List of pitch intervals
Appearance
(Redirected from Seventh (interval))
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Below is a list of intervals expressible 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
[edit]- 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 factored into 2 × 5 (and 9 into 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.
- Superparticular ratios are intervals that can be expressed as the ratio of two consecutive integers.
List
[edit]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). |
Cents | Note (from C) | Freq. ratio | Prime factors | Interval name | TET | Limit | M | S |
---|---|---|---|---|---|---|---|---|
0.00 |
C[2] | 1 : 1 | 1 : 1 | Unison,[3] monophony,[4] perfect prime,[3] tonic,[5] or fundamental | 1, 12 | 3 | M | |
0.03 |
65537 : 65536 | 65537 : 216 | Sixty-five-thousand-five-hundred-thirty-seventh harmonic | 65537 | S | |||
0.40 |
C![]() |
4375 : 4374 | 54×7 : 2×37 | Ragisma[3][6] | 7 | S | ||
0.72 |
E![]() ![]() ![]() ![]() ![]() |
2401 : 2400 | 74 : 25×3×52 | Breedsma[3][6] | 7 | S | ||
1.00 |
21/1200 | 21/1200 | Cent[7] | 1200 | ||||
1.20 |
21/1000 | 21/1000 | Millioctave | 1000 | ||||
1.95 |
B♯++ | 32805 : 32768 | 38×5 : 215 | Schisma[3][5] | 5 | |||
1.96 |
3:2÷(27/12) | 3 : 219/12 | Grad, Werckmeister[8] | |||||
3.99 |
101/1000 | 21/1000×51/1000 | Savart or eptaméride | 301.03 | ||||
7.71 |
B![]() |
225 : 224 | 32×52 : 25×7 | Septimal kleisma,[3][6] marvel comma | 7 | S | ||
8.11 |
B![]() |
15625 : 15552 | 56 : 26×35 | Kleisma or semicomma majeur[3][6] | 5 | |||
10.06 |
A![]() ![]() |
2109375 : 2097152 | 33×57 : 221 | Semicomma,[3][6] Fokker's comma[3] | 5 | |||
10.85 |
C![]() |
160 : 159 | 25×5 : 3×53 | Difference between 5:3 & 53:32 | 53 | S | ||
11.98 |
C![]() |
145 : 144 | 5×29 : 24×32 | Difference between 29:16 & 9:5 | 29 | S | ||
12.50 |
21/96 | 21/96 | Sixteenth tone | 96 | ||||
13.07 |
B![]() ![]() ![]() |
1728 : 1715 | 26×33 : 5×73 | Orwell comma[3][9] | 7 | |||
13.47 |
C![]() |
129 : 128 | 3×43 : 27 | Hundred-twenty-ninth harmonic | 43 | S | ||
13.79 |
D![]() ![]() |
126 : 125 | 2×32×7 : 53 | Small septimal semicomma,[6] small septimal comma,[3] starling comma | 7 | S | ||
14.37 |
C♭↑↑− | 121 : 120 | 112 : 23×3×5 | Undecimal seconds comma[3] | 11 | S | ||
16.67 |
C↑[a] | 21/72 | 21/72 | 1 step in 72 equal temperament | 72 | |||
18.13 |
C![]() |
96 : 95 | 25×3 : 5×19 | Difference between 19:16 & 6:5 | 19 | S | ||
19.55 |
D![]() |
2048 : 2025 | 211 : 34×52 | Diaschisma,[3][6] minor comma | 5 | |||
21.51 |
C+[2] | 81 : 80 | 34 : 24×5 | Syntonic comma,[3][5][6] major comma, komma, chromatic diesis, or comma of Didymus[3][6][10][11] | 5 | S | ||
22.64 |
21/53 | 21/53 | Holdrian comma, Holder's comma, 1 step in 53 equal temperament | 53 | ||||
23.46 |
B♯+++ | 531441 : 524288 | 312 : 219 | Pythagorean comma,[3][5][6][10][11] ditonic comma[3][6] | 3 | |||
25.00 |
21/48 | 21/48 | Eighth tone | 48 | ||||
26.84 |
C![]() |
65 : 64 | 5×13 : 26 | Sixty-fifth harmonic,[5] 13th-partial chroma[3] | 13 | S | ||
27.26 |
C![]() |
64 : 63 | 26 : 32×7 | Septimal comma,[3][6][11] Archytas' comma,[3] 63rd subharmonic | 7 | S | ||
29.27 |
21/41 | 21/41 | 1 step in 41 equal temperament | 41 | ||||
31.19 |
D![]() |
56 : 55 | 23×7 : 5×11 | Undecimal diesis,[3] Ptolemy's enharmonic:[5] difference between (11 : 8) and (7 : 5) tritone | 11 | S | ||
33.33 |
C![]() ![]() ![]() |
21/36 | 21/36 | Sixth tone | 36, 72 | |||
34.28 |
C![]() |
51 : 50 | 3×17 : 2×52 | Difference between 17:16 & 25:24 | 17 | S | ||
34.98 |
B![]() ![]() |
50 : 49 | 2×52 : 72 | Septimal sixth tone or jubilisma, Erlich's decatonic comma or tritonic diesis[3][6] | 7 | S | ||
35.70 |
D![]() ![]() |
49 : 48 | 72 : 24×3 | Septimal diesis, slendro diesis or septimal 1/6-tone[3] | 7 | S | ||
38.05 |
C![]() |
46 : 45 | 2×23 : 32×5 | Inferior quarter tone,[5] difference between 23:16 & 45:32 | 23 | S | ||
38.71 |
21/31 | 21/31 | 1 step in 31 equal temperament | 31 | ||||
38.91 |
C↓♯+ | 45 : 44 | 32×5 : 4×11 | Undecimal diesis or undecimal fifth tone | 11 | S | ||
40.00 |
21/30 | 21/30 | Fifth tone | 30 | ||||
41.06 |
D![]() |
