List of pitch intervals

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File:Meantone comparison.png
Comparison between tunings: Pythagorean, equal-tempered, 1/4-comma meantone, and others. For each, the common origin is arbitrarily chosen as C. The degrees are arranged in the order or the cycle of fifths; as in each of these tunings except just intonation all fifths are of the same size, the tunings appear as straight lines, the slope indicating the relative tempering with respect to Pythagorean, which has pure fifths (3:2, 702 cents). The Pythagorean Ab (at the left) is at 792 cents, G# (at the right) at 816 cents; the difference is the Pythagorean comma. Equal temperament by definition is such that Ab and G# are at the same level. 1/4 comma meantone produces the "just" major third (5:4, 386 cents, a syntonic comma lower than the Pythagorean one of 408 cents). 1/3 comma meantone produces the "just" minor third (6:5, 316 cents, a syntonic comma higher than the Pythagorean one of 294 cents). In both these meantone temperaments, the enharmony, here the difference between Ab and G#, is much larger than in Pythagorean, and with the flat degree higher than the sharp one.

Below is a list of intervals exprimable in terms of a prime limit (see Terminology), completed by a choice of intervals in various equal subdivisions of the octave or of other intervals.

For commonly encountered harmonic or melodic intervals between pairs of notes in contemporary Western music theory, without consideration of the way in which they are tuned, see Interval (music) § Main intervals.

Terminology

  • The prime limit[1] henceforth referred to simply as the limit, is the largest prime number occurring in the factorizations of the numerator and denominator of the frequency ratio describing a rational interval. For instance, the limit of the just perfect fourth (4 : 3) is 3, but the just minor tone (10 : 9) has a limit of 5, because 10 can be factorized into 2·5 (and 9 in 3·3). There exists another type of limit, the odd limit, a concept used by Harry Partch (bigger of odd numbers obtained after dividing numerator and denominator by highest possible powers of 2), but it is not used here. The term "limit" was devised by Partch.[1]
  • By definition, every interval in a given limit can also be part of a limit of higher order. For instance, a 3-limit unit can also be part of a 5-limit tuning and so on. By sorting the limit columns in the table below, all intervals of a given limit can be brought together (sort backwards by clicking the button twice).
  • Pythagorean tuning means 3-limit intonation—a ratio of numbers with prime factors no higher than three.
  • Just intonation means 5-limit intonation—a ratio of numbers with prime factors no higher than five.
  • Septimal, undecimal, tridecimal, and septendecimal mean, respectively, 7, 11, 13, and 17-limit intonation.
  • Meantone refers to meantone temperament, where the whole tone is the mean of the major third. In general, a meantone is constructed in the same way as Pythagorean tuning, as a stack of fifths: the tone is reached after two fifths, the major third after four, so that as all fifths are the same, the tone is the mean of the third. In a meantone temperament, each fifth is narrowed ("tempered") by the same small amount. The most common of meantone temperaments is the quarter-comma meantone, in which each fifth is tempered by 1/4 of the syntonic comma, so that after four steps the major third (as C-G-D-A-E) is a full syntonic comma lower than the Pythagorean one. The extremes of the meantone systems encountered in historical practice are the Pythagorean tuning, where the whole tone corresponds to 9:8, i.e (3:2)2/2, the mean of the major third (3:2)4/4, and the fifth (3:2) is not tempered; and the 1/3-comma meantone, where the fifth is tempered to the extent that three ascending fifths produce a pure minor third.(See Meantone temperaments). The music program Logic Pro uses also 1/2-comma meantone temperament.
  • Equal-tempered refers to X-tone equal temperament with intervals corresponding to X divisions per octave.
  • Tempered intervals however cannot be expressed in terms of prime limits and, unless exceptions, are not found in the table below.
  • The table can also be sorted by frequency ratio, by cents, or alphabetically.

List

Column Legend
TET X-tone equal temperament (12-tet, etc.).
Limit 3-limit intonation, or Pythagorean.
5-limit "just" intonation, or just.
7-limit intonation, or septimal.
11-limit intonation, or undecimal.
13-limit intonation, or tridecimal.
17-limit intonation, or septendecimal.
19-limit intonation, or novendecimal.
Higher limits.
M Meantone temperament or tuning.
S Superparticular ratio (no separate color code).
List of musical intervals
Cents Note (from C) Freq. ratio Prime factors Interval name TET Limit M S
0.00
 C[2] 1 : 1 1 : 1 <phonos file="Unison on C.mid">play</phonos> Unison,[3] monophony,[4] perfect prime,[3] tonic,[5] or fundamental 1, 12 3 M
0.40
 C7♯- 4375 : 4374 54·7 : 2·37 Audio file "Ragisma on C.mid" not found Ragisma[3][6] 7 S
0.72
 E7777triple flat+ 2401 : 2400 74 : 25·3·52 Audio file "Breedsma on C.mid" not found Breedsma[3][6] 7 S
1.00
 21/1200 21/1200 <phonos file="Cent on C.mid">play</phonos> Cent 1200
1.20
 21/1000 21/1000 <phonos file="Millioctave on C.mid">play</phonos> Millioctave 1000
1.95
 B♯++ 32805 : 32768 38·5 : 215 Audio file "Schisma on C.mid" not found Schisma[3][5] 5
3.99
