifth-tone U
41.06
Ddouble flat- 128 : 125 27 : 53 About this sound playEnharmonic diesis or 5-limit limma, minor diesis or diminished second,[4] minor diesis or diesis[2] 5 7 11 13 17 19
48.77
C7 upside-down 36 : 35 22·32 : 5·7 About this sound playSeptimal quarter tone, septimal diesis,[2][4] septimal comma[1] 7 11 13 17 19 S
50.00
Chalf sharp/Dthree quarter flat 21/24 : 1 About this sound playEqual-tempered quarter tone Q U
53.27
C 33 : 32 About this sound playThirty-third harmonic, undecimal comma, undecimal quarter-tone 11 13 17 19 S
62.96
C7- 28 : 27 About this sound playSeptimal minor second, small minor second 7 U
63.81
(3 : 2)1/11 : 1 About this sound playBeta scale step U
65.34
C13 upside down+ 27 : 26 33 : 2·13 About this sound playChromatic diesis,[9] tridecimal comma[2] 13 17 19 U S
66.67
21/18 : 1 About this sound playThird-tone U
70.67
C[1] 25 : 24 52 : 23·3 About this sound playJust chromatic semitone or minor chroma,[2] lesser chromatic semitone, small (just) semitone[7] or minor second,[3] minor chromatic semitone,[10] or minor semitone 5 7 11 13 17 19 S
78.00
(3 : 2)1/9 : 1 About this sound playAlpha scale step U
84.47
D7 21 : 20 3·7 : 22·5 About this sound playSeptimal chromatic semitone, minor semitone[2] 7 11 13 17 19 S
90.22
D--[1] 256 : 243 28 : 35 About this sound playPythagorean minor second or limma,[2][4][7] Pythagorean diatonic semitone, Low Semitone[11] 3 5 7 11 13 17 19
92.18
C+[1] 135 : 128 33·5 : 27 About this sound playGreater chromatic semitone, chromatic semitone, semitone medius, major chroma or major limma,[2] small limma,[7] major chromatic semitone[10] 5 7 11 13 17 19
98.95
D17 upside down 18 : 17 2·32 : 17 About this sound playJust minor semitone, Arabic lute index finger[2] 17 19 S
100.00
C/D 21/12 : 1 About this sound playEqual-tempered minor second or semitone E Q U
104.96
C17[1] 17 : 16 17 : 24 About this sound playMinor diatonic semitone, just major semitone, overtone semitone, 17th harmonic[2] 17 19 S
111.73
D-[1] 16 : 15 24 : 3·5 About this sound playJust diatonic semitone, large just semitone or major second,[3] major semitone, limma, minor diatonic semitone,[2] diatonic second[12] semitone,[11] diatonic semitone[7] 5 7 11 13 17 19 S
113.69
C++ 2187 : 2048 37 : 211 About this sound playapotome[2][7] or Pythagorean major semitone,[4] Pythagorean augmented prime, Pythagorean chromatic semitone, or Pythagorean apotome 3 5 7 11 13 17 19
116.72
181/19 : 51/19 About this sound playSecor U
119.44
C7 upside-down 15 : 14 3·5 : 2·7 About this sound playSeptimal diatonic semitone, major diatonic semitone[2] 7 11 13 17 19 S
133.24
D 27 : 25 33 : 52 About this sound playMinor second, semitone Maximus, large limma or Bohlen-Pierce small semitone,[2] high semitone[11] 5 7 11 13 17 19
150.00
Cthree quarter sharp/Dhalf flat 23/24 : 1 About this sound playEqual-tempered neutral second Q
150.64
D↓[1] 12 : 11 22·3 : 11 About this sound play3/4-tone or Undecimal neutral second,[2] trumpet three-quarter tone[7] 11 13 17 19 S
165.00
D↑-[1] 11 : 10 11 : 2·5 About this sound playGreater undecimal neutral second, 4/5-tone or Ptolemy's second[2] 11 13 17 19 S