128 : 125 | 27 : 53 | Enharmonic diesis or 5-limit limma, minor diesis,[6] diminished second,[5][6] minor diesis or diesis,[3] 125th subharmonic | 5 | |||
41.72 |
D![]() ![]() |
42 : 41 | 2×3×7 : 41 | Lesser 41-limit fifth tone | 41 | S | ||
42.75 |
C![]() |
41 : 40 | 41 : 23×5 | Greater 41-limit fifth tone | 41 | S | ||
43.83 |
C![]() |
40 : 39 | 23×5 : 3×13 | Tridecimal fifth tone | 13 | S | ||
44.97 |
C![]() ![]() |
39 : 38 | 3×13 : 2×19 | Superior quarter-tone,[5] novendecimal fifth tone | 19 | S | ||
46.17 |
D![]() ![]() ![]() |
38 : 37 | 2×19 : 37 | Lesser 37-limit quarter tone | 37 | S | ||
47.43 |
C![]() |
37 : 36 | 37 : 22×32 | Greater 37-limit quarter tone | 37 | S | ||
48.77 |
C![]() |
36 : 35 | 22×32 : 5×7 | Septimal quarter tone, septimal diesis,[3][6] septimal chroma,[2] superior quarter tone[5] | 7 | S | ||
49.98 |
246 : 239 | 3×41 : 239 | Just quarter tone[11] | 239 | ||||
50.00 |
C![]() ![]() |
21/24 | 21/24 | Equal-tempered quarter tone | 24 | |||
50.18 |
D![]() ![]() |
35 : 34 | 5×7 : 2×17 | ET quarter-tone approximation,[5] lesser 17-limit quarter tone | 17 | S | ||
50.72 |
B![]() |
59049 : 57344 | 310 : 213×7 | Harrison's comma (10 P5s – 1 H7)[3] | 7 | |||
51.68 |
C![]() |
34 : 33 | 2×17 : 3×11 | Greater 17-limit quarter tone | 17 | S | ||
53.27 |
C↑ | 33 : 32 | 3×11 : 25 | Thirty-third harmonic,[5] undecimal comma, undecimal quarter tone | 11 | S | ||
54.96 |
D![]() |
32 : 31 | 25 : 31 | Inferior quarter-tone,[5] thirty-first subharmonic | 31 | S | ||
56.55 |
B![]() ![]() |
529 : 512 | 232 : 29 | Five-hundred-twenty-ninth harmonic | 23 | |||
56.77 |
C![]() |
31 : 30 | 31 : 2×3×5 | Greater quarter-tone,[5] difference between 31:16 & 15:8 | 31 | S | ||
58.69 |
C![]() |
30 : 29 | 2×3×5 : 29 | Lesser 29-limit quarter tone | 29 | S | ||
60.75 |
C![]() ![]() |
29 : 28 | 29 : 22×7 | Greater 29-limit quarter tone | 29 | S | ||
62.96 |
D![]() |
28 : 27 | 22×7 : 33 | Septimal minor second, small minor second, inferior quarter tone[5] | 7 | S | ||
63.81 |
(3 : 2)1/11 | 31/11 : 21/11 | Beta scale step | 18.75 | ||||
65.34 |
C![]() |
27 : 26 | 33 : 2×13 | Chromatic diesis,[12] tridecimal comma[3] | 13 | S | ||
66.34 |
D![]() ![]() |
133 : 128 | 7×19 : 27 | One-hundred-thirty-third harmonic | 19 | |||
66.67 |
C![]() ![]() |
21/18 | 21/18 | Third tone | 18, 36, 72 | |||
67.90 |
D![]() ![]() |
26 : 25 | 2×13 : 52 | Tridecimal third tone, third tone[5] | 13 | S | ||
70.67 |
C♯[2] | 25 : 24 | 52 : 23×3 | Just chromatic semitone or minor chroma,[3] lesser chromatic semitone, small (just) semitone[11] or minor second,[4] minor chromatic semitone,[13] or minor semitone,[5] 2⁄7-comma meantone chromatic semitone, augmented unison | 5 | S | ||
73.68 |
D![]() |
24 : 23 | 23×3 : 23 | Lesser 23-limit semitone | 23 | S | ||
75.00 |
21/16 | 23/48 | 1 step in 16 equal temperament, 3 steps in 48 | 16, 48 | ||||
76.96 |
C![]() |
23 : 22 | 23 : 2×11 | Greater 23-limit semitone | 23 | S | ||
78.00 |
(3 : 2)1/9 | 31/9 : 21/9 | Alpha scale step | 15.39 | ||||
79.31 |
67 : 64 | 67 : 26 | Sixty-seventh harmonic[5] | 67 | ||||
80.54 |
C↑![]() |
22 : 21 | 2×11 : 3×7 | Hard semitone,[5] two-fifth tone small semitone | 11 | S | ||
84.47 |
D![]() |
21 : 20 | 3×7 : 22×5 | Septimal chromatic semitone, minor semitone[3] | 7 | S | ||
88.80 |
C![]() |
20 : 19 | 22×5 : 19 | Novendecimal augmented unison | 19 | S | ||
90.22 |
D♭−−[2] | 256 : 243 | 28 : 35 | Pythagorean minor second or limma,[3][6][11] Pythagorean diatonic semitone, Low Semitone[14] | 3 | |||
92.18 |
C♯+[2] | 135 : 128 | 33×5 : 27 | Greater chromatic semitone, chromatic semitone, semitone medius, major chroma or major limma,[3] small limma,[11] major chromatic semitone,[13] limma ascendant[5] | 5 | |||
93.60 |
D![]() |
19 : 18 | 19 : 2×9 | Novendecimal minor second | 19 | S | ||
97.36 |
D↓↓ | 128 : 121 | 27 : 112 | 121st subharmonic,[5][6] undecimal minor second | 11 | |||
98.95 |
D![]() |
18 : 17 | 2×32 : 17 | Just minor semitone, Arabic lute index finger[3] | 17 | S | ||
100.00 |
C♯/D♭ | 21/12 | 21/12 | Equal-tempered minor second or semitone | 12 | M | ||
104.96 |
C![]() |
