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.
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 | play Unison, [3] monophony, [4] perfect prime, [3] tonic, [5] or fundamental | 1, 12 | 3 | M | |
0.03 | 65537 : 65536 | 65537 : 216 | play Sixty-five-thousand-five-hundred-thirty-seventh harmonic | 65537 | S | |||
0.40 | C ♯− | 4375 : 4374 | 54×7 : 2×37 | play Ragisma [3] [6] | 7 | S | ||
0.72 | E + | 2401 : 2400 | 74 : 25×3×52 | play Breedsma [3] [6] | 7 | S | ||
1.00 | 21/1200 | 21/1200 | play Cent [7] | 1200 | ||||
1.20 | 21/1000 | 21/1000 | play Millioctave | 1000 | ||||
1.95 | B♯++ | 32805 : 32768 | 38×5 : 215 | play Schisma [3] [5] | 5 | |||
1.96 | 3:2÷(27/12) | 3 : 219/12 | Grad, Werckmeister [8] | |||||
3.99 | 101/1000 | 21/1000×51/1000 | play Savart or eptaméride | 301.03 | ||||
7.71 | B ♯ | 225 : 224 | 32×52 : 25×7 | play Septimal kleisma, [3] [6] marvel comma | 7 | S | ||
8.11 | B − | 15625 : 15552 | 56 : 26×35 | play Kleisma or semicomma majeur [3] [6] | 5 | |||
10.06 | A ++ | 2109375 : 2097152 | 33×57 : 221 | play Semicomma, [3] [6] Fokker's comma [3] | 5 | |||
10.85 | C | 160 : 159 | 25×5 : 3×53 | play Difference between 5:3 & 53:32 | 53 | S | ||
11.98 | C | 145 : 144 | 5×29 : 24×32 | play Difference between 29:16 & 9:5 | 29 | S | ||
12.50 | 21/96 | 21/96 | play Sixteenth tone | 96 | ||||
13.07 | B − | 1728 : 1715 | 26×33 : 5×73 | play Orwell comma [3] [9] | 7 | |||
13.47 | C | 129 : 128 | 3×43 : 27 | play Hundred-twenty-ninth harmonic | 43 | S | ||
13.79 | D | 126 : 125 | 2×32×7 : 53 | play Small septimal semicomma, [6] small septimal comma, [3] starling comma | 7 | S | ||
14.37 | C♭↑↑− | 121 : 120 | 112 : 23×3×5 | play Undecimal seconds comma [3] | 11 | S | ||
16.67 | C↑ [lower-alpha 1] | 21/72 | 21/72 | play 1 step in 72 equal temperament | 72 | |||
18.13 | C | 96 : 95 | 25×3 : 5×19 | play Difference between 19:16 & 6:5 | 19 | S | ||
19.55 | D -- [2] | 2048 : 2025 | 211 : 34×52 | play Diaschisma, [3] [6] minor comma | 5 | |||
21.51 | C+ [2] | 81 : 80 | 34 : 24×5 | play 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 | play Holdrian comma, Holder's comma, 1 step in 53 equal temperament | 53 | ||||
23.46 | B♯+++ | 531441 : 524288 | 312 : 219 | play Pythagorean comma, [3] [5] [6] [10] [11] ditonic comma [3] [6] | 3 | |||
25.00 | 21/48 | 21/48 | play Eighth tone | 48 | ||||
26.84 | C | 65 : 64 | 5×13 : 26 | play Sixty-fifth harmonic, [5] 13th-partial chroma [3] | 13 | S | ||
27.26 | C − | 64 : 63 | 26 : 32×7 | play Septimal comma, [3] [6] [11] Archytas' comma, [3] 63rd subharmonic | 7 | S | ||
29.27 | 21/41 | 21/41 | play 1 step in 41 equal temperament | 41 | ||||
31.19 | D ♭↓ | 56 : 55 | 23×7 : 5×11 | play Undecimal diesis, [3] Ptolemy's enharmonic: [5] difference between (11 : 8) and (7 : 5) tritone | 11 | S | ||
33.33 | C /D♭ [lower-alpha 1] | 21/36 | 21/36 | play Sixth tone | 36, 72 | |||
34.28 | C | 51 : 50 | 3×17 : 2×52 | play Difference between 17:16 & 25:24 | 17 | S | ||
34.98 | B ♯- | 50 : 49 | 2×52 : 72 | play Septimal sixth tone or jubilisma, Erlich's decatonic comma or tritonic diesis [3] [6] | 7 | S | ||
35.70 | D ♭ | 49 : 48 | 72 : 24×3 | play Septimal diesis, slendro diesis or septimal 1/6-tone [3] | 7 | S | ||
38.05 | C | 46 : 45 | 2×23 : 32×5 | play Inferior quarter tone, [5] difference between 23:16 & 45:32 | 23 | S | ||
38.71 | 21/31 | 21/31 | play 1 step in 31 equal temperament | 31 | ||||
38.91 | C↓♯+ | 45 : 44 | 32×5 : 4×11 | play Undecimal diesis or undecimal fifth tone | 11 | S | ||
40.00 | 21/30 | 21/30 | play Fifth tone | 30 | ||||
41.06 | D − | 128 : 125 | 27 : 53 | play 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 | play Lesser 41-limit fifth tone | 41 | S | ||
42.75 | C | 41 : 40 | 41 : 23×5 | play Greater 41-limit fifth tone | 41 | S | ||
43.83 | C ♯ | 40 : 39 | 23×5 : 3×13 | play Tridecimal fifth tone | 13 | S | ||
44.97 | C | 39 : 38 | 3×13 : 2×19 | play Superior quarter-tone, [5] novendecimal fifth tone | 19 | S | ||
46.17 | D - | 38 : 37 | 2×19 : 37 | play Lesser 37-limit quarter tone | 37 | S | ||