 101/1000 21/1000·51/1000 <phonos file="Savart on C.mid">play</phonos> Savart or eptaméride 301.03
7.71
 B7 upside-down♯ 225 : 224 32·52 : 25·7 Audio file "Septimal kleisma on C.mid" not found Septimal kleisma,[3][6] marvel comma 7 S
8.11
 Bdouble sharp- 15625 : 15552 56 : 26·35 <phonos file="Kleisma on C.mid">play</phonos> Kleisma or semicomma majeur[3][6] 5
10.06
 Adouble sharpdouble sharp++ 2109375 : 2097152 33·57 : 221 <phonos file="Semicomma on C.mid">play</phonos> Semicomma,[3][6] Fokker's comma[3] 5
11.98
 C29 145 : 144 5·29 : 24·32 Audio file "Johnston 29 on C.mid" not found Difference between 29:16 & 9:5 29 S
12.50
 21/96 21/96 Audio file "Sixteenth-tone on C.mid" not found Sixteenth tone 96
13.07
 B7 upside-down7 upside-down7 upside-down- 1728 : 1715 26·33 : 5·73 <phonos file="Orwell comma on C.mid">play</phonos> Orwell comma[3][7] 7
13.79
 Ddouble flat7 upside-down 126 : 125 2·32·7 : 53 <phonos file="Septimal semicomma on C.mid">play</phonos> Small septimal semicomma,[6] small septimal comma,[3] starling comma 7 S
14.37
 C♭↑↑- 121 : 120 112 : 23·3·5 Audio file "Undecimal neutral second comma on C.mid" not found Undecimal seconds comma[3] 11 S
16.67
 C↑[lower-alpha 1] 21/72 21/72 Audio file "1 step in 72-et on C.mid" not found 1 step in 72 equal temperament 72
18.13
 C19U 96 : 95 25·3 : 5·19 Audio file "Johnston U19 on C.mid" not found Difference between 19:16 & 6:5 19 S
19.55
 Ddouble flat--[2] 2048 : 2025 211 : 34·52 <phonos file="Diaschisma on C.mid">play</phonos> Diaschisma,[3][6] minor comma 5
21.51
 C+[2] 81 : 80 34 : 24·5 <phonos file="Syntonic comma on C.mid">play</phonos> Syntonic comma,[3][5][6] major comma, komma, chromatic diesis, or comma of Didymus[3][6][8][9] 5 S
22.64
 21/53 21/53 <phonos file="Holdrian comma on C.mid">play</phonos> Holdrian comma, Holder's comma, 1 step in 53 equal temperament 53
23.46
 B♯+++ 531441 : 524288 312 : 219 <phonos file="Pythagorean comma on C.mid">play</phonos> Pythagorean comma,[3][5][6][8][9] ditonic comma[3][6] 3
25.00
 21/48 21/48 <phonos file="Eighth-tone on C.mid">play</phonos> Eighth tone 48
26.84
 C13 65 : 64 5·13 : 26 Audio file "Sixty-fifth harmonic on C.mid" not found Sixty-fifth harmonic,[5] 13th-partial chroma[3] 13 S
27.26
 C7 upside-down- 64 : 63 26 : 32·7 <phonos file="Septimal comma on C.mid">play</phonos> Septimal comma,[3][6][9] Archytas' comma[3] 7 S
29.27
 21/41 21/41 Audio file "1 step in 41-et on C.mid" not found 1 step in 41 equal temperament 41
31.19
 D7♭↓ 56 : 55 23·7 : 5·11 Audio file "Ptolemy's enharmonic on C.mid" not found Ptolemy's enharmonic:[5] difference between (11 : 8) and (7 : 5) tritone 11 S
33.33
 CHalf up arrow.png/D♭Half down arrow.pngHalf down arrow.png[lower-alpha 1] 21/36 21/36 Audio file "Sixth-tone on C.mid" not found Sixth tone 36, 72
34.28
 C17 51 : 50 3·17 : 2·52 Audio file "Johnston 17 on C.mid" not found Difference between 17:16 & 25:24 17 S
34.98
 B7 upside-down7 upside-down♯- 50 : 49 2·52 : 72 Audio file "Septimal sixth-tone on C.mid" not found Septimal sixth tone or jubilisma, Erlich's decatonic comma or tritonic diesis[3][6] 7 S
35.70
 D77♭ 49 : 48 72 : 24·3 <phonos file="Septimal diesis on C.mid">play</phonos> Septimal diesis, slendro diesis or septimal 1/6-tone[3] 7 S
38.05
 C23 46 : 45 2·23 : 32·5 Audio file "Johnston 23 on C.mid" not found Inferior quarter tone,[5] difference between 23:16 & 45:32 23 S
38.71
 21/31 21/31 Audio file "1 step in 31-et on C.mid" not found 1 step in 31 equal temperament 31
38.91
 C↓♯+ 45 : 44 32·5 : 4·11 Audio file "Undecimal diesis on C.mid" not found Undecimal diesis or undecimal fifth tone 11 S
40.00
 21/30 21/30 Audio file "Fifth-tone on C.mid" not found Fifth tone 30
41.06
 Ddouble flat- 128 : 125 27 : 53 Audio file "5-limit limma on C.mid" not found Enharmonic diesis or 5-limit limma, minor diesis,[6] diminished second,[5][6] minor diesis or diesis[3] 5
41.72
 D41U7♭ 42 : 41 2·3·7 : 41 Audio file "Lesser 41-limit fifth tone on C.mid" not found Lesser 41-limit fifth tone 41 S
42.75
 C41 41 : 40 41 : 23·5 Audio file "Greater 41-limit fifth tone on C.mid" not found Greater 41-limit fifth tone 41 S
43.83
 C13 upside down♯ 40 : 39 23·5 : 3·13 Audio file "Tridecimal fifth tone on C.mid" not found Tridecimal fifth tone 13 S
44.97
 C19U13 39 : 38 3·13 : 2·19 Audio file "Novendecimal fifth tone on C.mid" not found Superior quarter-tone,[5] novendecimal fifth tone 19 S
46.17
 D37U19double flat- 38 : 37 2·19 : 37 Audio file "Lesser 37-limit quarter tone on C.mid" not found Lesser 37-limit quarter tone 37 S
47.43
 C37♯ 37 : 36 37 : 22·32 Audio file "Greater 37-limit quarter tone on C.mid" not found Greater 37-limit quarter tone 37 S
48.77
 C7 upside-down 36 : 35 22·32 : 5·7 <phonos file="Septimal quarter tone on C.mid">play</phonos> Septimal quarter tone, septimal diesis,[3][6] septimal comma,[2] superior quarter tone[5] 7 S
50.00
 Chalf sharp/Dthree quarter flat 21/24 21/24 <phonos file="Quarter tone on C.mid">play</phonos> Equal-tempered quarter tone 24
50.18
 D17 upside down7♭ 35 : 34 5·7 : 2·17 Audio file "Lesser septendecimal quarter tone on C.mid" not found ET quarter-tone approximation,[5] lesser 17-limit quarter tone 17 S
50.72
 B7 upside-down♯++ 59049 : 57344 310 : 213·7 Audio file "Harrison's comma on C.mid" not found Harrison's comma (9 P5s - 1 H7)[3] 7
51.68
 C17↓♯ 34 : 33 2·17 : 3·11 Audio file "Greater septendecimal quarter tone on C.mid" not found Greater 17-limit quarter tone 17 S
53.27
 C↑ 33 : 32 3·11 : 25 Audio file "Thirty-third harmonic on C.mid" not found Thirty-third harmonic,[5] undecimal comma, undecimal quarter tone 11 S