17 : 16 | 17 : 24 | Minor diatonic semitone, just major semitone, overtone semitone,[5] 17th harmonic,[3] limma[citation needed] | 17 | S | ||
111.45 |
25√5 | (5 : 1)1/25 | Studie II interval (compound just major third, 5:1, divided into 25 equal parts) | 25 | ||||
111.73 |
D♭-[2] | 16 : 15 | 24 : 3×5 | Just minor second,[15] just diatonic semitone, large just semitone or major second,[4] major semitone,[5] limma, minor diatonic semitone,[3] diatonic second[16] semitone,[14] diatonic semitone,[11] 1⁄6-comma meantone minor second | 5 | S | ||
113.69 |
C♯++ | 2187 : 2048 | 37 : 211 | Apotome[3][11] 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 | Secor | 10.28 | ||||
119.44 |
C![]() |
15 : 14 | 3×5 : 2×7 | Septimal diatonic semitone, major diatonic semitone,[3] Cowell semitone[5] | 7 | S | ||
125.00 |
25/48 | 25/48 | 5 steps in 48 equal temperament | 48 | ||||
128.30 |
D![]() ![]() |
14 : 13 | 2×7 : 13 | Lesser tridecimal 2/3-tone[17] | 13 | S | ||
130.23 |
C![]() |
69 : 64 | 3×23 : 26 | Sixty-ninth harmonic[5] | 23 | |||
133.24 |
D♭ | 27 : 25 | 33 : 52 | Semitone maximus, minor second, large limma or Bohlen-Pierce small semitone,[3] high semitone,[14] alternate Renaissance half-step,[5] large limma, acute minor second[citation needed] | 5 | |||
133.33 |
C♯![]() ![]() |
21/9 | 22/18 | Two-third tone | 9, 18, 36, 72 | |||
138.57 |
D![]() |
13 : 12 | 13 : 22×3 | Greater tridecimal 2/3-tone,[17] Three-quarter tone[5] | 13 | S | ||
150.00 |
C![]() ![]() |
23/24 | 21/8 | Equal-tempered neutral second | 8, 24 | |||
150.64 |
D↓[2] | 12 : 11 | 22×3 : 11 | 3⁄4 tone or Undecimal neutral second,[3][5] trumpet three-quarter tone,[11] middle finger [between frets][14] | 11 | S | ||
155.14 |
D![]() |
35 : 32 | 5×7 : 25 | Thirty-fifth harmonic[5] | 7 | |||
160.90 |
D−− | 800 : 729 | 25×52 : 36 | Grave whole tone,[3] neutral second, grave major second[citation needed] | 5 | |||
165.00 |
D↑♭−[2] | 11 : 10 | 11 : 2×5 | Greater undecimal minor/major/neutral second, 4/5-tone[6] or Ptolemy's second[3] | 11 | S | ||
171.43 |
21/7 | 21/7 | 1 step in 7 equal temperament | 7 | ||||
175.00 |
27/48 | 27/48 | 7 steps in 48 equal temperament | 48 | ||||
179.70 |
71 : 64 | 71 : 26 | Seventy-first harmonic[5] | 71 | ||||
180.45 |
E![]() |
65536 : 59049 | 216 : 310 | Pythagorean diminished third,[3][6] Pythagorean minor tone | 3 | |||
182.40 |
D−[2] | 10 : 9 | 2×5 : 32 | Small just whole tone or major second,[4] minor whole tone,[3][5] lesser whole tone,[16] minor tone,[14] minor second,[11] half-comma meantone major second | 5 | S | ||
200.00 |
D | 22/12 | 21/6 | Equal-tempered major second | 6, 12 | M | ||
203.91 |
D[2] | 9 : 8 | 32 : 23 | Pythagorean major second, Large just whole tone or major second[11] (sesquioctavan),[4] tonus, major whole tone,[3][5] greater whole tone,[16] major tone[14] | 3 | S | ||
215.89 |
D![]() |
145 : 128 | 5×29 : 27 | Hundred-forty-fifth harmonic | 29 | |||
223.46 |
E![]() |
256 : 225 | 28 : 32×52 | Just diminished third,[16] 225th subharmonic | 5 | |||
225.00 |
23/16 | 29/48 | 9 steps in 48 equal temperament | 16, 48 | ||||
227.79 |
73 : 64 | 73 : 26 | Seventy-third harmonic[5] | 73 | ||||
231.17 |
D![]() |
8 : 7 | 23 : 7 | Septimal major second,[4] septimal whole tone[3][5] | 7 | S | ||
240.00 |
21/5 | 21/5 | 1 step in 5 equal temperament | 5 | ||||
247.74 |
D![]() |
15 : 13 | 3×5 : 13 | Tridecimal 5⁄4 tone[3] | 13 | |||
250.00 |
D![]() ![]() |
25/24 | 25/24 | 5 steps in 24 equal temperament | 24 | |||
251.34 |
D![]() |
37 : 32 | 37 : 25 | Thirty-seventh harmonic[5] | 37 | |||
253.08 |
D♯− | 125 : 108 | 53 : 22×33 | Semi-augmented whole tone,[3] semi-augmented second[citation needed] | 5 | |||
262.37 |
E↓♭ | 64 : 55 | 26 : 5×11 | 55th subharmonic[5][6] | 11 | |||
266.87 |
E![]() |
7 : 6 | 7 : 2×3 | Septimal minor third[3][4][11] or Sub minor third[14] | 7 | S | ||
268.80 |
D![]() ![]() |
299 : 256 | 13×23 : 28 | Two-hundred-ninety-ninth harmonic | 23 | |||
274.58 |
D♯[2] | 75 : 64 | 3×52 : 26 | Just augmented second,[16] Augmented tone,[14] augmented second[5][13] | 5 | |||
275.00 |
211/48 | 211/48 | 11 steps in 48 equal temperament | 48 | ||||
