This page provides frequency tables for note lists. Pitch names follow the linear music code: a pound/hash sign (#) indicates a sharp while an exclamation point (!) indicates a flat.
The relevant unit of measurement for tunings is the cent.
In equal-tempered tunings, the ratio between two consecutive degrees of the chromatic scale is everywhere
C D! D E! E F F#/G! G A! A B! B Cents 0 100 200 300 400 500 600 700 800 900 1000 1100
Table 1-1: Equal-Tempered Tuning.
The equal-tempered perfect fifth of 700 cents is 2 cents narrower than the ratio 3:2. The equal-tempered major third of 400 cents is 14 cents wider than the ratio 5:4.
Octave C C# D E! E F F# G A! A B! B 0 16.35 17.32 18.35 19.45 20.60 21.83 23.12 24.50 25.96 27.50 29.14 30.87 1 32.70 34.65 36.71 38.89 41.20 43.65 46.25 49.00 51.91 55 58.27 61.74 2 65.41 69.30 73.42 77.78 82.41 87.31 92.50 98.00 103.8 110 116.5 123.5 3 130.8 138.6 146.8 155.6 164.8 174.6 185.0 196.0 207.7 220 233.1 246.9 4 261.6 277.2 293.7 311.1 329.6 349.2 370.0 392.0 415.3 440 466.2 493.9 5 523.3 554.4 587.3 622.3 659.3 698.5 740.0 784.0 830.6 880 932.3 987.8 6 1047 1109 1175 1245 1319 1397 1480 1568 1661 1760 1865 1976 7 2093 2217 2349 2489 2637 2794 2960 3136 3322 3520 3729 3951 8 4186 4435 4699 4978 5274 5588 5920 6272 6645 7040 7459 7902 9 8372 8870 9397 9956 10548 11175 11840 12544 13290 14080 14917 15804
Table 1-1: Frequencies for equal-tempered tuning based on A4 = 440 Hz.
According to the Wikipedia entry on Meantone Temperament, Pythagorean tuning generates “all non-octave intervals” from a “stack of perfect fifths”. Each of these fifths is tuned in an exact ratio of 3:2, or 702 cents. Table 2-1 details which ratios go with which pitch names.
C D! D E! E F F# G! G A! A B! B Ratios 1:1 256:243 9:8 32:27 81:64 4:3 729:512 1024:729 3:2 128:81 27:16 16:9 243:128 Cents 0 90 204 996 408 498 612 588 702 792 906 996 1110
Table 2-1: Ratios for Pythagorean tuning.
Notice that each successive ratio in the sequence C, G, D, A, E, B, F# adds a factor of 3 to the numerator; the denominator always being a power of 2. Similarly, each successive ratio in the sequence D, F, B!, E!, A!, D! G! adds a factor of 3 to the denominator; in these cases the numerators are always powers of 2. Notice also that F# and G! have different ratios. F# is sharper than G! by 23 cents.
The Pythagorean major third from C to E is 408 cents, which is 21 cents wider than the ratio 5:4.
Octave C D! D E! E F F# G! G A! A B! B 0 16.35 17.22 18.39 19.38 20.69 21.80 23.28 22.97 24.53 25.84 27.59 29.07 31.04 1 32.70 34.45 36.79 38.76 41.39 43.60 46.56 45.93 49.05 51.67 55.18 58.13 62.08 2 65.40 68.90 73.58 77.51 82.77 87.20 93.12 91.87 98.10 103.3 110.4 116.3 124.2 3 130.8 137.8 147.2 155.0 165.5 174.4 186.2 183.7 196.2 206.7 220.7 232.5 248.3 4 261.6 275.6 294.3 310.0 331.1 348.8 372.5 367.5 392.4 413.4 441.5 465.1 496.6 5 523.2 551.2 588.6 620.1 662.2 697.6 744.9 734.9 784.8 826.8 882.9 930.1 993.3 6 1046 1102 1177 1240 1324 1395 1490 1470 1570 1654 1766 1860 1987 7 2093 2205 2354 2480 2649 2790 2980 2940 3139 3307 3532 3721 3973 8 4186 4410 4709 4961 5297 5581 5960 5879 6278 6614 7063 7441 7946 9 8371 8819 9418 9921 10595 11162 11919 11759 12557 13229 14126 14882 15892
Table 2-2: Frequencies for Pythagorean tuning based on C4 = 261.6 Hz.