54.96
 D31U♭- 32 : 31 25 : 31 Audio file "Thirty-first subharmonic on C.mid" not found Inferior quarter-tone,[5] thirty-first subharmonic 31 S
56.77
 C31 31 : 30 31 : 2·3·5 Audio file "Johnston 31 on C.mid" not found Inferior quarter-tone,[5] difference between 31:16 & 15:8 31 S
58.69
 C29U♯ 30 : 29 2·3·5 : 29 Audio file "Lesser 29-limit quarter tone on C.mid" not found Lesser 29-limit quarter tone 29 S
60.75
 C297 upside-down 29 : 28 29 : 22·7 Audio file "Greater 29-limit quarter tone on C.mid" not found Greater 29-limit quarter tone 29 S
62.96
 C7♯- 28 : 27 22·7 : 33 <phonos file="Septimal minor second on C.mid">play</phonos> Septimal minor second, small minor second, inferior quarter tone[5] 7 S
63.81
 (3 : 2)1/11 31/11 : 21/11 <phonos file="Beta scale step on C.mid">play</phonos> Beta scale step 18.75
65.34
 C13 upside down♯+ 27 : 26 33 : 2·13 Audio file "Chromatic diesis on C.mid" not found Chromatic diesis,[10] tridecimal comma[3] 13 S
66.67
 Cx14px↑/C♯Half down arrow.png[lower-alpha 1] 21/18 21/18 Audio file "Third-tone on C.mid" not found Third tone 18, 36, 72
67.90
 D13double flat- 26 : 25 2·13 : 52 <phonos file="Tridecimal third tone on C.mid">play</phonos> Tridecimal third tone, third tone[5] 13 S
70.67
 C♯[2] 25 : 24 52 : 23·3 <phonos file="Just chromatic semitone on C.mid">play</phonos> Just chromatic semitone or minor chroma,[3] lesser chromatic semitone, small (just) semitone[9] or minor second,[4] minor chromatic semitone,[11] or minor semitone,[5] 2/7-comma meantone chromatic semitone 5 S
73.68
 D23U♭- 24 : 23 23·3 : 23 Audio file "Lesser 23-limit semitone on C.mid" not found Lesser 23-limit semitone 23 S
76.96
 C23↓♯+ 23 : 22 23 : 2·11 Audio file "Greater 23-limit semitone on C.mid" not found Greater 23-limit semitone 23 S
78.00
 (3 : 2)1/9 31/9 : 21/9 <phonos file="Alpha scale step on C.mid">play</phonos> Alpha scale step 15.39
79.31
 67 : 64 67 : 26 Audio file "Sixty-seventh harmonic on C.mid" not found Sixty-seventh harmonic[5] 67
80.54
 C↑7 upside-down- 22 : 21 2·11 : 3·7 Audio file "Undecimal two-fifth tone on C.mid" not found Hard semitone,[5] two-fifth tone small semitone 11 S
84.47
 D7♭ 21 : 20 3·7 : 22·5 <phonos file="Septimal chromatic semitone on C.mid">play</phonos> Septimal chromatic semitone, minor semitone[3] 7 S
88.80
 C19U♯ 20 : 19 22·5 : 19 Audio file "Novendecimal augmented unison on C.mid" not found Novendecimal augmented unison 19 S
90.22
 D♭--[2] 256 : 243 28 : 35 <phonos file="Pythagorean minor semitone on C.mid">play</phonos> Pythagorean minor second or limma,[3][6][9] Pythagorean diatonic semitone, Low Semitone[12] 3
92.18
 C♯+[2] 135 : 128 33·5 : 27 <phonos file="Greater chromatic semitone on C.mid">play</phonos> Greater chromatic semitone, chromatic semitone, semitone medius, major chroma or major limma,[3] small limma,[9] major chromatic semitone,[11] limma ascendant[5] 5
93.60
 D19♭- 19 : 18 19 : 2·9 Novendecimal minor secondAudio file "Novendecimal minor second on C.mid" not found 19 S
98.95
 D17 upside down♭ 18 : 17 2·32 : 17 <phonos file="Just minor semitone on C.mid">play</phonos> Just minor semitone, Arabic lute index finger[3] 17 S
100.00
 C♯/D♭ 21/12 21/12 <phonos file="Minor second on C.mid">play</phonos> Equal-tempered minor second or semitone 12 M
104.96
 C17♯[2] 17 : 16 17 : 24 <phonos file="Just major semitone on C.mid">play</phonos> Minor diatonic semitone, just major semitone, overtone semitone,[5] 17th harmonic,[3] limma[citation needed] 17 S
111.73
 D♭-[2] 16 : 15 24 : 3·5 <phonos file="Just diatonic semitone on C.mid">play</phonos> Just minor second,[13] just diatonic semitone, large just semitone or major second,[4] major semitone,[5] limma, minor diatonic semitone,[3] diatonic second[14] semitone,[12] diatonic semitone,[9] 1/6-comma meantone minor second 5 S
113.69
 C♯++ 2187 : 2048 37 : 211 <phonos file="Pythagorean apotome on C.mid">play</phonos> apotome[3][9] or Pythagorean major semitone,[6] Pythagorean augmented unison, Pythagorean chromatic semitone, or Pythagorean apotome 3
116.72
 (18 : 5)1/19 21/19·32/19 : 51/19 <phonos file="Secor on C.mid">play</phonos> Secor 10.28
119.44
 C7 upside-down♯ 15 : 14 3·5 : 2·7 <phonos file="Septimal diatonic semitone on C.mid">play</phonos> Septimal diatonic semitone, major diatonic semitone,[3] Cowell semitone[5] 7 S
128.30
 D13 upside down7 14 : 13 2·7 : 13 <phonos file="Lesser tridecimal two-third tone on C.mid">play</phonos> Lesser tridecimal 2/3-tone[15] 13 S
130.23
 C23♯+ 69 : 64 3·23 : 26 Audio file "Sixty-ninth harmonic on C.mid" not found Sixty-ninth harmonic[5] 23
133.24
 D♭ 27 : 25 33 : 52 <phonos file="Semitone Maximus on C.mid">play</phonos> Semitone maximus, minor second, large limma or Bohlen-Pierce small semitone,[3] high semitone,[12] alternate Renaissance half-step,[5] large limma, acute minor second[citation needed] 5
133.33
 C♯Half up arrow.png/D♭Half up arrow.png[lower-alpha 1] 21/9 22/18 Audio file "Two-third tone on C.mid" not found Two-third tone 9, 18, 36, 72
138.57
 D13♭- 13 : 12 13 : 22·3 <phonos file="Greater tridecimal two-third tone on C.mid">play</phonos> Greater tridecimal 2/3-tone,[15] Three-quarter tone[5] 13 S
150.00
 Cthree quarter sharp/Dhalf flat 23/24 21/8 <phonos file="Neutral second on C.mid">play</phonos> Equal-tempered neutral second 8, 24
150.64
 D↓[2] 12 : 11 22·3 : 11 <phonos file="Neutral second on C.mid">play</phonos> 3/4-tone or Undecimal neutral second,[3][5] trumpet three-quarter tone[9] 11 S
155.14