289.21 |
E![]() |
13 : 11 | 13 : 11 | Tridecimal minor third[3] | 13 | |||
294.13 |
E♭−[2] | 32 : 27 | 25 : 33 | Pythagorean minor third[3][5][6][14][16] semiditone, or 27th subharmonic | 3 | |||
297.51 |
E![]() |
19 : 16 | 19 : 24 | 19th harmonic,[3] 19-limit minor third, overtone minor third[5] | 19 | |||
300.00 |
D♯/E♭ | 23/12 | 21/4 | Equal-tempered minor third | 4, 12 | M | ||
301.85 |
D![]() |
25 : 21[5] | 52 : 3×7 | Quasi-equal-tempered minor third, 2nd 7-limit minor third, Bohlen-Pierce second[3][6] | 7 | |||
310.26 |
6:5÷(81:80)1/4 | 22 : 53/4 | Quarter-comma meantone minor third | M | ||||
311.98 |
(3 : 2)4/9 | 34/9 : 24/9 | Alpha scale minor third | 3.85 | ||||
315.64 |
E♭[2] | 6 : 5 | 2×3 : 5 | Just minor third,[3][4][5][11][16] minor third,[14] 1⁄3-comma meantone minor third | 5 | M | S | |
317.60 |
D♯++ | 19683 : 16384 | 39 : 214 | Pythagorean augmented second[3][6] | 3 | |||
320.14 |
E![]() |
77 : 64 | 7×11 : 26 | Seventy-seventh harmonic[5] | 11 | |||
325.00 |
213/48 | 213/48 | 13 steps in 48 equal temperament | 48 | ||||
336.13 |
D![]() ![]() |
17 : 14 | 17 : 2×7 | Superminor third[18] | 17 | |||
337.15 |
E♭+ | 243 : 200 | 35 : 23×52 | Acute minor third[3] | 5 | |||
342.48 |
E![]() |
39 : 32 | 3×13 : 25 | Thirty-ninth harmonic[5] | 13 | |||
342.86 |
22/7 | 22/7 | 2 steps in 7 equal temperament | 7 | ||||
342.91 |
E![]() |
128 : 105 | 27 : 3×5×7 | 105th subharmonic,[5] septimal neutral third[6] | 7 | |||
347.41 |
E↑♭−[2] | 11 : 9 | 11 : 32 | Undecimal neutral third[3][5] | 11 | |||
350.00 |
D![]() ![]() |
27/24 | 27/24 | Equal-tempered neutral third | 24 | |||
354.55 |
E↓+ | 27 : 22 | 33 : 2×11 | Zalzal's wosta[6] 12:11 X 9:8[14] | 11 | |||
359.47 |
E![]() |
16 : 13 | 24 : 13 | Tridecimal neutral third[3] | 13 | |||
364.54 |
79 : 64 | 79 : 26 | Seventy-ninth harmonic[5] | 79 | ||||
364.81 |
E− | 100 : 81 | 22×52 : 34 | Grave major third[3] | 5 | |||
375.00 |
25/16 | 215/48 | 15 steps in 48 equal temperament | 16, 48 | ||||
384.36 |
F♭−− | 8192 : 6561 | 213 : 38 | Pythagorean diminished fourth,[3][6] Pythagorean 'schismatic' third[5] | 3 | |||
386.31 |
E[2] | 5 : 4 | 5 : 22 | Just major third,[3][4][5][11][16] major third,[14] quarter-comma meantone major third | 5 | M | S | |
397.10 |
E![]() ![]() |
161 : 128 | 7×23 : 27 | One-hundred-sixty-first harmonic | 23 | |||
400.00 |
E | 24/12 | 21/3 | Equal-tempered major third | 3, 12 | M | ||
402.47 |
E![]() ![]() |
323 : 256 | 17×19 : 28 | Three-hundred-twenty-third harmonic | 19 | |||
407.82 |
E+[2] | 81 : 64 | 34 : 26 | Pythagorean major third,[3][5][6][14][16] ditone | 3 | |||
417.51 |
F![]() |
14 : 11 | 2×7 : 11 | Undecimal diminished fourth or major third[3] | 11 | |||
425.00 |
217/48 | 217/48 | 17 steps in 48 equal temperament | 48 | ||||
427.37 |
F♭[2] | 32 : 25 | 25 : 52 | Just diminished fourth,[16] diminished fourth,[5][13] 25th subharmonic | 5 | |||
429.06 |
E![]() |
41 : 32 | 41 : 25 | Forty-first harmonic[5] | 41 | |||
435.08 |
E![]() |
9 : 7 | 32 : 7 | Septimal major third,[3][5] Bohlen-Pierce third,[3] Super major Third[14] | 7 | |||
444.77 |
F↓ | 128 : 99 | 27 : 9×11 | 99th subharmonic[5][6] | 11 | |||
450.00 |
E![]() ![]() |
29/24 | 29/24 | 9 steps in 24 equal temperament | 24 | |||
450.05 |
83 : 64 | 83 : 26 | Eighty-third harmonic[5] | 83 | ||||
454.21 |
F♭![]() |
13 : 10 | 13 : 2×5 | Tridecimal major third or diminished fourth | 13 | |||
456.99 |
E♯[2] | 125 : 96 | 53 : 25×3 | Just augmented third, augmented third[5] | 5 | |||
462.35 |
E![]() ![]() |
64 : 49 | 26 : 72 | 49th subharmonic[5][6] | 7 | |||
470.78 |
F![]() |
21 : 16 | 3×7 : 24 | Twenty-first harmonic, narrow fourth,[3] septimal fourth,[5] wide augmented third,[citation needed] H7 on G | 7 | |||
475.00 |
219/48 | 219/48 | 19 steps in 48 equal temperament | 48 | ||||
478.49 |
E♯+ | 675 : 512 | 33×52 : 29 | Six-hundred-seventy-fifth harmonic, wide augmented third[3] | 5 | |||
480.00 |
22/5 | 22/5 | 2 steps in 5 equal temperament | 5 | ||||
491.27 |
E![]() |
85 : 64 | 5×17 : 26 | Eighty-fifth harmonic[5] | 17 | |||
498.04 |
F[2] | 4 : 3 | 22 : 3 | Perfect fourth,[3][5][16] Pythagorean perfect fourth, Just perfect fourth or diatessaron[4] | 3 | S | ||