According to Wikipedia, Meantone Temperament generates “generates all non-octave intervals from a stack of tempered perfect fifths” (italics mine). Meantone temperament has many flavors. These include equal temperament, where a perfect fifth has 700 cents, and Pythagorean tuning, where a perfect fifth has 702 cents. However the temperament most commonly referred to as “meantone” is the “quarter-comma” flavor, where a perfect fifth has 697 cents (5 cents narrow than the ratio 3:2).
C D! D E! E F F# G! G A! A B! B Cents 0 115 194 309 388 503 687 618 697 812 891 1006 1190
Table 3-1: Quarter-comma meantone tuning.
The quarter-comma meantone major third from C to E is 388 cents, which is only 2 cents wider than the ratio 5:4.
Octave C D! D E! E F F# G! G A! A B! B 0 16.35 17.47 18.29 19.54 20.46 21.86 22.88 23.36 24.45 26.13 27.35 29.23 30.60 1 32.70 34.95 36.58 39.09 40.91 43.72 45.77 46.73 48.91 52.27 54.71 58.47 61.20 2 65.40 69.89 73.16 78.18 81.83 87.45 91.53 93.46 97.82 104.5 109.4 116.9 122.4 3 130.8 139.8 146.3 156.4 163.7 174.9 183.1 186.9 195.6 209.1 218.8 233.9 244.8 4 261.6 279.6 292.6 312.7 327.3 349.8 366.1 373.8 391.3 418.2 437.7 467.7 489.6 5 523.2 559.1 585.2 625.4 654.6 699.6 732.3 747.6 782.6 836.3 875.4 935.5 979.1 6 1046 1118 1170 1251 1309 1399 1465 1495 1565 1673 1751 1871 1958 7 2093 2237 2341 2502 2619 2798 2929 2991 3130 3345 3501 3742 3917 8 4186 4473 4682 5003 5237 5597 5858 5981 6260 6690 7003 7484 7833 9 8371 8946 9364 10007 10474 11194 11716 11962 12521 13381 14006 14968 15666
Table 3-2: Frequencies for “quarter-comma meantone” tuning based on C4 = 261.6 Hz.
According to Wikipedia, Just Intonation includes “any musical tuning in which the frequencies of notes are related by ratios of small whole numbers”.
C D! D E! E F F# G! G A! A B! B Ratios 1:1 16:15 9:8 6:5 5:4 4:3 45:32 64:45 3:2 8:5 5:3 9:5 15:8 Cents 0 112 204 316 386 498 590 610 702 814 884 1018 1088
Table 3-1: Ratios for 5-Limit tuning.
The just “5-limit” tuning presented here comes from the same Wikipedia entry. It assigns ratios to pitch names as detailed in Table 4-1.
Octave C D! D E! E F F# G! G A! A B! B 0 16.35 17.44 18.39 19.62 20.44 21.80 22.99 23.25 24.53 26.16 27.25 29.43 30.66 1 32.70 34.88 36.79 39.24 40.88 43.60 45.98 46.51 49.05 52.32 54.50 58.86 61.31 2 65.40 69.76 73.58 78.48 81.75 87.20 91.97 93.01 98.10 104.6 109.0 117.7 122.6 3 130.8 139.5 147.2 157.0 163.5 174.4 183.9 186.0 196.2 209.3 218.0 235.4 245.3 4 261.6 279.0 294.3 313.9 327 348.8 367.9 372.1 392.4 418.6 436.0 470.9 490.5 5 523.2 558.1 588.6 627.8 654 697.6 735.8 744.1 784.8 837.1 872.0 941.8 981.0 6 1046 1116 1177 1256 1308 1395 1472 1488 1570 1674 1744 1884 1962 7 2093 2232 2354 2511 2616 2790 2943 2976 3139 3348 3488 3767 3924 8 4186 4465 4709 5023 5232 5581 5886 5953 6278 6697 6976 7534 7848 9 8371 8929 9418 10045 10464 11162 11772 11906 12557 13394 13952 15068 15696
Table 4-2: Frequencies for just “5-limit” tuning based on C4 = 261.6 Hz.
According to Wikipedia, Just Intonation includes “any musical tuning in which the frequencies of notes are related by ratios of small whole numbers”.