 D7 35 : 32 5·7 : 25 Audio file "Thirty-fifth harmonic on C.mid" not found Thirty-fifth harmonic[5] 7
160.90
 D-- 800 : 729 25·52 : 36 Audio file "Grave whole tone on C.mid" not found Grave whole tone,[3] neutral second, grave major second[citation needed] 5
165.00
 D↑♭-[2] 11 : 10 11 : 2·5 <phonos file="Greater undecimal neutral second on C.mid">play</phonos> Greater undecimal minor/major/neutral second, 4/5-tone or Ptolemy's second[3] 11 S
171.43
 21/7 21/7 <phonos file="1 step in 7-et on C.mid">play</phonos> 1 step in 7 equal temperament 7
179.70
 71 : 64 71 : 26 Audio file "Seventy-first harmonic on C.mid" not found Seventy-first harmonic[5] 71
180.45
 Edouble flat--- 65536 : 59049 216 : 310 <phonos file="Minor tone on C.mid">play</phonos> Pythagorean diminished third,[3][6] Pythagorean minor tone 3
182.40
 D-[2] 10 : 9 2·5 : 32 <phonos file="Minor tone on C.mid">play</phonos> Small just whole tone or major second,[4] minor whole tone,[3][5] lesser whole tone,[14] minor tone,[12] minor second,[9] half-comma meantone major second 5 S
200.00
 D 22/12 21/6 <phonos file="Major second on C.mid">play</phonos> Equal-tempered major second 6, 12 M
203.91
 D[2] 9 : 8 32 : 23 <phonos file="Major tone on C.mid">play</phonos> Pythagorean major second, Large just whole tone or major second[9] (sesquioctavan),[4] tonus, major whole tone,[3][5] greater whole tone,[14] major tone[12] 3 S
223.46
 Edouble flat-[2] 256 : 225 28 : 32·52 Audio file "Just diminished third on C.mid" not found Just diminished third[14] 5
227.79
 73 : 64 73 : 26 Audio file "Seventy-third harmonic on C.mid" not found Seventy-third harmonic[5] 73
231.17
 D7 upside-down-[2] 8 : 7 23 : 7 <phonos file="Septimal major second on C.mid">play</phonos> Septimal major second,[4] septimal whole tone[3][5] 7 S
240.00
 21/5 21/5 <phonos file="1 step in 5-et on C.mid">play</phonos> 1 step in 5 equal temperament 5
251.34
 D37♯ 37 : 32 37 : 25 Audio file "Thirty-seventh harmonic on C.mid" not found Thirty-seventh harmonic[5] 37
253.08
 D♯- 125 : 108 53 : 22·33 Audio file "Semi-augmented whole tone on C.mid" not found Semi-augmented whole tone,[3] semi-augmented second[citation needed] 5
266.87
 E7♭[2] 7 : 6 7 : 2·3 <phonos file="Septimal minor third on C.mid">play</phonos> Septimal minor third[3][4][9] or Sub minor third[12] 7 S
274.58
 D♯[2] 75 : 64 3·52 : 26 <phonos file="Just augmented second on C.mid">play</phonos> Just augmented second,[14] Augmented tone,[12] augmented second[5][11] 5
294.13
 E♭-[2] 32 : 27 25 : 33 <phonos file="Pythagorean minor third on C.mid">play</phonos> Pythagorean minor third[3][5][6][12][14] or semiditone 3
297.51
 E19♭[2] 19 : 16 19 : 24 Audio file "19th harmonic on C.mid" not found 19th harmonic,[3] 19-limit minor third, overtone minor third[5] 19
300.00
 D♯/E♭ 23/12 21/4 Audio file "Minor third on C.mid" not found Equal-tempered minor third 4, 12 M
310.26
 6:5÷(81:80)1/4 22 : 53/4 Audio file "Quarter-comma meantone minor third on C.mid" not found Quarter-comma meantone minor third M
311.98
 (3 : 2)4/9 34/9 : 24/9 Audio file "Alpha scale minor third on C.mid" not found Alpha scale minor third 3.85
315.64
 E♭[2] 6 : 5 2·3 : 5 Audio file "Just minor third on C.mid" not found Just minor third,[3][4][5][9][14] minor third,[12] 1/3-comma meantone minor third 5 M S
317.60
 D♯++ 19683 : 16384 39 : 214 Audio file "Pythagorean augmented second on C.mid" not found Pythagorean augmented second[3][6] 3
320.14
 E7♭↑ 77 : 64 7·11 : 26 Audio file "Seventy-seventh harmonic on C.mid" not found Seventy-seventh harmonic[5] 11
336.13
 D177 upside-down♯- 17 : 14 17 : 2·7 Audio file "Superminor third on C.mid" not found Superminor third[16] 17
337.15
 E♭+ 243 : 200 35 : 23·52 Audio file "Acute minor third on C.mid" not found Acute minor third[3] 5
342.48
 E13♭ 39 : 32 3·13 : 25 Audio file "Thirty-ninth harmonic on C.mid" not found Thirty-ninth harmonic[5] 13
342.86
 22/7 22/7 Audio file "2 steps in 7-et on C.mid" not found 2 steps in 7 equal temperament 7
347.41
 E↑♭-[2] 11 : 9 11 : 32 Audio file "Undecimal neutral third on C.mid" not found Undecimal neutral third[3] 11
350.00
 Dthree quarter sharp/Ehalf flat 27/24 27/24 Audio file "Neutral third on C.mid" not found Equal-tempered neutral third 24
359.47
 E13 upside down[2] 16 : 13 24 : 13 Audio file "Tridecimal neutral third on C.mid" not found Tridecimal neutral third[3] 13
364.54
 79 : 64 79 : 26 Audio file "Seventy-ninth harmonic on C.mid" not found Seventy-ninth harmonic[5] 79
364.81
 E- 100 : 81 22·52 : 34 Audio file "Grave major third on C.mid" not found Grave major third[3] 5
384.36
 F♭-- 8192 : 6561 213 : 38 Audio file "Pythagorean diminished fourth on C.mid" not found Pythagorean diminished fourth,[3][6] Pythagorean 'schismatic' third[5] 3
386.31
 E[2] 5 : 4 5 : 22 Audio file "Just major third on C.mid" not found Just major third,[3][4][5][9][14] major third,[12] quarter-comma meantone major third 5 M S
400.00
 E 24/12 21/3 Audio file "Major third on C.mid" not found Equal-tempered major third 3, 12 M
407.82
 E+[2] 81 : 64 34 : 26 Audio file "Pythagorean major third on C.mid" not found Pythagorean major third,[3][5][6][12][14] ditone 3
417.51
 F7↓+[2] 14 : 11 2·7 : 11 Audio file "Undecimal major third on C.mid" not found Undecimal diminished fourth or major third[3] 11
427.37
 F♭[2] 32 : 25 25 : 52 Audio file "Just diminished fourth on C.mid" not found Just diminished fourth,[14] diminished fourth[5][11] 5
429.06
 E41 41 : 32 41 : 25 Audio file "Forty-first harmonic on C.mid" not found Forty-first harmonic[5] 41
435.08