500.00 |
F | 25/12 | 25/12 | Equal-tempered perfect fourth | 12 | M | ||
501.42 |
F![]() |
171 : 128 | 32×19 : 27 | One-hundred-seventy-first harmonic | 19 | |||
510.51 |
(3 : 2)8/11 | 38/11 : 28/11 | Beta scale perfect fourth | 18.75 | ||||
511.52 |
F![]() |
43 : 32 | 43 : 25 | Forty-third harmonic[5] | 43 | |||
514.29 |
23/7 | 23/7 | 3 steps in 7 equal temperament | 7 | ||||
519.55 |
F+[2] | 27 : 20 | 33 : 22×5 | 5-limit wolf fourth, acute fourth,[3] imperfect fourth[16] | 5 | |||
521.51 |
E♯+++ | 177147 : 131072 | 311 : 217 | Pythagorean augmented third[3][6] (F+ (pitch)) | 3 | |||
525.00 |
27/16 | 221/48 | 21 steps in 48 equal temperament | 16, 48 | ||||
531.53 |
F![]() |
87 : 64 | 3×29 : 26 | Eighty-seventh harmonic[5] | 29 | |||
536.95 |
F↓♯+ | 15 : 11 | 3×5 : 11 | Undecimal augmented fourth[3] | 11 | |||
550.00 |
F![]() ![]() |
211/24 | 211/24 | 11 steps in 24 equal temperament | 24 | |||
551.32 |
F↑[2] | 11 : 8 | 11 : 23 | eleventh harmonic,[5] undecimal tritone,[5] lesser undecimal tritone, undecimal semi-augmented fourth[3] | 11 | |||
563.38 |
F![]() |
18 : 13 | 2×9 : 13 | Tridecimal augmented fourth[3] | 13 | |||
568.72 |
F♯[2] | 25 : 18 | 52 : 2×32 | Just augmented fourth[3][5] | 5 | |||
570.88 |
89 : 64 | 89 : 26 | Eighty-ninth harmonic[5] | 89 | ||||
575.00 |
223/48 | 223/48 | 23 steps in 48 equal temperament | 48 | ||||
582.51 |
G![]() |
7 : 5 | 7 : 5 | Lesser septimal tritone, septimal tritone[3][4][5] Huygens' tritone or Bohlen-Pierce fourth,[3] septimal fifth,[11] septimal diminished fifth[19] | 7 | |||
588.27 |
G♭−− | 1024 : 729 | 210 : 36 | Pythagorean diminished fifth,[3][6] low Pythagorean tritone[5] | 3 | |||
590.22 |
F♯+[2] | 45 : 32 | 32×5 : 25 | Just augmented fourth, just tritone,[4][11] tritone,[6] diatonic tritone,[3] 'augmented' or 'false' fourth,[16] high 5-limit tritone,[5] 1⁄6-comma meantone augmented fourth | 5 | |||
595.03 |
G![]() ![]() |
361 : 256 | 192 : 28 | Three-hundred-sixty-first harmonic | 19 | |||
600.00 |
F♯/G♭ | 26/12 | 21/2=√2 | Equal-tempered tritone | 2, 12 | M | ||
609.35 |
G![]() ![]() |
91 : 64 | 7×13 : 26 | Ninety-first harmonic[5] | 13 | |||
609.78 |
G♭−[2] | 64 : 45 | 26 : 32×5 | Just tritone,[4] 2nd tritone,[6] 'false' fifth,[16] diminished fifth,[13] low 5-limit tritone,[5] 45th subharmonic | 5 | |||
611.73 |
F♯++ | 729 : 512 | 36 : 29 | Pythagorean tritone,[3][6] Pythagorean augmented fourth, high Pythagorean tritone[5] | 3 | |||
617.49 |
F♯![]() |
10 : 7 | 2×5 : 7 | Greater septimal tritone, septimal tritone,[4][5] Euler's tritone[3] | 7 | |||
625.00 |
225/48 | 225/48 | 25 steps in 48 equal temperament | 48 | ||||
628.27 |
F![]() |
23 : 16 | 23 : 24 | Twenty-third harmonic,[5] classic diminished fifth[citation needed] | 23 | |||
631.28 |
G♭[2] | 36 : 25 | 22×32 : 52 | Just diminished fifth[5] | 5 | |||
646.99 |
F![]() |
93 : 64 | 3×31 : 26 | Ninety-third harmonic[5] | 31 | |||
648.68 |
G↓[2] | 16 : 11 | 24 : 11 | ` undecimal semi-diminished fifth[3] | 11 | |||
650.00 |
F![]() ![]() |
213/24 | 213/24 | 13 steps in 24 equal temperament | 24 | |||
665.51 |
G![]() |
47 : 32 | 47 : 25 | Forty-seventh harmonic[5] | 47 | |||
675.00 |
29/16 | 227/48 | 27 steps in 48 equal temperament | 16, 48 | ||||
678.49 |
A![]() |
262144 : 177147 | 218 : 311 | Pythagorean diminished sixth[3][6] | 3 | |||
680.45 |
G− | 40 : 27 | 23×5 : 33 | 5-limit wolf fifth,[5] or diminished sixth, grave fifth,[3][6][11] imperfect fifth,[16] | 5 | |||
683.83 |
G![]() |
95 : 64 | 5×19 : 26 | Ninety-fifth harmonic[5] | 19 | |||
684.82 |
E![]() ![]() ![]() ![]() |
12167 : 8192 | 233 : 213 | 12167th harmonic | 23 | |||
685.71 |
24/7 : 1 | 4 steps in 7 equal temperament | ||||||
691.20 |
3:2÷(81:80)1/2 | 2×51/2 : 3 | Half-comma meantone perfect fifth | M | ||||
694.79 |
3:2÷(81:80)1/3 | 21/3×51/3 : 31/3 | 1⁄3-comma meantone perfect fifth | M | ||||
695.81 |
3:2÷(81:80)2/7 | 21/7×52/7 : 31/7 | 2⁄7-comma meantone perfect fifth | M | ||||
696.58 |
3:2÷(81:80)1/4 | 51/4 | Quarter-comma meantone perfect fifth | M | ||||
697.65 |
3:2÷(81:80)1/5 | 31/5×51/5 : 21/5 | 1⁄5-comma meantone perfect fifth | M | ||||