47.43 | C ♯ | 37 : 36 | 37 : 22×32 | play Greater 37-limit quarter tone | 37 | S | ||
48.77 | C | 36 : 35 | 22×32 : 5×7 | play Septimal quarter tone, septimal diesis, [3] [6] septimal chroma, [2] superior quarter tone [5] | 7 | S | ||
49.98 | 246 : 239 | 3×41 : 239 | play Just quarter tone [11] | 239 | ||||
50.00 | C /D | 21/24 | 21/24 | play Equal-tempered quarter tone | 24 | |||
50.18 | D ♭ | 35 : 34 | 5×7 : 2×17 | play ET quarter-tone approximation, [5] lesser 17-limit quarter tone | 17 | S | ||
50.72 | B ♯++ | 59049 : 57344 | 310 : 213×7 | play Harrison's comma (10 P5s – 1 H7) [3] | 7 | |||
51.68 | C ↓♯ | 34 : 33 | 2×17 : 3×11 | play Greater 17-limit quarter tone | 17 | S | ||
53.27 | C↑ | 33 : 32 | 3×11 : 25 | play Thirty-third harmonic, [5] undecimal comma, undecimal quarter tone | 11 | S | ||
54.96 | D ♭- | 32 : 31 | 25 : 31 | play Inferior quarter-tone, [5] thirty-first subharmonic | 31 | S | ||
56.55 | B ♯+ | 529 : 512 | 232 : 29 | play Five-hundred-twenty-ninth harmonic | 23 | |||
56.77 | C | 31 : 30 | 31 : 2×3×5 | play Greater quarter-tone, [5] difference between 31:16 & 15:8 | 31 | S | ||
58.69 | C ♯ | 30 : 29 | 2×3×5 : 29 | play Lesser 29-limit quarter tone | 29 | S | ||
60.75 | C | 29 : 28 | 29 : 22×7 | play Greater 29-limit quarter tone | 29 | S | ||
62.96 | D ♭- | 28 : 27 | 22×7 : 33 | play Septimal minor second, small minor second, inferior quarter tone [5] | 7 | S | ||
63.81 | (3 : 2)1/11 | 31/11 : 21/11 | play Beta scale step | 18.75 | ||||
65.34 | C ♯+ | 27 : 26 | 33 : 2×13 | play Chromatic diesis, [12] tridecimal comma [3] | 13 | S | ||
66.34 | D ♭ | 133 : 128 | 7×19 : 27 | play One-hundred-thirty-third harmonic | 19 | |||
66.67 | C ↑/C♯ [lower-alpha 1] | 21/18 | 21/18 | play Third tone | 18, 36, 72 | |||
67.90 | D - | 26 : 25 | 2×13 : 52 | play Tridecimal third tone, third tone [5] | 13 | S | ||
70.67 | C♯ [2] | 25 : 24 | 52 : 23×3 | play 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 | play Lesser 23-limit semitone | 23 | S | ||
75.00 | 21/16 | 23/48 | play 1 step in 16 equal temperament, 3 steps in 48 | 16, 48 | ||||
76.96 | C ↓♯+ | 23 : 22 | 23 : 2×11 | play Greater 23-limit semitone | 23 | S | ||
78.00 | (3 : 2)1/9 | 31/9 : 21/9 | play Alpha scale step | 15.39 | ||||
79.31 | 67 : 64 | 67 : 26 | play Sixty-seventh harmonic [5] | 67 | ||||
80.54 | C↑ - | 22 : 21 | 2×11 : 3×7 | play Hard semitone, [5] two-fifth tone small semitone | 11 | S | ||
84.47 | D ♭ | 21 : 20 | 3×7 : 22×5 | play Septimal chromatic semitone, minor semitone [3] | 7 | S | ||
88.80 | C ♯ | 20 : 19 | 22×5 : 19 | play Novendecimal augmented unison | 19 | S | ||
90.22 | D♭−− [2] | 256 : 243 | 28 : 35 | play Pythagorean minor second or limma, [3] [6] [11] Pythagorean diatonic semitone, Low Semitone [14] | 3 | |||
92.18 | C♯+ [2] | 135 : 128 | 33×5 : 27 | play 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 play | 19 | S | ||
97.36 | D↓↓ | 128 : 121 | 27 : 112 | play 121st subharmonic, [5] [6] undecimal minor second | 11 | |||
98.95 | D ♭ | 18 : 17 | 2×32 : 17 | play Just minor semitone, Arabic lute index finger [3] | 17 | S | ||
100.00 | C♯/D♭ | 21/12 | 21/12 | play Equal-tempered minor second or semitone | 12 | M | ||
104.96 | C ♯ [2] | 17 : 16 | 17 : 24 | play 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 | play Studie II interval (compound just major third, 5:1, divided into 25 equal parts) | 25 | ||||
111.73 | D♭- [2] | 16 : 15 | 24 : 3×5 | play 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 | play 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 | play Secor | 10.28 | ||||
119.44 | C ♯ | 15 : 14 | 3×5 : 2×7 | play Septimal diatonic semitone, major diatonic semitone, [3] Cowell semitone [5] | 7 | S | ||
125.00 | 25/48 | 25/48 | play 5 steps in 48 equal temperament | 48 | ||||
128.30 | D | 14 : 13 | 2×7 : 13 | play Lesser tridecimal 2/3-tone [17] | 13 | S | ||
130.23 | C ♯+ | 69 : 64 | 3×23 : 26 | play Sixty-ninth harmonic [5] | 23 | |||
133.24 | D♭ | 27 : 25 | 33 : 52 | play 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♯ /D♭ [lower-alpha 1] | 21/9 | 22/18 | play Two-third tone | 9, 18, 36, 72 | |||