171.43
21/7 : 1 About this sound play1 step in 7 equal temperament U
180.45
Edouble flat--- 65536 : 59049 216 : 310 About this sound playPythagorean diminished third,[2][4] Pythagorean minor tone 3 5 7 11 13 17 19
182.40
D-[1] 10 : 9 2·5 : 32 About this sound playSmall just whole tone or major second,[3] minor whole tone,[2] lesser whole tone,[12] minor tone,[11] minor second[7] 5 7 11 13 17 19 S
200.00
D 22/12 : 1 About this sound playEqual-tempered major second E Q
203.91
D[1] 9 : 8 32 : 23 About this sound playPythagorean major second, Large just whole tone or major second[7] (sesquioctavan),[3] tonus, major whole tone,[2] greater whole tone,[12] major tone[11] 3 5 7 11 13 17 19 S
223.46
Edouble flat-[1] 256 : 225 28 : 32·52 About this sound playJust diminished third[12] 5 7 11 13 17 19
231.17
D7 upside-down-[1] 8 : 7 23 : 7 About this sound playSeptimal major second,[3] septimal whole tone[2] 7 11 13 17 19 S
240.00
21/5 : 1 About this sound play1 step in 5 equal temperament U
266.87
E7[1] 7 : 6 7 : 2·3 About this sound playSeptimal minor third[2][3][7] or Sub minor Third[11] 7 11 13 17 19 S
274.58
D[1] 75 : 64 3·52 : 26 About this sound playJust augmented second,[12] Augmented Tone,[11] augmented second[10] 5 7 11 13 17 19
294.13
E-[1] 32 : 27 25 : 33 About this sound playPythagorean minor third[2][4][11][12] or semiditone 3 5 7 11 13 17 19
297.51
E19[1] 19 : 16 19 : 24 About this sound play19th harmonic,[2] 19-limit minor third 19
300.00
D/E 23/12 : 1 About this sound playEqual-tempered minor third E Q
310.26
6:5÷(81:80)1/4 About this sound playQuarter-comma meantone minor third M
311.98
(3 : 2)4/9 : 1 About this sound playAlpha scale minor third
315.64
E[1] 6 : 5 2·3 : 5 About this sound playJust minor third,[2][3][7][12] minor third[11] 5 7 11 13 17 19 S
317.60
D++ 19683 : 16384 39 : 214 About this sound playPythagorean augmented second[2][4] 3 5 7 11 13 17 19
342.86
22/7 : 1 About this sound play2 steps in 7 equal temperament
347.41
E↑-[1] 11 : 9 11 : 32 About this sound playUndecimal neutral third[2] 11 13 17 19
350.00
Dthree quarter sharp/Ehalf flat 27/24 : 1 About this sound playEqual-tempered neutral third Q
359.47
E13 upside down[1] 16 : 13 24 : 13 About this sound playTridecimal neutral third[2] 13 17 19
384.36
F-- 8192 : 6561 213 : 38 About this sound playPythagorean diminished fourth[2][4] 3 5 7 11 13 17 19
386.31
E[1] 5 : 4 5 : 22 About this sound playJust major third,[2][3][7][12] major third[11] 5 7 11 13 17 19 M S
400.00
E 24/12 : 1 About this sound playEqual-tempered major third E Q
407.82
E+[1] 81 : 64 34 : 26 About this sound playPythagorean major third,[2][4][11][12] ditone 3 5 7 11 13 17 19
417.51
F7+[1] 14 : 11 2·7 : 11 About this sound playUndecimal diminished fourth or major third[2] 11 13 17 19
427.37
F[1] 32 : 25 25 : 52 About this sound playJust diminished fourth,[12] diminished fourth[10] 5 7 11 13 17 19
435.08
E7 upside-down[1] 9 : 7 32 : 7 About this sound playSeptimal major third or Bohlen-Pierce third,[2] Super major Third[11] 7 11 13 17 19
456.99
E[1] 125 : 96 53 : 25·3 About this sound playJust augmented third 5 7 11 13 17 19