698.37 |
3:2÷(81:80)1/6 | 31/3×51/6 : 21/3 | 1⁄6-comma meantone perfect fifth | M | ||||
700.00 |
G | 27/12 | 27/12 | Equal-tempered perfect fifth | 12 | M | ||
701.89 |
231/53 | 231/53 | 53-TET perfect fifth | 53 | ||||
701.96 |
G[2] | 3 : 2 | 3 : 2 | Perfect fifth,[3][5][16] Pythagorean perfect fifth, Just perfect fifth or diapente,[4] fifth,[14] Just fifth[11] | 3 | S | ||
702.44 |
224/41 | 224/41 | 41-TET perfect fifth | 41 | ||||
703.45 |
217/29 | 217/29 | 29-TET perfect fifth | 29 | ||||
719.90 |
97 : 64 | 97 : 26 | Ninety-seventh harmonic[5] | 97 | ||||
720.00 |
23/5 : 1 | 3 steps in 5 equal temperament | 5 | |||||
721.51 |
A![]() |
1024 : 675 | 210 : 33×52 | Narrow diminished sixth[3] | 5 | |||
725.00 |
229/48 | 229/48 | 29 steps in 48 equal temperament | 48 | ||||
729.22 |
G![]() |
32 : 21 | 24 : 3×7 | 21st subharmonic,[5][6] septimal diminished sixth | 7 | |||
733.23 |
F![]() ![]() ![]() |
391 : 256 | 17×23 : 28 | Three-hundred-ninety-first harmonic | 23 | |||
737.65 |
A![]() ![]() |
49 : 32 | 7×7 : 25 | Forty-ninth harmonic[5] | 7 | |||
743.01 |
A![]() |
192 : 125 | 26×3 : 53 | Classic diminished sixth[3] | 5 | |||
750.00 |
G![]() ![]() |
215/24 | 215/24 | 15 steps in 24 equal temperament | 24 | |||
755.23 |
G↑ | 99 : 64 | 32×11 : 26 | Ninety-ninth harmonic[5] | 11 | |||
764.92 |
A![]() |
14 : 9 | 2×7 : 32 | Septimal minor sixth[3][5] | 7 | |||
772.63 |
G♯ | 25 : 16 | 52 : 24 | Just augmented fifth[5][16] | 5 | |||
775.00 |
231/48 | 231/48 | 31 steps in 48 equal temperament | 48 | ||||
781.79 |
π : 2 | Wallis product | ||||||
782.49 |
G![]() |
11 : 7 | 11 : 7 | Undecimal minor sixth,[5] undecimal augmented fifth,[3] Fibonacci numbers | 11 | |||
789.85 |
101 : 64 | 101 : 26 | Hundred-first harmonic[5] | 101 | ||||
792.18 |
A♭−[2] | 128 : 81 | 27 : 34 | Pythagorean minor sixth,[3][5][6] 81st subharmonic | 3 | |||
798.40 |
A![]() ![]() |
203 : 128 | 7×29 : 27 | Two-hundred-third harmonic | 29 | |||
800.00 |
G♯/A♭ | 28/12 | 22/3 | Equal-tempered minor sixth | 3, 12 | M | ||
806.91 |
G![]() |
51 : 32 | 3×17 : 25 | Fifty-first harmonic[5] | 17 | |||
813.69 |
A♭[2] | 8 : 5 | 23 : 5 | Just minor sixth[3][4][11][16] | 5 | |||
815.64 |
G♯++ | 6561 : 4096 | 38 : 212 | Pythagorean augmented fifth,[3][6] Pythagorean 'schismatic' sixth[5] | 3 | |||
823.80 |
103 : 64 | 103 : 26 | Hundred-third harmonic[5] | 103 | ||||
825.00 |
211/16 | 233/48 | 33 steps in 48 equal temperament | 16, 48 | ||||
832.18 |
G![]() |
207 : 128 | 32×23 : 27 | Two-hundred-seventh harmonic | 23 | |||
833.09 |
(51/2+1)/2 | φ : 1 | Golden ratio (833 cents scale) | |||||
835.19 |
A♭+ | 81 : 50 | 34 : 2×52 | Acute minor sixth[3] | 5 | |||
840.53 |
A![]() |
13 : 8 | 13 : 23 | Tridecimal neutral sixth,[3] overtone sixth,[5] thirteenth harmonic | 13 | |||
848.83 |
A![]() |
209 : 128 | 11×19 : 27 | Two-hundred-ninth harmonic | 19 | |||
850.00 |
G![]() ![]() |
217/24 | 217/24 | Equal-tempered neutral sixth | 24 | |||
852.59 |
A↓+[2] | 18 : 11 | 2×32 : 11 | Undecimal neutral sixth,[3][5] Zalzal's neutral sixth | 11 | |||
857.09 |
A![]() |
105 : 64 | 3×5×7 : 26 | Hundred-fifth harmonic[5] | 7 | |||
857.14 |
25/7 | 25/7 | 5 steps in 7 equal temperament | 7 | ||||
862.85 |
A− | 400 : 243 | 24×52 : 35 | Grave major sixth[3] | 5 | |||
873.50 |
A![]() |
53 : 32 | 53 : 25 | Fifty-third harmonic[5] | 53 | |||
875.00 |
235/48 | 235/48 | 35 steps in 48 equal temperament | 48 | ||||
879.86 |
A↓![]() |
128 : 77 | 27 : 7×11 | 77th subharmonic[5][6] | 11 | |||
882.40 |
B![]() |
32768 : 19683 | 215 : 39 | Pythagorean diminished seventh[3][6] | 3 | |||
884.36 |
A[2] | 5 : 3 | 5 : 3 | Just major sixth,[3][4][5][11][16] Bohlen-Pierce sixth,[3] 1⁄3-comma meantone major sixth | 5 | M | ||
889.76 |
107 : 64 | 107 : 26 | Hundred-seventh harmonic[5] | 107 | ||||
892.54 |
B![]() ![]() ![]() ![]() |
6859 : 4096 | 193 : 212 | 6859th harmonic | 19 | |||
900.00 |
A | 29/12 | 23/4 | Equal-tempered major sixth | 4, 12 | M | ||
902.49 |
A![]() |
32 : 19 | 25 : 19 | 19th subharmonic[5][6] | 19 | |||
905.87 |
A+[2] | 27 : 16 | 33 : 24 | Pythagorean major sixth[3][5][11][16] | 3 | |||