138.57 | D ♭- | 13 : 12 | 13 : 22×3 | play Greater tridecimal 2/3-tone, [17] Three-quarter tone [5] | 13 | S | ||
150.00 | C /D | 23/24 | 21/8 | play Equal-tempered neutral second | 8, 24 | |||
150.64 | D↓ [2] | 12 : 11 | 22×3 : 11 | play 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 | play Thirty-fifth harmonic [5] | 7 | |||
160.90 | D−− | 800 : 729 | 25×52 : 36 | play Grave whole tone, [3] neutral second, grave major second[ citation needed ] | 5 | |||
165.00 | D↑♭− [2] | 11 : 10 | 11 : 2×5 | play Greater undecimal minor/major/neutral second, 4/5-tone [6] or Ptolemy's second [3] | 11 | S | ||
171.43 | 21/7 | 21/7 | play 1 step in 7 equal temperament | 7 | ||||
175.00 | 27/48 | 27/48 | play 7 steps in 48 equal temperament | 48 | ||||
179.70 | 71 : 64 | 71 : 26 | play Seventy-first harmonic [5] | 71 | ||||
180.45 | E −−− | 65536 : 59049 | 216 : 310 | play Pythagorean diminished third, [3] [6] Pythagorean minor tone | 3 | |||
182.40 | D− [2] | 10 : 9 | 2×5 : 32 | play 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 | play Equal-tempered major second | 6, 12 | M | ||
203.91 | D [2] | 9 : 8 | 32 : 23 | play 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 | play Hundred-forty-fifth harmonic | 29 | |||
223.46 | E − [2] | 256 : 225 | 28 : 32×52 | play Just diminished third, [16] 225th subharmonic | 5 | |||
225.00 | 23/16 | 29/48 | play 9 steps in 48 equal temperament | 16, 48 | ||||
227.79 | 73 : 64 | 73 : 26 | play Seventy-third harmonic [5] | 73 | ||||
231.17 | D − [2] | 8 : 7 | 23 : 7 | play Septimal major second, [4] septimal whole tone [3] [5] | 7 | S | ||
240.00 | 21/5 | 21/5 | play 1 step in 5 equal temperament | 5 | ||||
247.74 | D ♯ | 15 : 13 | 3×5 : 13 | play Tridecimal 5⁄4 tone [3] | 13 | |||
250.00 | D /E | 25/24 | 25/24 | play 5 steps in 24 equal temperament | 24 | |||
251.34 | D ♯ | 37 : 32 | 37 : 25 | play Thirty-seventh harmonic [5] | 37 | |||
253.08 | D♯− | 125 : 108 | 53 : 22×33 | play Semi-augmented whole tone, [3] semi-augmented second[ citation needed ] | 5 | |||
262.37 | E↓♭ | 64 : 55 | 26 : 5×11 | play 55th subharmonic [5] [6] | 11 | |||
266.87 | E ♭ [2] | 7 : 6 | 7 : 2×3 | play Septimal minor third [3] [4] [11] or Sub minor third [14] | 7 | S | ||
268.80 | D | 299 : 256 | 13×23 : 28 | play Two-hundred-ninety-ninth harmonic | 23 | |||
274.58 | D♯ [2] | 75 : 64 | 3×52 : 26 | play Just augmented second, [16] Augmented tone, [14] augmented second [5] [13] | 5 | |||
275.00 | 211/48 | 211/48 | play 11 steps in 48 equal temperament | 48 | ||||
289.21 | E ↓♭ | 13 : 11 | 13 : 11 | play Tridecimal minor third [3] | 13 | |||
294.13 | E♭− [2] | 32 : 27 | 25 : 33 | play Pythagorean minor third [3] [5] [6] [14] [16] semiditone, or 27th subharmonic | 3 | |||
297.51 | E ♭ [2] | 19 : 16 | 19 : 24 | play 19th harmonic, [3] 19-limit minor third, overtone minor third [5] | 19 | |||
300.00 | D♯/E♭ | 23/12 | 21/4 | play Equal-tempered minor third | 4, 12 | M | ||
301.85 | D ♯- | 25 : 21 [5] | 52 : 3×7 | play 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 | play Quarter-comma meantone minor third | M | ||||
311.98 | (3 : 2)4/9 | 34/9 : 24/9 | play Alpha scale minor third | 3.85 | ||||
315.64 | E♭ [2] | 6 : 5 | 2×3 : 5 | play 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 | play Pythagorean augmented second [3] [6] | 3 | |||
320.14 | E ♭↑ | 77 : 64 | 7×11 : 26 | play Seventy-seventh harmonic [5] | 11 | |||
325.00 | 213/48 | 213/48 | play 13 steps in 48 equal temperament | 48 | ||||
336.13 | D ♯- | 17 : 14 | 17 : 2×7 | play Superminor third [18] | 17 | |||
337.15 | E♭+ | 243 : 200 | 35 : 23×52 | play Acute minor third [3] | 5 | |||
342.48 | E ♭ | 39 : 32 | 3×13 : 25 | play Thirty-ninth harmonic [5] | 13 | |||
342.86 | 22/7 | 22/7 | play 2 steps in 7 equal temperament | 7 | ||||
342.91 | E ♭- | 128 : 105 | 27 : 3×5×7 | play 105th subharmonic, [5] septimal neutral third [6] | 7 | |||
347.41 | E↑♭− [2] | 11 : 9 | 11 : 32 | play Undecimal neutral third [3] [5] | 11 | |||