470.78
F7+[1] 21 : 16 3·7 : 24 About this sound play About this sound playTwenty-first harmonic, narrow fourth[2] 7 11 13 17 19
480.00
22/5 : 1 About this sound play2 steps in 5 equal temperament
498.04
F[1] 4 : 3 22 : 3 About this sound playPerfect fourth,[2][12] Pythagorean perfect fourth, Just perfect fourth or diatessaron[3] 3 5 7 11 13 17 19 S
500.00
F 25/12 : 1 About this sound playEqual-tempered perfect fourth E Q
510.51
(3 : 2)8/11 : 1 About this sound playBeta scale perfect fourth
514.29
23/7 : 1 About this sound play3 steps in 7 equal temperament
519.55
F+[1] 27 : 20 33 : 22·5 About this sound play5-limit wolf fourth, acute fourth,[2] imperfect fourth[12] 5 7 11 13 17 19
521.51
E+++ 177147 : 131072 311 : 217 About this sound playPythagorean augmented third[2][4] (F+ (pitch)) 3 5 7 11 13 17 19
551.32
F↑[1] 11 : 8 11 : 23 About this sound playeleventh harmonic, lesser undecimal tritone, undecimal semi-augmented fourth[2] 11 13 17 19
568.72
F[1] 25 : 18 52 : 2·32 About this sound playJust augmented fourth[2] 5 7 11 13 17 19
582.51
G7[1] 7 : 5 About this sound playLesser septimal tritone, septimal tritone[2][3] Huygens' tritone or Bohlen-Pierce fourth,[2] septimal fifth,[7] septimal diminished fifth[13] 7 11 13 17 19
588.27
G-- 1024 : 729 210 : 36 About this sound playPythagorean diminished fifth[2][4] 3 5 7 11 13 17 19
590.22
F+[1] 45 : 32 32·5 : 25 About this sound playJust augmented fourth, just tritone,[3][7] tritone,[4] diatonic tritone,[2] 'augmented' or 'false' fourth[12] 5 7 11 13 17 19
600.00
F/G 26/12 : 1 About this sound playEqual-tempered tritone E Q
609.78
G-[1] 64 : 45 26 : 32·5 About this sound playJust tritone,[3] 2nd tritone,[4] 'false' fifth,[12] diminished fifth[10] 5 7 11 13 17 19
611.73
F#++ 729 : 512 36 : 29 About this sound playPythagorean tritone,[2][4] Pythagorean augmented fourth 3 5 7 11 13 17 19
617.49
F7 upside-down[1] 10 : 7 2·5 : 7 About this sound playGreater septimal tritone, septimal tritone,[3] Euler's tritone[2] 7 11 13 17 19
631.28
G[1] 36 : 25 22·32 : 52 About this sound playJust diminished fifth 5 7 11 13 17 19
648.68
G↓[1] 16 : 11 24 : 11 About this sound playInversion of eleventh harmonic, undecimal semi-diminished fifth[2] 11 13 17 19
678.49
Adouble flat--- 262144 : 177147 218 : 311 About this sound playPythagorean diminished sixth[2][4] 3 5 7 11 13 17 19
680.45
G- 40 : 27 23·5 : 33 About this sound play5-limit wolf fifth or diminished sixth, grave fifth,[2][4][7] imperfect fifth,[12] 5 7 11 13 17 19
696.58
3:2÷(81:80)1/4 About this sound playQuarter-comma meantone perfect fifth M
700.00
G 27/12 : 1 About this sound playEqual-tempered perfect fifth E Q
701.96
G[1] 3 : 2 3 : 2 About this sound playPerfect fifth,[2][12] Pythagorean perfect fifth, Just perfect fifth or diapente,[3] fifth,[11] Just fifth[7] 3 5 7 11 13 17 19 S
764.92
A7[1] 14 : 9 2·7 : 32 About this sound playSeptimal minor sixth[2] 7 11 13 17 19
772.63
G 25 : 16 52 : 24 About this sound playJust augmented fifth[12] 5 7 11 13 17 19
782.49
G↑-[1] 11 : 7 11 : 7 About this sound playUndecimal minor sixth, undecimal augmented fifth[2] 11 13 17 19