921.82 |
109 : 64 | 109 : 26 | Hundred-ninth harmonic[5] | 109 | ||||
925.00 |
237/48 | 237/48 | 37 steps in 48 equal temperament | 48 | ||||
925.42 |
B![]() |
128 : 75 | 27 : 3×52 | Just diminished seventh,[16] diminished seventh,[5][13] 75th subharmonic | 5 | |||
925.79 |
A![]() ![]() |
437 : 256 | 19×23 : 28 | Four-hundred-thirty-seventh harmonic | 23 | |||
933.13 |
A![]() |
12 : 7 | 22×3 : 7 | Septimal major sixth[3][4][5] | 7 | |||
937.63 |
A↑ | 55 : 32 | 5×11 : 25 | Fifty-fifth harmonic[5][20] | 11 | |||
950.00 |
A![]() ![]() |
219/24 | 219/24 | 19 steps in 24 equal temperament | 24 | |||
953.30 |
A![]() |
111 : 64 | 3×37 : 26 | Hundred-eleventh harmonic[5] | 37 | |||
955.03 |
A♯[2] | 125 : 72 | 53 : 23×32 | Just augmented sixth[5] | 5 | |||
957.21 |
(3 : 2)15/11 | 315/11 : 215/11 | 15 steps in Beta scale | 18.75 | ||||
960.00 |
24/5 | 24/5 | 4 steps in 5 equal temperament | 5 | ||||
968.83 |
B![]() |
7 : 4 | 7 : 22 | Septimal minor seventh,[4][5][11] harmonic seventh,[3][11] augmented sixth[citation needed] | 7 | |||
975.00 |
213/16 | 239/48 | 39 steps in 48 equal temperament | 16, 48 | ||||
976.54 |
A♯+[2] | 225 : 128 | 32×52 : 27 | Just augmented sixth[16] | 5 | |||
984.21 |
113 : 64 | 113 : 26 | Hundred-thirteenth harmonic[5] | 113 | ||||
996.09 |
B♭−[2] | 16 : 9 | 24 : 32 | Pythagorean minor seventh,[3] Small just minor seventh,[4] lesser minor seventh,[16] just minor seventh,[11] Pythagorean small minor seventh[5] | 3 | |||
999.47 |
B![]() |
57 : 32 | 3×19 : 25 | Fifty-seventh harmonic[5] | 19 | |||
1000.00 |
A♯/B♭ | 210/12 | 25/6 | Equal-tempered minor seventh | 6, 12 | M | ||
1014.59 |
A![]() |
115 : 64 | 5×23 : 26 | Hundred-fifteenth harmonic[5] | 23 | |||
1017.60 |
B♭[2] | 9 : 5 | 32 : 5 | Greater just minor seventh,[16] large just minor seventh,[4][5] Bohlen-Pierce seventh[3] | 5 | |||
1019.55 |
A♯+++ | 59049 : 32768 | 310 : 215 | Pythagorean augmented sixth[3][6] | 3 | |||
1025.00 |
241/48 | 241/48 | 41 steps in 48 equal temperament | 48 | ||||
1028.57 |
26/7 | 26/7 | 6 steps in 7 equal temperament | 7 | ||||
1029.58 |
B![]() |
29 : 16 | 29 : 24 | Twenty-ninth harmonic,[5] minor seventh[citation needed] | 29 | |||
1035.00 |
B↓[2] | 20 : 11 | 22×5 : 11 | Lesser undecimal neutral seventh, large minor seventh[3] | 11 | |||
1039.10 |
B♭+ | 729 : 400 | 36 : 24×52 | Acute minor seventh[3] | 5 | |||
1044.44 |
B![]() |
117 : 64 | 32×13 : 26 | Hundred-seventeenth harmonic[5] | 13 | |||
1044.86 |
B![]() |
64 : 35 | 26 : 5×7 | 35th subharmonic,[5] septimal neutral seventh[6] | 7 | |||
1049.36 |
B↑♭−[2] | 11 : 6 | 11 : 2×3 | 21⁄4-tone or Undecimal neutral seventh,[3] undecimal 'median' seventh[5] | 11 | |||
1050.00 |
A![]() ![]() |
221/24 | 27/8 | Equal-tempered neutral seventh | 8, 24 | |||
1059.17 |
59 : 32 | 59 : 25 | Fifty-ninth harmonic[5] | 59 | ||||
1066.76 |
B− | 50 : 27 | 2×52 : 33 | Grave major seventh[3] | 5 | |||
1071.70 |
B![]() ![]() |
13 : 7 | 13 : 7 | Tridecimal neutral seventh[21] | 13 | |||
1073.78 |
B![]() ![]() |
119 : 64 | 7×17 : 26 | Hundred-nineteenth harmonic[5] | 17 | |||
1075.00 |
243/48 | 243/48 | 43 steps in 48 equal temperament | 48 | ||||
1086.31 |
C′♭−− | 4096 : 2187 | 212 : 37 | Pythagorean diminished octave[3][6] | 3 | |||
1088.27 |
B[2] | 15 : 8 | 3×5 : 23 | Just major seventh,[3][5][11][16] small just major seventh,[4] 1⁄6-comma meantone major seventh | 5 | |||
1095.04 |
C![]() |
32 : 17 | 25 : 17 | 17th subharmonic[5][6] | 17 | |||
1100.00 |
B | 211/12 | 211/12 | Equal-tempered major seventh | 12 | M | ||
1102.64 |
B↑↑♭- | 121 : 64 | 112 : 26 | Hundred-twenty-first harmonic[5] | 11 | |||
1107.82 |
C′♭− | 256 : 135 | 28 : 33×5 | Octave − major chroma,[3] 135th subharmonic, narrow diminished octave[citation needed] | 5 | |||
1109.78 |
B+[2] | 243 : 128 | 35 : 27 | Pythagorean major seventh[3][5][6][11] | 3 | |||
1116.88 |
61 : 32 | 61 : 25 | Sixty-first harmonic[5] | 61 | ||||
1125.00 |
215/16 | 245/48 | 45 steps in 48 equal temperament | 16, 48 | ||||
1129.33 |
C′♭[2] | 48 : 25 | 24×3 : 52 | Classic diminished octave,[3][6] large just major seventh[4] | 5 | |||
1131.02 |
B![]() |