350.00 | D /E | 27/24 | 27/24 | play Equal-tempered neutral third | 24 | |||
354.55 | E↓+ | 27 : 22 | 33 : 2×11 | play Zalzal's wosta [6] 12:11 X 9:8 [14] | 11 | |||
359.47 | E [2] | 16 : 13 | 24 : 13 | play Tridecimal neutral third [3] | 13 | |||
364.54 | 79 : 64 | 79 : 26 | play Seventy-ninth harmonic [5] | 79 | ||||
364.81 | E− | 100 : 81 | 22×52 : 34 | play Grave major third [3] | 5 | |||
375.00 | 25/16 | 215/48 | play 15 steps in 48 equal temperament | 16, 48 | ||||
384.36 | F♭−− | 8192 : 6561 | 213 : 38 | play Pythagorean diminished fourth, [3] [6] Pythagorean 'schismatic' third [5] | 3 | |||
386.31 | E [2] | 5 : 4 | 5 : 22 | play 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 | play One-hundred-sixty-first harmonic | 23 | |||
400.00 | E | 24/12 | 21/3 | play Equal-tempered major third | 3, 12 | M | ||
402.47 | E | 323 : 256 | 17×19 : 28 | play Three-hundred-twenty-third harmonic | 19 | |||
407.82 | E+ [2] | 81 : 64 | 34 : 26 | play Pythagorean major third, [3] [5] [6] [14] [16] ditone | 3 | |||
417.51 | F ↓+ [2] | 14 : 11 | 2×7 : 11 | play Undecimal diminished fourth or major third [3] | 11 | |||
425.00 | 217/48 | 217/48 | play 17 steps in 48 equal temperament | 48 | ||||
427.37 | F♭ [2] | 32 : 25 | 25 : 52 | play Just diminished fourth, [16] diminished fourth, [5] [13] 25th subharmonic | 5 | |||
429.06 | E | 41 : 32 | 41 : 25 | play Forty-first harmonic [5] | 41 | |||
435.08 | E [2] | 9 : 7 | 32 : 7 | play Septimal major third, [3] [5] Bohlen-Pierce third, [3] Super major Third [14] | 7 | |||
444.77 | F↓ | 128 : 99 | 27 : 9×11 | play 99th subharmonic [5] [6] | 11 | |||
450.00 | E /F | 29/24 | 29/24 | play 9 steps in 24 equal temperament | 24 | |||
450.05 | 83 : 64 | 83 : 26 | play Eighty-third harmonic [5] | 83 | ||||
454.21 | F♭ | 13 : 10 | 13 : 2×5 | play Tridecimal major third or diminished fourth | 13 | |||
456.99 | E♯ [2] | 125 : 96 | 53 : 25×3 | play Just augmented third, augmented third [5] | 5 | |||
462.35 | E - | 64 : 49 | 26 : 72 | play 49th subharmonic [5] [6] | 7 | |||
470.78 | F + [2] | 21 : 16 | 3×7 : 24 | play Twenty-first harmonic, narrow fourth, [3] septimal fourth, [5] wide augmented third,[ citation needed ] H7 on G | 7 | |||
475.00 | 219/48 | 219/48 | play 19 steps in 48 equal temperament | 48 | ||||
478.49 | E♯+ | 675 : 512 | 33×52 : 29 | play Six-hundred-seventy-fifth harmonic, wide augmented third [3] | 5 | |||
480.00 | 22/5 | 22/5 | play 2 steps in 5 equal temperament | 5 | ||||
491.27 | E ♯ | 85 : 64 | 5×17 : 26 | play Eighty-fifth harmonic [5] | 17 | |||
498.04 | F [2] | 4 : 3 | 22 : 3 | play Perfect fourth, [3] [5] [16] Pythagorean perfect fourth, Just perfect fourth or diatessaron [4] | 3 | S | ||
500.00 | F | 25/12 | 25/12 | play Equal-tempered perfect fourth | 12 | M | ||
501.42 | F + | 171 : 128 | 32×19 : 27 | play One-hundred-seventy-first harmonic | 19 | |||
510.51 | (3 : 2)8/11 | 38/11 : 28/11 | play Beta scale perfect fourth | 18.75 | ||||
511.52 | F | 43 : 32 | 43 : 25 | play Forty-third harmonic [5] | 43 | |||
514.29 | 23/7 | 23/7 | play 3 steps in 7 equal temperament | 7 | ||||
519.55 | F+ [2] | 27 : 20 | 33 : 22×5 | play 5-limit wolf fourth, acute fourth, [3] imperfect fourth [16] | 5 | |||
521.51 | E♯+++ | 177147 : 131072 | 311 : 217 | play Pythagorean augmented third [3] [6] (F+ (pitch)) | 3 | |||
525.00 | 27/16 | 221/48 | play 21 steps in 48 equal temperament | 16, 48 | ||||
531.53 | F + | 87 : 64 | 3×29 : 26 | play Eighty-seventh harmonic [5] | 29 | |||
536.95 | F↓♯+ | 15 : 11 | 3×5 : 11 | play Undecimal augmented fourth [3] | 11 | |||
550.00 | F /G | 211/24 | 211/24 | play 11 steps in 24 equal temperament | 24 | |||
551.32 | F↑ [2] | 11 : 8 | 11 : 23 | play eleventh harmonic, [5] undecimal tritone, [5] lesser undecimal tritone, undecimal semi-augmented fourth [3] | 11 | |||
563.38 | F ♯+ | 18 : 13 | 2×9 : 13 | play Tridecimal augmented fourth [3] | 13 | |||
568.72 | F♯ [2] | 25 : 18 | 52 : 2×32 | play Just augmented fourth [3] [5] | 5 | |||
570.88 | 89 : 64 | 89 : 26 | play Eighty-ninth harmonic [5] | 89 | ||||