 E7 upside-down[2] 9 : 7 32 : 7 Audio file "Septimal major third on C.mid" not found Septimal major third,[3][5] Bohlen-Pierce third,[3] Super major Third[12] 7
450.05
 83 : 64 83 : 26 Audio file "Eighty-third harmonic on C.mid" not found Eighty-third harmonic[5] 83
454.21
 F♭13 13 : 10 13 : 2·5 Audio file "Tridecimal major third on C.mid" not found Tridecimal major third or diminished fourth 13
456.99
 E♯[2] 125 : 96 53 : 25·3 Audio file "Just augmented third on C.mid" not found Just augmented third, augmented third[5] 5
470.78
 F7+[2] 21 : 16 3·7 : 24 Audio file "Twenty-first harmonic on C.mid" not found Twenty-first harmonic, narrow fourth,[3] septimal fourth,[5] wide augmented third,[citation needed] H7 on G 7
478.49
 E♯+ 675 : 512 33·52 : 29 Audio file "Wide augmented third on C.mid" not found Wide augmented third[3] 5
480.00
 22/5 22/5 Audio file "2 steps in 5-et on C.mid" not found 2 steps in 5 equal temperament 5
491.27
 E17♯ 85 : 64 5·17 : 26 Audio file "Eighty-fifth harmonic on C.mid" not found Eighty-fifth harmonic[5] 17
498.04
 F[2] 4 : 3 22 : 3 Audio file "Just perfect fourth on C.mid" not found Perfect fourth,[3][5][14] Pythagorean perfect fourth, Just perfect fourth or diatessaron[4] 3 S
500.00
 F 25/12 25/12 Audio file "Perfect fourth on C.mid" not found Equal-tempered perfect fourth 12 M
510.51
 (3 : 2)8/11 38/11 : 28/11 Audio file "Beta scale perfect fourth on C.mid" not found Beta scale perfect fourth 18.75
511.52
 43 : 32 43 : 25 Audio file "Forty-third harmonic on C.mid" not found Forty-third harmonic[5] 43
514.29
 23/7 23/7 Audio file "3 steps in 7-et on C.mid" not found 3 steps in 7 equal temperament 7
519.55
 F+[2] 27 : 20 33 : 22·5 Audio file "Wolf fourth on C.mid" not found 5-limit wolf fourth, acute fourth,[3] imperfect fourth[14] 5
521.51
 E♯+++ 177147 : 131072 311 : 217 Audio file "Pythagorean augmented third on C.mid" not found Pythagorean augmented third[3][6] (F+ (pitch)) 3
531.53
 F29+ 87 : 64 3·29 : 26 Audio file "Eighty-seventh harmonic on C.mid" not found Eighty-seventh harmonic[5] 29
551.32
 F↑[2] 11 : 8 11 : 23 Audio file "Eleventh harmonic on C.mid" not found eleventh harmonic,[5] undecimal tritone,[5] lesser undecimal tritone, undecimal semi-augmented fourth[3] 11
568.72
 F♯[2] 25 : 18 52 : 2·32 Audio file "Just augmented fourth on C.mid" not found Just augmented fourth[3][5] 5
570.88
 89 : 64 89 : 26 Audio file "Eighty-ninth harmonic on C.mid" not found Eighty-ninth harmonic[5] 89
582.51
 G7♭[2] 7 : 5 7 : 5 Audio file "Lesser septimal tritone on C.mid" not found Lesser septimal tritone, septimal tritone[3][4][5] Huygens' tritone or Bohlen-Pierce fourth,[3] septimal fifth,[9] septimal diminished fifth[17] 7
588.27
 G♭-- 1024 : 729 210 : 36 Audio file "Diminished fifth tritone on C.mid" not found Pythagorean diminished fifth,[3][6] low Pythagorean tritone[5] 3
590.22
 F♯+[2] 45 : 32 32·5 : 25 Audio file "Just augmented fourth on C.mid" not found Just augmented fourth, just tritone,[4][9] tritone,[6] diatonic tritone,[3] 'augmented' or 'false' fourth,[14] high 5-limit tritone,[5] 1/6-comma meantone augmented fourth 5
600.00
 F♯/G♭ 26/12 21/2=√2 Audio file "Tritone on C.mid" not found Equal-tempered tritone 2, 12 M
609.35
 G137♭ 91 : 64 7·13 : 26 Audio file "Ninety-first harmonic on C.mid" not found Ninety-first harmonic[5] 13
609.78
 G♭-[2] 64 : 45 26 : 32·5 Audio file "Just tritone on C.mid" not found Just tritone,[4] 2nd tritone,[6] 'false' fifth,[14] diminished fifth,[11] low 5-limit tritone[5] 5
611.73
 F#++ 729 : 512 36 : 29 Audio file "Pythagorean augmented fourth on C.mid" not found Pythagorean tritone,[3][6] Pythagorean augmented fourth, high Pythagorean tritone[5] 3
617.49
 F♯7 upside-down[2] 10 : 7 2·5 : 7 Audio file "Greater septimal tritone on C.mid" not found Greater septimal tritone, septimal tritone,[4][5] Euler's tritone[3] 7
628.27
 F23♯+ 23 : 16 23 : 24 Audio file "Twenty-third harmonic on C.mid" not found Twenty-third harmonic,[5] classic diminished fifth[citation needed] 23
631.28
 G♭[2] 36 : 25 22·32 : 52 Audio file "Just diminished fifth on C.mid" not found Just diminished fifth[5] 5
646.99
 F31♯+ 93 : 64 3·31 : 26 Audio file "Ninety-third harmonic on C.mid" not found Ninety-third harmonic[5] 31
648.68
 G↓[2] 16 : 11 24 : 11 Audio file "Eleventh harmonic inverse on C.mid" not found Inversion of eleventh harmonic, undecimal semi-diminished fifth[3] 11
665.51
 47 : 32 47 : 25 Audio file "Forty-seventh harmonic on C.mid" not found Forty-seventh harmonic[5] 47
678.49
 Adouble flat--- 262144 : 177147 218 : 311 Audio file "Pythagorean diminished sixth on C.mid" not found Pythagorean diminished sixth[3][6] 3
680.45
 G- 40 : 27 23·5 : 33 Audio file "Wolf fifth on C.mid" not found 5-limit wolf fifth,[5] or diminished sixth, grave fifth,[3][6][9] imperfect fifth,[14] 5
683.83
 G19 95 : 64 5·19 : 26 Audio file "Ninety-fifth harmonic on C.mid" not found Ninety-fifth harmonic[5] 19
691.20
 3:2÷(81:80)1/2 2·51/2 : 3 Audio file "Half-comma meantone perfect fifth on C.mid" not found Half-comma meantone perfect fifth M
694.79
 3:2÷(81:80)1/3 21/3·51/3 : 31/3 Audio file "Third-comma meantone perfect fifth on C.mid" not found 1/3-comma meantone perfect fifth M
695.81
 3:2÷(81:80)2/7 21/7·52/7 : 31/7 Audio file "Two seventh-comma meantone perfect fifth on C.mid" not found 2/7-comma meantone perfect fifth M
696.58
 3:2÷(81:80)1/4 51/4 Audio file "Quarter-comma meantone perfect fifth on C.mid" not found Quarter-comma meantone perfect fifth M
697.65