792.18
A-[1] 128 : 81 27 : 34 About this sound playPythagorean minor sixth[2][4] 3 5 7 11 13 17 19
800.00
G/A 28/12 : 1 About this sound playEqual-tempered minor sixth E Q
813.69
A[1] 8 : 5 23 : 5 About this sound playJust minor sixth[2][3][7][12] 5 7 11 13 17 19
815.64
G++ 6561 : 4096 38 : 212 About this sound playPythagorean augmented fifth[2][4] 3 5 7 11 13 17 19
833.09
233 : 144 233 : 24·32 About this sound playGolden ratio
840.53
A13[1] 13 : 8 13 : 23 About this sound playTridecimal neutral sixth,[2] overtone sixth, thirteenth harmonic 13 17 19
850.00
Gthree quarter sharp/Ahalf flat 217/24 : 1 About this sound playEqual-tempered neutral sixth Q
852.59
A↓[1] 18 : 11 2·32 : 11 About this sound playUndecimal neutral sixth,[2] Zalzal's neutral sixth 11 13 17 19
857.14
25/7 : 1 About this sound play5 steps in 7 equal temperament
882.40
Bdouble flat--- 32768 : 19683 215 : 39 About this sound playPythagorean diminished seventh[2][4] 3 5 7 11 13 17 19
884.36
A[1] 5 : 3 5 : 3 About this sound playJust major sixth,[2][3][7][12] Bohlen-Pierce sixth[2] 5 7 11 13 17 19
900.00
A 29/12 : 1 About this sound playEqual-tempered major sixth E Q
905.87
A+[1] 27 : 16 33 : 24 About this sound playPythagorean major sixth[2][7][12] 3 5 7 11 13 17 19
925.42
Bdouble flat-[1] 128 : 75 27 : 3·52 About this sound playJust diminished seventh,[12] diminished seventh[10] 5 7 11 13 17 19
933.13
A7 upside-down[1] 12 : 7 22·3 : 7 About this sound playSeptimal major sixth[2][3] 7 11 13 17 19
955.03
A[1] 125 : 72 53 : 23·32 About this sound playJust augmented sixth 5 7 11 13 17 19
957.21
(3 : 2)15/11 : 1 About this sound play15 steps in Beta scale
960.00
24/5 : 1 About this sound play4 steps in 5 equal temperament
968.83
B7[1] 7 : 4 7 : 22 About this sound playSeptimal minor seventh,[3][7] harmonic seventh[2][7] 7 11 13 17 19
976.54
A+[1] 225 : 128 32·52 : 27 About this sound playJust augmented sixth[12] 5 7 11 13 17 19
996.09
B-[1] 16 : 9 24 : 32 About this sound playPythagorean minor seventh,[2] Small just minor seventh,[3] lesser minor seventh,[12] just minor seventh[7] 3 5 7 11 13 17 19
1000.00
A/B 210/12 : 1 About this sound playEqual-tempered minor seventh E Q
1017.60
B[1] 9 : 5 32 : 5 About this sound playGreater just minor seventh,[12] large just minor seventh,[3] Bohlen-Pierce seventh[2] 5 7 11 13 17 19
1019.55
A+++ 59049 : 32768 310 : 215 About this sound playPythagorean augmented sixth[2][4] 3 5 7 11 13 17 19
1028.57
26/7 : 1 About this sound play6 steps in 7 equal temperament
1035.00
B↓[1] 20 : 11 22·5 : 11 About this sound playLesser undecimal neutral seventh, large minor seventh[2] 11 13 17 19
1049.36
B↑-[1] 11 : 6 11 : 2·3 About this sound play21/4-tone or Undecimal neutral seventh[2] 11 13 17 19
1050.00
Athree quarter sharp/Bhalf flat 221/24 : 1 About this sound playEqual-tempered neutral seventh Q
1086.31
C-- 4096 : 2187 212 : 37 About this sound playPythagorean diminished octave[2][4] 3 5 7 11 13 17 19
1088.27
B[1] 15 : 8 3·5 : 23 About this sound playJust major seventh,[2][7][12] small just major seventh,[3] 5 7 11 13 17 19
1100.00