575.00 | 223/48 | 223/48 | play 23 steps in 48 equal temperament | 48 | ||||
582.51 | G ♭ [2] | 7 : 5 | 7 : 5 | play 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 | play Pythagorean diminished fifth, [3] [6] low Pythagorean tritone [5] | 3 | |||
590.22 | F♯+ [2] | 45 : 32 | 32×5 : 25 | play 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 | play Three-hundred-sixty-first harmonic | 19 | |||
600.00 | F♯/G♭ | 26/12 | 21/2=√2 | play Equal-tempered tritone | 2, 12 | M | ||
609.35 | G ♭ | 91 : 64 | 7×13 : 26 | play Ninety-first harmonic [5] | 13 | |||
609.78 | G♭− [2] | 64 : 45 | 26 : 32×5 | play 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 | play Pythagorean tritone, [3] [6] Pythagorean augmented fourth, high Pythagorean tritone [5] | 3 | |||
617.49 | F♯ [2] | 10 : 7 | 2×5 : 7 | play Greater septimal tritone, septimal tritone, [4] [5] Euler's tritone [3] | 7 | |||
625.00 | 225/48 | 225/48 | play 25 steps in 48 equal temperament | 48 | ||||
628.27 | F ♯+ | 23 : 16 | 23 : 24 | play Twenty-third harmonic, [5] classic diminished fifth[ citation needed ] | 23 | |||
631.28 | G♭ [2] | 36 : 25 | 22×32 : 52 | play Just diminished fifth [5] | 5 | |||
646.99 | F ♯+ | 93 : 64 | 3×31 : 26 | play Ninety-third harmonic [5] | 31 | |||
648.68 | G↓ [2] | 16 : 11 | 24 : 11 | play ` undecimal semi-diminished fifth [3] | 11 | |||
650.00 | F /G | 213/24 | 213/24 | play 13 steps in 24 equal temperament | 24 | |||
665.51 | G | 47 : 32 | 47 : 25 | play Forty-seventh harmonic [5] | 47 | |||
675.00 | 29/16 | 227/48 | play 27 steps in 48 equal temperament | 16, 48 | ||||
678.49 | A −−− | 262144 : 177147 | 218 : 311 | play Pythagorean diminished sixth [3] [6] | 3 | |||
680.45 | G− | 40 : 27 | 23×5 : 33 | play 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 | play Ninety-fifth harmonic [5] | 19 | |||
684.82 | E ++ | 12167 : 8192 | 233 : 213 | play 12167th harmonic | 23 | |||
685.71 | 24/7 : 1 | play 4 steps in 7 equal temperament | ||||||
691.20 | 3:2÷(81:80)1/2 | 2×51/2 : 3 | play Half-comma meantone perfect fifth | M | ||||
694.79 | 3:2÷(81:80)1/3 | 21/3×51/3 : 31/3 | play 1⁄3-comma meantone perfect fifth | M | ||||
695.81 | 3:2÷(81:80)2/7 | 21/7×52/7 : 31/7 | play 2⁄7-comma meantone perfect fifth | M | ||||
696.58 | 3:2÷(81:80)1/4 | 51/4 | play Quarter-comma meantone perfect fifth | M | ||||
697.65 | 3:2÷(81:80)1/5 | 31/5×51/5 : 21/5 | play 1⁄5-comma meantone perfect fifth | M | ||||
698.37 | 3:2÷(81:80)1/6 | 31/3×51/6 : 21/3 | play 1⁄6-comma meantone perfect fifth | M | ||||
700.00 | G | 27/12 | 27/12 | play Equal-tempered perfect fifth | 12 | M | ||
701.89 | 231/53 | 231/53 | play 53-TET perfect fifth | 53 | ||||
701.96 | G [2] | 3 : 2 | 3 : 2 | play 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 | play 41-TET perfect fifth | 41 | ||||
703.45 | 217/29 | 217/29 | play 29-TET perfect fifth | 29 | ||||
719.90 | 97 : 64 | 97 : 26 | play Ninety-seventh harmonic [5] | 97 | ||||
720.00 | 23/5 : 1 | play 3 steps in 5 equal temperament | 5 | |||||
721.51 | A − | 1024 : 675 | 210 : 33×52 | play Narrow diminished sixth [3] | 5 | |||
725.00 | 229/48 | 229/48 | play 29 steps in 48 equal temperament | 48 | ||||
729.22 | G - | 32 : 21 | 24 : 3×7 | play 21st subharmonic, [5] [6] septimal diminished sixth | 7 | |||
733.23 | F + | 391 : 256 | 17×23 : 28 | play Three-hundred-ninety-first harmonic | 23 | |||
737.65 | A ♭+ | 49 : 32 | 7×7 : 25 | play Forty-ninth harmonic [5] | 7 | |||
743.01 | A | 192 : 125 | 26×3 : 53 | play Classic diminished sixth [3] | 5 | |||
750.00 | G /A | 215/24 | 215/24 | play 15 steps in 24 equal temperament | 24 | |||
755.23 | G↑ | 99 : 64 | 32×11 : 26 | play Ninety-ninth harmonic [5] | 11 | |||
764.92 | A ♭ [2] | 14 : 9 | 2×7 : 32 | play Septimal minor sixth [3] [5] | 7 | |||
772.63 | G♯ | 25 : 16 | 52 : 24 | play Just augmented fifth [5] [16] | 5 | |||
775.00 | 231/48 | 231/48 | play 31 steps in 48 equal temperament | 48 | ||||
781.79 | π : 2 | play Wallis product | ||||||
782.49 | G ↑- [2] | 11 : 7 | 11 : 7 | play Undecimal minor sixth, [5] undecimal augmented fifth, [3] Fibonacci numbers | 11 | |||