 3:2÷(81:80)1/5 31/5·51/5 : 21/5 Audio file "Fifth-comma meantone perfect fifth on C.mid" not found 1/5-comma meantone perfect fifth M
698.37
 3:2÷(81:80)1/6 31/3·51/6 : 21/3 Audio file "Sixth-comma meantone perfect fifth on C.mid" not found 1/6-comma meantone perfect fifth M
700.00
 G 27/12 27/12 Audio file "Perfect fifth on C.mid" not found Equal-tempered perfect fifth 12 M
701.89
 231/53 231/53 Audio file "53-TET perfect fifth on C.mid" not found 53-TET perfect fifth 53
701.96
 G[2] 3 : 2 3 : 2 Audio file "Just perfect fifth on C.mid" not found Perfect fifth,[3][5][14] Pythagorean perfect fifth, Just perfect fifth or diapente,[4] fifth,[12] Just fifth[9] 3 S
702.44
 224/41 224/41 Audio file "41-TET perfect fifth on C.mid" not found 41-TET perfect fifth 41
703.45
 217/29 217/29 Audio file "29-TET perfect fifth on C.mid" not found 29-TET perfect fifth 29
719.90
 97 : 64 97 : 26 Audio file "Ninety-seventh harmonic on C.mid" not found Ninety-seventh harmonic[5] 97
721.51
 Adouble flat- 1024 : 675 210 : 33·52 Audio file "Narrow diminished sixth on C.mid" not found Narrow diminished sixth[3] 5
737.65
 A77♭+ 49 : 32 7·7 : 25 Audio file "Forty-ninth harmonic on C.mid" not found Forty-ninth harmonic[5] 7
743.01
 Adouble flat 192 : 125 26·3 : 53 Audio file "Classic diminished sixth on C.mid" not found Classic diminished sixth[3] 5
755.23
 G↑ 99 : 64 32·11 : 26 Audio file "Ninety-ninth harmonic on C.mid" not found Ninety-ninth harmonic[5] 11
764.92
 A7♭[2] 14 : 9 2·7 : 32 Audio file "Septimal minor sixth on C.mid" not found Septimal minor sixth[3][5] 7
772.63
 G♯ 25 : 16 52 : 24 Audio file "Just augmented fifth on C.mid" not found Just augmented fifth[5][14] 5
782.49
 G↑-[2] 11 : 7 11 : 7 Audio file "Undecimal minor sixth on C.mid" not found Undecimal minor sixth,[5] undecimal augmented fifth,[3] pi 11
789.85
 101 : 64 101 : 26 Audio file "Hundred-first harmonic on C.mid" not found Hundred-first harmonic[5] 101
792.18
 A♭-[2] 128 : 81 27 : 34 Audio file "Pythagorean minor sixth on C.mid" not found Pythagorean minor sixth[3][5][6] 3
800.00
 G♯/A♭ 28/12 22/3 Audio file "Minor sixth on C.mid" not found Equal-tempered minor sixth 3, 12 M
806.91
 G17♯ 51 : 32 3·17 : 25 Audio file "Fifty-first harmonic on C.mid" not found Fifty-first harmonic[5] 17
813.69
 A♭[2] 8 : 5 23 : 5 Audio file "Just minor sixth on C.mid" not found Just minor sixth[3][4][9][14] 5
815.64
 G♯++ 6561 : 4096 38 : 212 Audio file "Pythagorean augmented fifth on C.mid" not found Pythagorean augmented fifth,[3][6] Pythagorean 'schismatic' sixth[5] 3
823.80
 103 : 64 103 : 26 Audio file "Hundred-third harmonic on C.mid" not found Hundred-third harmonic[5] 103
833.09
 51/2+1 : 2 Audio file "Golden ratio on C.mid" not found Golden ratio (833 cents scale)
833.11
 233 : 144 233 : 24·32 Audio file "Golden ratio 233 144 on C.mid" not found Golden ratio approximation (833 cents scale) 233
835.19
 A♭+ 81 : 50 34 : 2·52 Audio file "Acute minor sixth on C.mid" not found Acute minor sixth[3] 5
840.53
 A13♭[2] 13 : 8 13 : 23 Audio file "Tridecimal neutral sixth on C.mid" not found Tridecimal neutral sixth,[3] overtone sixth,[5] thirteenth harmonic 13
850.00
 Gthree quarter sharp/Ahalf flat 217/24 217/24 Audio file "Neutral sixth on C.mid" not found Equal-tempered neutral sixth 24
852.59
 A↓[2] 18 : 11 2·32 : 11 Audio file "Undecimal neutral sixth on C.mid" not found Undecimal neutral sixth,[3][5] Zalzal's neutral sixth 11
857.10
 A7+ 105 : 64 3·5·7 : 26 Audio file "Hundred-fifth harmonic on C.mid" not found Hundred-fifth harmonic[5] 7
857.14
 25/7 25/7 Audio file "5 steps in 7-et on C.mid" not found 5 steps in 7 equal temperament 7
862.85
 A- 400 : 243 24·52 : 35 Audio file "Grave major sixth on C.mid" not found Grave major sixth[3] 5
873.51
 53 : 32 53 : 25 Audio file "Fifty-third harmonic on C.mid" not found Fifty-third harmonic[5] 53
882.40
 Bdouble flat--- 32768 : 19683 215 : 39 Audio file "Pythagorean diminished seventh on C.mid" not found Pythagorean diminished seventh[3][6] 3
884.36
 A[2] 5 : 3 5 : 3 Audio file "Just major sixth on C.mid" not found Just major sixth,[3][4][5][9][14] Bohlen-Pierce sixth,[3] 1/3-comma meantone major sixth 5 M
889.76
 107 : 64 107 : 26 Audio file "Hundred-seventh harmonic on C.mid" not found Hundred-seventh harmonic[5] 107
900.00
 A 29/12 23/4 Audio file "Dim seventh on C.mid" not found Equal-tempered major sixth 4, 12 M
905.87
 A+[2] 27 : 16 33 : 24 Audio file "Pythagorean major sixth on C.mid" not found Pythagorean major sixth[3][5][9][14] 3
921.82
 109 : 64 109 : 26 Audio file "Hundred-ninth harmonic on C.mid" not found Hundred-ninth harmonic[5] 109
925.42
 Bdouble flat-[2] 128 : 75 27 : 3·52 Audio file "Just diminished seventh on C.mid" not found Just diminished seventh,[14] diminished seventh[5][11] 5
933.13
 A7 upside-down[2] 12 : 7 22·3 : 7 Audio file "Septimal major sixth on C.mid" not found Septimal major sixth[3][4][5] 7
937.63
 A↑ 55 : 32 5·11 : 25 Audio file "Fifty-fifth harmonic on C.mid" not found Fifty-fifth harmonic[5][18] 11
953.30
 A37♯+ 111 : 64 3·37 : 26 Audio file "Hundred-eleventh harmonic on C.mid" not found Hundred-eleventh harmonic[5] 37
955.03
 A♯[2] 125 : 72 53 : 23·32 Audio file "Just augmented sixth on C.mid" not found Just augmented sixth[5] 5
957.21
 (3 : 2)15/11 315/11 : 215/11 Audio file "Beta scale 15 steps on C.mid" not found 15 steps in Beta scale 18.75
960.00
 24/5 24/5 Audio file "4 steps in 5-et on C.mid" not found 4 steps in 5 equal temperament 5
968.83
 B7♭[2] 7 : 4 7 : 22 Audio file "Harmonic seventh on C.mid" not found Septimal minor seventh,[4][5][9] harmonic seventh,[3][9] augmented sixth[citation needed] 7