B 211/12 : 1 About this sound playEqual-tempered major seventh E Q
1109.78
B+[1] 243 : 128 35 : 27 About this sound playPythagorean major seventh[2][4][7] 3 5 7 11 13 17 19
1129.33
C[1] 48 : 25 24·3 : 52 About this sound playClassic diminished octave,[2][4] large just major seventh[3] 5 7 11 13 17 19
1158.94
B[1] 125 : 64 53 : 26 About this sound playJust augmented seventh, 125th harmonic 5 7 11 13 17 19
1200.00
C' 2 : 1 2 : 1 About this sound playOctave[2][7] or diapason[3] E Q 2 3 5 7 11 13 17 19 M U S
1223.46
B+++ 531441 : 262144 312 : 218 About this sound playPythagorean augmented seventh[2][4] 3 5 7 11 13 17 19
1901.96
G' 3 : 1 3 : 1 About this sound playTritave or just perfect twelfth 3 5 7 11 13 17 19 U
2400.00
C" 4 : 1 22 : 1 About this sound playFifteenth or two octaves E Q 2 3 5 7 11 13 17 19 M
3986.31
E 10 : 1 5·2 : 4÷4 About this sound playDecade, compound just major third 5 7 11 13 17 19 M U

See also[edit]

References[edit]

  1. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah ai aj ak al am an ao ap aq ar as at au av aw ax ay az ba bb bc bd be bf bg bh bi Fonville, John. 1991. "Ben Johnston's Extended Just Intonation: A Guide for Interpreters". Perspectives of New Music 29, no. 2 (Summer): 106–37.
  2. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah ai aj ak al am an ao ap aq ar as at au av aw ax ay az ba bb bc bd be bf bg bh bi bj bk bl bm bn bo bp bq br bs bt bu bv bw bx by bz ca cb cc cd ce cf cg ch ci cj ck cl cm cn "List of intervals", Huygens-Fokker Foundation. The Foundation uses "classic" to indicate "just" or leaves off any adjective, as in "major sixth".
  3. ^ a b c d e f g h i j k l m n o p q r s t u v w x Partch, Harry (1979). Genesis of a Music, p.68-69. ISBN 978-0-306-80106-8.
  4. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah ai aj ak Haluška, Ján (2003). The Mathematical Theory of Tone Systems, p.xxv-xxix. ISBN 978-0-8247-4714-5.
  5. ^ "Orwell Temperaments", Xenharmony.org.
  6. ^ a b Partch (1979), p.70.
  7. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa Alexander John Ellis (1885). On the musical scales of various nations, p.488. s.n.
  8. ^ "Anatomy of an Octave", KyleGann.com.
  9. ^ William Smythe Babcock Mathews (1895). Pronouncing dictionary and condensed encyclopedia of musical terms, p.13. ISBN 1-112-44188-3.
  10. ^ a b c d e f Anger, Joseph Humfrey (1912). A treatise on harmony, with exercises, Volume 3, p.xiv-xv. W. Tyrrell.
  11. ^ a b c d e f g h i j k l m Hermann Ludwig F. von Helmholtz (Alexander John Ellis, trans.) (1875). On the sensations of tone as a physiological basis for the theory of music, p.644. ISBN .
  12. ^ a b c d e f g h i j k l m n o p q r s t u v w x y Paul, Oscar (1885). A manual of harmony for use in music-schools and seminaries and for self-instruction, p.165. Theodore Baker, trans. G. Schirmer. Paul uses "natural" for "just".
  13. ^ Sabat, Marc and von Schweinitz, Wolfgang (2004). "[www.newmusicbox.org/assets/72/HelmholtzEllisLegend.pdf The Extended Helmholtz-Ellis JI Pitch Notation]" [PDF], NewMusicBox.org. Accessed: 04:12, 15 March 2014 (UTC).

External links[edit]