789.85 | 101 : 64 | 101 : 26 | play Hundred-first harmonic [5] | 101 | ||||
792.18 | A♭− [2] | 128 : 81 | 27 : 34 | play Pythagorean minor sixth, [3] [5] [6] 81st subharmonic | 3 | |||
798.40 | A ♭+ | 203 : 128 | 7×29 : 27 | play Two-hundred-third harmonic | 29 | |||
800.00 | G♯/A♭ | 28/12 | 22/3 | play Equal-tempered minor sixth | 3, 12 | M | ||
806.91 | G ♯ | 51 : 32 | 3×17 : 25 | play Fifty-first harmonic [5] | 17 | |||
813.69 | A♭ [2] | 8 : 5 | 23 : 5 | play Just minor sixth [3] [4] [11] [16] | 5 | |||
815.64 | G♯++ | 6561 : 4096 | 38 : 212 | play Pythagorean augmented fifth, [3] [6] Pythagorean 'schismatic' sixth [5] | 3 | |||
823.80 | 103 : 64 | 103 : 26 | play Hundred-third harmonic [5] | 103 | ||||
825.00 | 211/16 | 233/48 | play 33 steps in 48 equal temperament | 16, 48 | ||||
832.18 | G ♯+ | 207 : 128 | 32×23 : 27 | play Two-hundred-seventh harmonic | 23 | |||
833.09 | (51/2+1)/2 | φ : 1 | play Golden ratio (833 cents scale) | |||||
835.19 | A♭+ | 81 : 50 | 34 : 2×52 | play Acute minor sixth [3] | 5 | |||
840.53 | A ♭ [2] | 13 : 8 | 13 : 23 | play Tridecimal neutral sixth, [3] overtone sixth, [5] thirteenth harmonic | 13 | |||
848.83 | A ♭↑ | 209 : 128 | 11×19 : 27 | play Two-hundred-ninth harmonic | 19 | |||
850.00 | G /A | 217/24 | 217/24 | play Equal-tempered neutral sixth | 24 | |||
852.59 | A↓+ [2] | 18 : 11 | 2×32 : 11 | play Undecimal neutral sixth, [3] [5] Zalzal's neutral sixth | 11 | |||
857.09 | A + | 105 : 64 | 3×5×7 : 26 | play Hundred-fifth harmonic [5] | 7 | |||
857.14 | 25/7 | 25/7 | play 5 steps in 7 equal temperament | 7 | ||||
862.85 | A− | 400 : 243 | 24×52 : 35 | play Grave major sixth [3] | 5 | |||
873.50 | A | 53 : 32 | 53 : 25 | play Fifty-third harmonic [5] | 53 | |||
875.00 | 235/48 | 235/48 | play 35 steps in 48 equal temperament | 48 | ||||
879.86 | A↓ | 128 : 77 | 27 : 7×11 | play 77th subharmonic [5] [6] | 11 | |||
882.40 | B −−− | 32768 : 19683 | 215 : 39 | play Pythagorean diminished seventh [3] [6] | 3 | |||
884.36 | A [2] | 5 : 3 | 5 : 3 | play 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 | play Hundred-seventh harmonic [5] | 107 | ||||
892.54 | B | 6859 : 4096 | 193 : 212 | play 6859th harmonic | 19 | |||
900.00 | A | 29/12 | 23/4 | play Equal-tempered major sixth | 4, 12 | M | ||
902.49 | A | 32 : 19 | 25 : 19 | play 19th subharmonic [5] [6] | 19 | |||
905.87 | A+ [2] | 27 : 16 | 33 : 24 | play Pythagorean major sixth [3] [5] [11] [16] | 3 | |||
921.82 | 109 : 64 | 109 : 26 | play Hundred-ninth harmonic [5] | 109 | ||||
925.00 | 237/48 | 237/48 | play 37 steps in 48 equal temperament | 48 | ||||
925.42 | B − [2] | 128 : 75 | 27 : 3×52 | play Just diminished seventh, [16] diminished seventh, [5] [13] 75th subharmonic | 5 | |||
925.79 | A + | 437 : 256 | 19×23 : 28 | play Four-hundred-thirty-seventh harmonic | 23 | |||
933.13 | A [2] | 12 : 7 | 22×3 : 7 | play Septimal major sixth [3] [4] [5] | 7 | |||
937.63 | A↑ | 55 : 32 | 5×11 : 25 | play Fifty-fifth harmonic [5] [20] | 11 | |||
950.00 | A /B | 219/24 | 219/24 | play 19 steps in 24 equal temperament | 24 | |||
953.30 | A ♯+ | 111 : 64 | 3×37 : 26 | play Hundred-eleventh harmonic [5] | 37 | |||
955.03 | A♯ [2] | 125 : 72 | 53 : 23×32 | play Just augmented sixth [5] | 5 | |||
957.21 | (3 : 2)15/11 | 315/11 : 215/11 | play 15 steps in Beta scale | 18.75 | ||||
960.00 | 24/5 | 24/5 | play 4 steps in 5 equal temperament | 5 | ||||
968.83 | B ♭ [2] | 7 : 4 | 7 : 22 | play Septimal minor seventh, [4] [5] [11] harmonic seventh, [3] [11] augmented sixth[ citation needed ] | 7 | |||
975.00 | 213/16 | 239/48 | play 39 steps in 48 equal temperament | 16, 48 | ||||
976.54 | A♯+ [2] | 225 : 128 | 32×52 : 27 | play Just augmented sixth [16] | 5 | |||
984.21 | 113 : 64 | 113 : 26 | play Hundred-thirteenth harmonic [5] | 113 | ||||
996.09 | B♭− [2] | 16 : 9 | 24 : 32 | play 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 | play Fifty-seventh harmonic [5] | 19 | |||
1000.00 | A♯/B♭ | 210/12 | 25/6 | play Equal-tempered minor seventh | 6, 12 | M | ||