976.54
 A♯+[2] 225 : 128 32·52 : 27 Audio file "Just augmented sixth on C.mid" not found Just augmented sixth[14] 5
984.22
 113 : 64 113 : 26 Audio file "Hundred-thirteenth harmonic on C.mid" not found Hundred-thirteenth harmonic[5] 113
996.09
 B♭-[2] 16 : 9 24 : 32 Audio file "Lesser just minor seventh on C.mid" not found Pythagorean minor seventh,[3] Small just minor seventh,[4] lesser minor seventh,[14] just minor seventh,[9] Pythagorean small minor seventh[5] 3
999.47
 B19♭ 57 : 32 3·19 : 25 Audio file "Fifty-seventh harmonic on C.mid" not found Fifty-seventh harmonic[5] 19
1000.00
 A♯/B♭ 210/12 25/6 Audio file "Minor seventh on C.mid" not found Equal-tempered minor seventh 6, 12 M
1014.59
 A23♯+ 115 : 64 5·23 : 26 Audio file "Hundred-fifteenth harmonic on C.mid" not found Hundred-fifteenth harmonic[5] 23
1017.60
 B♭[2] 9 : 5 32 : 5 Audio file "Greater just minor seventh on C.mid" not found Greater just minor seventh,[14] large just minor seventh,[4][5] Bohlen-Pierce seventh[3] 5
1019.55
 A♯+++ 59049 : 32768 310 : 215 Audio file "Pythagorean augmented sixth on C.mid" not found Pythagorean augmented sixth[3][6] 3
1028.57
 26/7 26/7 Audio file "6 steps in 7-et on C.mid" not found 6 steps in 7 equal temperament 7
1029.58
 B29♭ 29 : 16 29 : 24 Audio file "Twenty-ninth harmonic on C.mid" not found Twenty-ninth harmonic,[5] minor seventh[citation needed] 29
1035.00
 B↓[2] 20 : 11 22·5 : 11 Audio file "Lesser undecimal neutral seventh on C.mid" not found Lesser undecimal neutral seventh, large minor seventh[3] 11
1039.10
 B♭+ 729 : 400 36 : 24·52 Audio file "Acute minor seventh on C.mid" not found Acute minor seventh[3] 5
1044.44
 A13♭ 117 : 64 32·13 : 26 Audio file "Hundred-seventeenth harmonic on C.mid" not found Hundred-seventeenth harmonic[5] 13
1049.36
 B↑♭-[2] 11 : 6 11 : 2·3 Audio file "Neutral seventh on C.mid" not found 21/4-tone or Undecimal neutral seventh,[3] undecimal 'median' seventh[5] 11
1050.00
 Athree quarter sharp/Bhalf flat 221/24 27/8 Audio file "Neutral seventh on C.mid" not found Equal-tempered neutral seventh 8, 24
1059.17
 59 : 32 59 : 25 Audio file "Fifty-ninth harmonic on C.mid" not found Fifty-ninth harmonic[5] 59
1066.76
 B- 50 : 27 2·52 : 33 Audio file "Grave major seventh on C.mid" not found Grave major seventh[3] 5
1073.78
 B717 119 : 64 7·17 : 26 Audio file "Hundred-nineteenth harmonic on C.mid" not found Hundred-nineteenth harmonic[5] 17
1086.31
 C♭-- 4096 : 2187 212 : 37 Audio file "Pythagorean diminished octave on C.mid" not found Pythagorean diminished octave[3][6] 3
1088.27
 B[2] 15 : 8 3·5 : 23 Audio file "Just major seventh on C.mid" not found Just major seventh,[3][5][9][14] small just major seventh,[4] 1/6-comma meantone major seventh 5
1100.00
 B 211/12 211/12 Audio file "Major seventh on C.mid" not found Equal-tempered major seventh 12 M
1102.64
 B↑↑♭- 121 : 64 112 : 26 Audio file "Hundred-twenty-first harmonic on C.mid" not found Hundred-twenty-first harmonic[5] 11
1107.82
 C'♭- 256 : 135 28 : 33·5 Audio file "Octave minus major chroma on C.mid" not found Octave − major chroma,[3] narrow diminished octave[citation needed] 5
1109.78
 B+[2] 243 : 128 35 : 27 Audio file "Pythagorean major seventh on C.mid" not found Pythagorean major seventh[3][5][6][9] 3
1116.89
 61 : 32 61 : 25 Audio file "Sixty-first harmonic on C.mid" not found Sixty-first harmonic[5] 61
1129.33
 C'♭[2] 48 : 25 24·3 : 52 Audio file "Classic diminished octave on C.mid" not found Classic diminished octave,[3][6] large just major seventh[4] 5
1131.02
 B41 123 : 64 3·41 : 26 Audio file "Hundred-twenty-third harmonic on C.mid" not found Hundred-twenty-third harmonic[5] 41
1137.04
 B7 upside-down 27 : 14 33 : 2·7 Audio file "Septimal major seventh on C.mid" not found Septimal major seventh[5] 7
1145.04
 B31 31 : 16 31 : 24 Audio file "Thirty-first harmonic on C.mid" not found Thirty-first harmonic,[5] augmented seventh[citation needed] 31
1151.23
 C7 35 : 18 5·7 : 2·32 Audio file "Septimal supermajor seventh on C.mid" not found Septimal supermajor seventh, septimal quarter tone inverted 7
1158.94
 B♯[2] 125 : 64 53 : 26 Audio file "Just augmented seventh on C.mid" not found Just augmented seventh,[5] 125th harmonic 5
1172.74
 C7+ 63 : 32 32·7 : 25 Audio file "Sixty-third harmonic on C.mid" not found Sixty-third harmonic[5] 7
1178.49
 C'- 160 : 81 25·5 : 34 Audio file "Octave minus syntonic comma on C.mid" not found Octave − syntonic comma,[3] semi-diminished octave[citation needed] 5
1186.42
 127 : 64 127 : 26 Audio file "Hundred-twenty-seventh harmonic on C.mid" not found Hundred-twenty-seventh harmonic[5] 127
1200.00
 C' 2 : 1 2 : 1 Audio file "Perfect octave on C.mid" not found Octave[3][9] or diapason[4] 1, 12 3 M S
1223.46
 B♯+++ 531441 : 262144 312 : 218 Audio file "Pythagorean augmented seventh on C.mid" not found Pythagorean augmented seventh[3][6] 3
1525.86
 21/2+1 Audio file "Silver ratio on C.mid" not found Silver ratio
1901.96
 G' 3 : 1 3 : 1 Audio file "Tritave on C.mid" not found Tritave or just perfect twelfth 3
2400.00
 C" 4 : 1 22 : 1 Audio file "Perfect fifteenth on C.mid" not found Fifteenth or two octaves 1, 12 3 M
3986.31
 E''' 10 : 1 5·2 : 1 Audio file "Decade on C.mid" not found Decade, compound just major third 5 M

See also

Notes

  1. ↑ 1.0 1.1 1.2 1.3 Maneri-Sims notation

References

  1. ↑ 1.0 1.1 Fox, Christopher (2003). "Microtones and Microtonalities", Contemporary Music Review, v. 22, pt. 1-2. (Abingdon, Oxfordshire, UK: Routledge): p.13.