123 : 64 | 3×41 : 26 | Hundred-twenty-third harmonic[5] | 41 | |||
1137.04 |
B![]() |
27 : 14 | 33 : 2×7 | Septimal major seventh[5] | 7 | |||
1138.04 |
C![]() ![]() |
247 : 128 | 13×19 : 27 | Two-hundred-forty-seventh harmonic | 19 | |||
1145.04 |
B![]() |
31 : 16 | 31 : 24 | Thirty-first harmonic,[5] augmented seventh[citation needed] | 31 | |||
1146.73 |
C↓ | 64 : 33 | 26 : 3×11 | 33rd subharmonic[6] | 11 | |||
1150.00 |
B![]() ![]() |
223/24 | 223/24 | 23 steps in 24 equal temperament | 24 | |||
1151.23 |
C![]() |
35 : 18 | 5×7 : 2×32 | Septimal supermajor seventh, septimal quarter tone inverted | 7 | |||
1158.94 |
B♯[2] | 125 : 64 | 53 : 26 | Just augmented seventh,[5] 125th harmonic | 5 | |||
1172.74 |
C![]() |
63 : 32 | 32×7 : 25 | Sixty-third harmonic[5] | 7 | |||
1175.00 |
247/48 | 247/48 | 47 steps in 48 equal temperament | 48 | ||||
1178.49 |
C′− | 160 : 81 | 25×5 : 34 | Octave − syntonic comma,[3] semi-diminished octave[citation needed] | 5 | |||
1179.59 |
B![]() |
253 : 128 | 11×23 : 27 | Two-hundred-fifty-third harmonic[5] | 23 | |||
1186.42 |
127 : 64 | 127 : 26 | Hundred-twenty-seventh harmonic[5] | 127 | ||||
1200.00 |
C′ | 2 : 1 | 2 : 1 | Octave[3][11] or diapason[4] | 1, 12 | 3 | M | S |
See also
[edit]Notes
[edit]- ^ a b c d Maneri-Sims notation
References
[edit]- ^ a b Fox, Christopher (2003). "Microtones and Microtonalities", Contemporary Music Review, v. 22, pt. 1–2. (Abingdon, Oxfordshire, UK: Routledge): p. 13.
- ^ 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): 106–137.
- ^ 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 co cp cq cr cs ct cu cv cw cx cy cz da db dc dd de df dg dh di "List of intervals", Huygens-Fokker Foundation. The Foundation uses "classic" to indicate "just" or leaves off any adjective, as in "major sixth".
- ^ 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. pp. 68–69. ISBN 978-0-306-80106-8.
- ^ 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 co cp cq cr cs ct cu cv cw cx cy cz da db dc dd de df dg dh di dj dk dl dm dn do dp dq dr ds dt du dv dw dx dy dz "Anatomy of an Octave", Kyle Gann (1998). Gann leaves off "just" but includes "5-limit". He uses "median" for "neutral".
- ^ 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 Haluška, Ján (2003). The Mathematical Theory of Tone Systems, pp. xxv–xxix. ISBN 978-0-8247-4714-5.
- ^ Ellis, Alexander J.; Hipkins, Alfred J. (1884). "Tonometrical Observations on Some Existing Non-Harmonic Musical Scales". Proceedings of the Royal Society of London. 37 (232–234): 368–385. doi:10.1098/rspl.1884.0041. JSTOR 114325. S2CID 122407786.
- ^ "Logarithmic Interval Measures", Huygens-Fokker Foundation. Accessed 2015-06-06.
- ^ "Orwell Temperaments", Xenharmony.org.
- ^ a b Partch 1979, p. 70
- ^ 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 Alexander John Ellis (March 1885). On the musical scales of various nations, p. 488. Journal of the Society of Arts, vol. XXXII, no. 1688
- ^ William Smythe Babcock Mathews (1895). Pronouncing Dictionary and Condensed Encyclopedia of Musical Terms, p. 13. ISBN 1-112-44188-3.
- ^ a b c d e f Anger, Joseph Humfrey (1912). A Treatise on Harmony, with Exercises, Volume 3, pp. xiv–xv. W. Tyrrell.
- ^ a b c d e f g h i j k l m n o 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. [ISBN unspecified]
- ^ A. R. Meuss (2004). Intervals, Scales, Tones and the Concert Pitch C. Temple Lodge Publishing. p. 15. ISBN 1902636465.
- ^ 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".
- ^ a b "13th-harmonic", 31et.com.
- ^ Brabner, John H. F. (1884). The National Encyclopaedia, vol. 13, p. 182. London. [ISBN unspecified]
- ^ Sabat, Marc and von Schweinitz, Wolfgang (2004). "The Extended Helmholtz-Ellis JI Pitch Notation" [PDF], NewMusicBox. Accessed: 15 March 2014.
- ^ Hermann L. F. von Helmholtz (2007). On the Sensations of Tone, p. 456. ISBN 978-1-60206-639-7.
- ^ "Gallery of Just Intervals", Xenharmonic Wiki.
External links
[edit]- "Names of seven-limit commas", XenHarmony.org. (Archived copy)
- "List of Overtones", Xenharmonic Wiki.
- "All Known Musical Intervals" (by Dale Pond), Svpvril.com.