1014.59 | A ♯+ | 115 : 64 | 5×23 : 26 | play Hundred-fifteenth harmonic [5] | 23 | |||
1017.60 | B♭ [2] | 9 : 5 | 32 : 5 | play Greater just minor seventh, [16] large just minor seventh, [4] [5] Bohlen-Pierce seventh [3] | 5 | |||
1019.55 | A♯+++ | 59049 : 32768 | 310 : 215 | play Pythagorean augmented sixth [3] [6] | 3 | |||
1025.00 | 241/48 | 241/48 | play 41 steps in 48 equal temperament | 48 | ||||
1028.57 | 26/7 | 26/7 | play 6 steps in 7 equal temperament | 7 | ||||
1029.58 | B ♭ | 29 : 16 | 29 : 24 | play Twenty-ninth harmonic, [5] minor seventh[ citation needed ] | 29 | |||
1035.00 | B↓ [2] | 20 : 11 | 22×5 : 11 | play Lesser undecimal neutral seventh, large minor seventh [3] | 11 | |||
1039.10 | B♭+ | 729 : 400 | 36 : 24×52 | play Acute minor seventh [3] | 5 | |||
1044.44 | B ♭ | 117 : 64 | 32×13 : 26 | play Hundred-seventeenth harmonic [5] | 13 | |||
1044.86 | B ♭- | 64 : 35 | 26 : 5×7 | play 35th subharmonic, [5] septimal neutral seventh [6] | 7 | |||
1049.36 | B↑♭− [2] | 11 : 6 | 11 : 2×3 | play 21⁄4-tone or Undecimal neutral seventh, [3] undecimal 'median' seventh [5] | 11 | |||
1050.00 | A /B | 221/24 | 27/8 | play Equal-tempered neutral seventh | 8, 24 | |||
1059.17 | 59 : 32 | 59 : 25 | play Fifty-ninth harmonic [5] | 59 | ||||
1066.76 | B− | 50 : 27 | 2×52 : 33 | play Grave major seventh [3] | 5 | |||
1071.70 | B ♭- | 13 : 7 | 13 : 7 | play Tridecimal neutral seventh [21] | 13 | |||
1073.78 | B | 119 : 64 | 7×17 : 26 | play Hundred-nineteenth harmonic [5] | 17 | |||
1075.00 | 243/48 | 243/48 | play 43 steps in 48 equal temperament | 48 | ||||
1086.31 | C′♭−− | 4096 : 2187 | 212 : 37 | play Pythagorean diminished octave [3] [6] | 3 | |||
1088.27 | B [2] | 15 : 8 | 3×5 : 23 | play 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 | play 17th subharmonic [5] [6] | 17 | |||
1100.00 | B | 211/12 | 211/12 | play Equal-tempered major seventh | 12 | M | ||
1102.64 | B↑↑♭- | 121 : 64 | 112 : 26 | play Hundred-twenty-first harmonic [5] | 11 | |||
1107.82 | C′♭− | 256 : 135 | 28 : 33×5 | play Octave − major chroma, [3] 135th subharmonic, narrow diminished octave[ citation needed ] | 5 | |||
1109.78 | B+ [2] | 243 : 128 | 35 : 27 | play Pythagorean major seventh [3] [5] [6] [11] | 3 | |||
1116.88 | 61 : 32 | 61 : 25 | play Sixty-first harmonic [5] | 61 | ||||
1125.00 | 215/16 | 245/48 | play 45 steps in 48 equal temperament | 16, 48 | ||||
1129.33 | C′♭ [2] | 48 : 25 | 24×3 : 52 | play Classic diminished octave, [3] [6] large just major seventh [4] | 5 | |||
1131.02 | B | 123 : 64 | 3×41 : 26 | play Hundred-twenty-third harmonic [5] | 41 | |||
1137.04 | B | 27 : 14 | 33 : 2×7 | play Septimal major seventh [5] | 7 | |||
1138.04 | C ♭ | 247 : 128 | 13×19 : 27 | play Two-hundred-forty-seventh harmonic | 19 | |||
1145.04 | B | 31 : 16 | 31 : 24 | play Thirty-first harmonic, [5] augmented seventh[ citation needed ] | 31 | |||
1146.73 | C↓ | 64 : 33 | 26 : 3×11 | play 33rd subharmonic [6] | 11 | |||
1150.00 | B /C | 223/24 | 223/24 | play 23 steps in 24 equal temperament | 24 | |||
1151.23 | C | 35 : 18 | 5×7 : 2×32 | play Septimal supermajor seventh, septimal quarter tone inverted | 7 | |||
1158.94 | B♯ [2] | 125 : 64 | 53 : 26 | play Just augmented seventh, [5] 125th harmonic | 5 | |||
1172.74 | C + | 63 : 32 | 32×7 : 25 | play Sixty-third harmonic [5] | 7 | |||
1175.00 | 247/48 | 247/48 | play 47 steps in 48 equal temperament | 48 | ||||
1178.49 | C′− | 160 : 81 | 25×5 : 34 | play Octave − syntonic comma, [3] semi-diminished octave[ citation needed ] | 5 | |||
1179.59 | B ↑ | 253 : 128 | 11×23 : 27 | play Two-hundred-fifty-third harmonic [5] | 23 | |||
1186.42 | 127 : 64 | 127 : 26 | play Hundred-twenty-seventh harmonic [5] | 127 | ||||
1200.00 | C′ | 2 : 1 | 2 : 1 | play Octave [3] [11] or diapason [4] | 1, 12 | 3 | M | S |
In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals consist of tones from a single harmonic series of an implied fundamental. For example, in the diagram, if the notes G3 and C4 are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just fourth.