  2. ↑ 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 2.30 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 2.40 2.41 2.42 2.43 2.44 2.45 2.46 2.47 2.48 2.49 2.50 2.51 2.52 2.53 2.54 2.55 2.56 2.57 2.58 2.59 2.60 Fonville, John. 1991. "Ben Johnston's Extended Just Intonation: A Guide for Interpreters". Perspectives of New Music 29, no. 2 (Summer): 106–37.
  3. ↑ 3.000 3.001 3.002 3.003 3.004 3.005 3.006 3.007 3.008 3.009 3.010 3.011 3.012 3.013 3.014 3.015 3.016 3.017 3.018 3.019 3.020 3.021 3.022 3.023 3.024 3.025 3.026 3.027 3.028 3.029 3.030 3.031 3.032 3.033 3.034 3.035 3.036 3.037 3.038 3.039 3.040 3.041 3.042 3.043 3.044 3.045 3.046 3.047 3.048 3.049 3.050 3.051 3.052 3.053 3.054 3.055 3.056 3.057 3.058 3.059 3.060 3.061 3.062 3.063 3.064 3.065 3.066 3.067 3.068 3.069 3.070 3.071 3.072 3.073 3.074 3.075 3.076 3.077 3.078 3.079 3.080 3.081 3.082 3.083 3.084 3.085 3.086 3.087 3.088 3.089 3.090 3.091 3.092 3.093 3.094 3.095 3.096 3.097 3.098 3.099 3.100 3.101 3.102 3.103 3.104 3.105 3.106 3.107 "List of intervals", Huygens-Fokker Foundation. The Foundation uses "classic" to indicate "just" or leaves off any adjective, as in "major sixth".
  4. ↑ 4.00 4.01 4.02 4.03 4.04 4.05 4.06 4.07 4.08 4.09 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 Partch, Harry (1979). Genesis of a Music, p.68-69. ISBN 978-0-306-80106-8.
  5. ↑ 5.000 5.001 5.002 5.003 5.004 5.005 5.006 5.007 5.008 5.009 5.010 5.011 5.012 5.013 5.014 5.015 5.016 5.017 5.018 5.019 5.020 5.021 5.022 5.023 5.024 5.025 5.026 5.027 5.028 5.029 5.030 5.031 5.032 5.033 5.034 5.035 5.036 5.037 5.038 5.039 5.040 5.041 5.042 5.043 5.044 5.045 5.046 5.047 5.048 5.049 5.050 5.051 5.052 5.053 5.054 5.055 5.056 5.057 5.058 5.059 5.060 5.061 5.062 5.063 5.064 5.065 5.066 5.067 5.068 5.069 5.070 5.071 5.072 5.073 5.074 5.075 5.076 5.077 5.078 5.079 5.080 5.081 5.082 5.083 5.084 5.085 5.086 5.087 5.088 5.089 5.090 5.091 5.092 5.093 5.094 5.095 5.096 5.097 5.098 5.099 5.100 5.101 5.102 5.103 5.104 5.105 5.106 5.107 5.108 5.109 5.110 5.111 5.112 5.113 5.114 5.115 5.116 "Anatomy of an Octave", KyleGann.com. Gann leaves off "just" but includes "5-limit".
  6. ↑ 6.00 6.01 6.02 6.03 6.04 6.05 6.06 6.07 6.08 6.09 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22 6.23 6.24 6.25 6.26 6.27 6.28 6.29 6.30 6.31 6.32 6.33 6.34 6.35 6.36 6.37 Haluška, Ján (2003). The Mathematical Theory of Tone Systems, p.xxv-xxix. ISBN 978-0-8247-4714-5.
  7. ↑ "Orwell Temperaments", Xenharmony.org.
  8. ↑ 8.0 8.1 Partch (1979), p.70.
  9. ↑ 9.00 9.01 9.02 9.03 9.04 9.05 9.06 9.07 9.08 9.09 9.10 9.11 9.12 9.13 9.14 9.15 9.16 9.17 9.18 9.19 9.20 9.21 9.22 9.23 9.24 9.25 9.26 Alexander John Ellis (1885). On the musical scales of various nations, p.488. s.n.
  10. ↑ William Smythe Babcock Mathews (1895). Pronouncing dictionary and condensed encyclopedia of musical terms, p.13. ISBN 1-112-44188-3.
  11. ↑ 11.0 11.1 11.2 11.3 11.4 11.5 Anger, Joseph Humfrey (1912). A treatise on harmony, with exercises, Volume 3, p.xiv-xv. W. Tyrrell.
  12. ↑ 12.00 12.01 12.02 12.03 12.04 12.05 12.06 12.07 12.08 12.09 12.10 12.11 12.12 Hermann Ludwig F. von Helmholtz (Alexander John Ellis, trans.) (1875). "Additions by the translator", On the sensations of tone as a physiological basis for the theory of music, p.644. No ISBN specified.
  13. ↑ Lua error in package.lua at line 80: module 'strict' not found.
  14. ↑ 14.00 14.01 14.02 14.03 14.04 14.05 14.06 14.07 14.08 14.09 14.10 14.11 14.12 14.13 14.14 14.15 14.16 14.17 14.18 14.19 14.20 14.21 14.22 14.23 14.24 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".
  15. ↑ 15.0 15.1 "13th-harmonic", 31et.com.
  16. ↑ Brabner, John H. F. (1884). The National Encyclopaedia, Vol.13, p.182. London. [ISBN unspecified]
  17. ↑ Sabat, Marc and von Schweinitz, Wolfgang (2004). "The Extended Helmholtz-Ellis JI Pitch Notation" [PDF], NewMusicBox.org. Accessed: 04:12, 15 March 2014 (UTC).
  18. ↑ Hermann L. F Von Helmholtz (2007). On the Sensations of Tone, p.456. ISBN 978-1-60206-639-7.

External links

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