C D! D E! E F F# G A! A B! B Ratio 1:1 17:16 9:8 19:16 5:4 21:16 11:8 3:2 25:16 13:8 7:4 15:8 Cents 0 105 204 298 386 471 551 702 773 841 969 1088
Table 5-1: Ratios for overtone tuning.
This just tuning assigns ratios to pitch names as detailed in Table 5-1. These ratios come exclusively the overtone series; hence the denominator of each ratio is a power of 2.
Octave C D! D E! E F F# G A! A B! B 0 16.35 17.37 18.39 19.42 20.44 21.46 22.48 24.53 25.55 26.57 28.61 30.66 1 32.70 34.74 36.79 38.83 40.88 42.92 44.96 49.05 51.09 53.14 57.23 61.31 2 65.40 69.49 73.58 77.66 81.75 85.84 89.93 98.10 102.2 106.3 114.5 122.6 3 130.8 139.0 147.2 155.3 163.5 171.7 179.9 196.2 204.4 212.6 228.9 245.3 4 261.6 278.0 294.3 310.7 327 343.4 359.7 392.4 408.8 425.1 457.8 490.5 5 523.2 555.9 588.6 621.3 654 686.7 719.4 784.8 817.5 850.2 915.6 981.0 6 1046 1112 1177 1243 1308 1373 1439 1570 1635 1700 1831 1962 7 2093 2224 2354 2485 2616 2747 2878 3139 3270 3401 3662 3924 8 4186 4447 4709 4970 5232 5494 5755 6278 6540 6802 7325 7848 9 8371 8894 9418 9941 10464 10987 11510 12557 13080 13603 14650 15696
Table 5: Frequencies for just overtone tuning based on C4 = 261.6 Hz.
According to Wikipedia, Just Intonation includes “any musical tuning in which the frequencies of notes are related by ratios of small whole numbers”.
Pitch C D! D D# E! E F F# G! G G# A B!! B! B Ratio 1:1 12:11 8:7 7:6 14:11 21:16 4:3 11:8 16:11 3:2 32:21 11:7 12:7 7:4 11:6 Cents 0 151 231 267 418 471 498 551 649 702 729 782 933 969 1049
Table 5-1: Ratios for 7-11 tuning.
I made up the just 7-11 tuning for the Note-List Instructions. I have no idea if anyone else has used it before and took no trouble to research the issue. My initial idea was to create a tuning with open consonances (ratios involving 3) and semi-consonances (ratios involving 7), but no soft consonances (ratios involving 5). Since this initial prescription produced only 9 ratios, I added in several ratios involving 11. The selected ratios are detailed in Table 6-1.
Octave C D! D D# E! E F F# G! G G# A B!! B! B 0 16.35 17.84 18.69 19.08 20.81 21.46 21.80 22.48 23.78 24.53 24.91 25.69 28.03 28.61 29.98 1 32.70 35.67 37.37 38.15 41.62 42.92 43.60 44.96 47.56 49.05 49.83 51.39 56.06 57.23 59.95 2 65.40 71.35 74.74 76.30 83.24 85.84 87.20 89.93 95.13 98.10 99.66 102.8 112.1 114.5 119.9 3 130.8 142.7 149.5 152.6 166.5 171.7 174.4 179.9 190.3 196.2 199.3 205.5 224.2 228.9 239.8 4 261.6 285.4 299.0 305.2 332.9 343.4 348.8 359.7 380.5 392.4 398.6 411.1 448.5 457.8 479.6 5 523.2 570.8 597.9 610.4 665.9 686.7 697.6 719.4 761.0 784.8 797.3 822.2 896.9 915.6 959.2 6 1046 1142 1196 1221 1332 1373 1395 1439 1522 1570 1595 1644 1794 1831 1918 7 2093 2283 2392 2442 2664 2747 2790 2878 3044 3139 3189 3289 3588 3662 3837 8 4186 4566 4784 4883 5327 5494 5581 5755 6088 6278 6378 6577 7175 7325 7674 9 8371 9132 9567 9766 10654 10987 11162 11510 12176 12557 12756 13155 14351 14650 15347
Table 6-2: Frequencies for just 7-11 tuning based on C4 = 261.6 Hz.
| © Charles Ames | Page created: 2013-02-20 | Last updated: 2017-08-15 |