In music, there are two common meanings for tuning:
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ≈702 cents wide.
Meantone temperament is a musical temperament, that is a tuning system, obtained by narrowing the fifths so that their ratio is slightly less than 3:2, in order to push the thirds closer to pure. Meantone temperaments are constructed the same way as Pythagorean tuning, as a stack of equal fifths, but it is a temperament in that the fifths are not pure.
In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.
In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80. Two notes that differ by this interval would sound different from each other even to untrained ears, but would be close enough that they would be more likely interpreted as out-of-tune versions of the same note than as different notes. The comma is also referred to as a Didymean comma because it is the amount by which Didymus corrected the Pythagorean major third to a just major third.
In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament. More broadly, it is also used to refer to similar intervals produced by other tuning systems, including Pythagorean and most meantone temperaments.
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale. For example, C is adjacent to C♯; the interval between them is a semitone.
In music theory, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions. The minor third is one of two commonly occurring thirds. It is called minor because it is the smaller of the two: the major third spans an additional semitone. For example, the interval from A to C is a minor third, as the note C lies three semitones above A. Coincidentally, there are three staff positions from A to C. Diminished and augmented thirds span the same number of staff positions, but consist of a different number of semitones. The minor third is a skip melodically.
Regular temperament is any tempered system of musical tuning such that each frequency ratio is obtainable as a product of powers of a finite number of generators, or generating frequency ratios. For instance, in 12-TET, the system of music most commonly used in the Western world, the generator is a tempered fifth, which is the basis behind the circle of fifths.
A schismatic temperament is a musical tuning system that results from tempering the schisma of 32805:32768 to a unison. It is also called the schismic temperament, Helmholtz temperament, or quasi-Pythagorean temperament.
In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways. Strictly speaking, there are only two kinds of comma, the syntonic comma, "the difference between a just major 3rd and four just perfect 5ths less two octaves", and the Pythagorean comma, "the difference between twelve 5ths and seven octaves". The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F♯ tuned using the D-based Pythagorean tuning system, and another F♯ tuned using the D-based quarter-comma meantone tuning system. Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B♭ and A♯ are both approximated by the same interval although they are a septimal kleisma apart.
Twelve-tone equal temperament (12-TET) is the musical system that divides the octave into 12 parts, all of which are equally tempered on a logarithmic scale, with a ratio equal to the 12th root of 2. That resulting smallest interval, 1⁄12 the width of an octave, is called a semitone or half step.
In music, 53 equal temperament, called 53 TET, 53 EDO, or 53 ET, is the tempered scale derived by dividing the octave into 53 equal steps. Play (help·info) Each step represents a frequency ratio of 21⁄53, or 22.6415 cents, an interval sometimes called the Holdrian comma.
In musical tuning theory, a Pythagorean interval is a musical interval with frequency ratio equal to a power of two divided by a power of three, or vice versa. For instance, the perfect fifth with ratio 3/2 (equivalent to 31/ 21) and the perfect fourth with ratio 4/3 (equivalent to 22/ 31) are Pythagorean intervals.
Music theory analyzes the pitch, timing, and structure of music. It uses mathematics to study elements of music such as tempo, chord progression, form, and meter. The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of set theory, abstract algebra and number theory.
In musical tuning, a temperament is a tuning system that slightly compromises the pure intervals of just intonation to meet other requirements. Most modern Western musical instruments are tuned in the equal temperament system. Tempering is the process of altering the size of an interval by making it narrower or wider than pure. "Any plan that describes the adjustments to the sizes of some or all of the twelve fifth intervals in the circle of fifths so that they accommodate pure octaves and produce certain sizes of major thirds is called a temperament." Temperament is especially important for keyboard instruments, which typically allow a player to play only the pitches assigned to the various keys, and lack any way to alter pitch of a note in performance. Historically, the use of just intonation, Pythagorean tuning and meantone temperament meant that such instruments could sound "in tune" in one key, or some keys, but would then have more dissonance in other keys.
In music, 41 equal temperament, abbreviated 41-TET, 41-EDO, or 41-ET, is the tempered scale derived by dividing the octave into 41 equally sized steps. Play (help·info) Each step represents a frequency ratio of 21/41, or 29.27 cents, an interval close in size to the septimal comma. 41-ET can be seen as a tuning of the schismatic, magic and miracle temperaments. It is the second smallest equal temperament, after 29-ET, whose perfect fifth is closer to just intonation than that of 12-ET. In other words, is a better approximation to the ratio than either or .
A regular diatonic tuning is any musical scale consisting of "tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's. In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size which makes it a Linear temperament with the tempered fifth as a generator.
Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by products of integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), such as 2−3·31·51 = 15/8.