Relative cent

From Xenharmonic Wiki
Jump to: navigation, search

Relative cent (rct, ) is a logarithmic interval size measure based on a given equal-stepped tonal system. Its size is 1 percent of the distance between adjacent pitches.

Given any N EDO, the size of an interval in relative cents is N/12 times its size in cents; or equivalently, 100 N times its logarithm base 2. Hence in 7edo, the octave is 700 relative cents, in 53edo, 5300 relative cents and so forth.

An existing example is the turkish cent, which is the relative cent of 106edo. The iota, the relative cent for 17edo, has been proposed by George Secor and Margo Schulter for use with 17edo, and Tútim Dennsuul has advocated the purdal, which divides the octave into 9900 parts. The millioctave is another such measure, as it can be viewed as the relative cent measure for 10edo.

Measuring the error of an approximation of an interval in an edo in terms of relative cents gives the relative error, which so long as the corresponding val is used is additive. For instance, the fifth of 12edo is 1.995 cents flat, or -1.955 cents sharp, which is therefore also its error in relative cents. The fifth of 41edo is 1.654 relative cents sharp. Thus for 53=41+12, the fifth is -1.955 + 1.654 = -0.301 relative cents sharp, and hence (-0.301)*(12/53) = -0.068 cents sharp, which is to say 0.068 cents flat.

The problem of the relative cent, though, is that 3/2 is 58.496 percent of the octave, which means the relative goodness of 3/2 to it is very heavily dependent on which edo you divide by 100. But if you increase the divisor, the following situation emerges:

Divisor Octave 3/2 *12
101 1212 f¢ 59.081° 708.975 f¢
102 1224 M¢ 59.666 715.994 Μ¢
103 1236 60.251 723.014
104 1248 60.836 730.033
105 1260 61.421 737.053
106 1272 π¢ 62.006 744.072 π¢
107 1284 62.591 751.092
108 1296 (90012) 63.176 (53.21412) 758.111 (532.14112)
109 1308 63.761 765.131
110 1320 64.346 772.1505
111 1332 64.931 779.17
112 1344 (6C014) lb 65.516 (49.73114) 786.19 (402.29214) lb
113 1356 66.101 793.209
114 1368 66.686 800.229
115 1380 67.271 807.248
116 1392 ν¢ 67.856 814.268 ν¢
117 1404 68.441 821.287
118 1416 69.026 828.307
119 1428 μ¢ 69.6105 835.3265 μ¢
120 1440 (X0012) dF 70.1955 (5X.24212) 842.346 (5X2.41X12) dF
121 1452 ¢2 70.7805 849.366 ¢2
122 1464 71.365 856.385
123 1476 ξ¢ 71.95 863.405 ξ¢
124 1488 72.535 870.424
125 1500 ¢3 73.12 877.444 ¢3
126 1512 (7A014) 73.705 (53.9C314) 884.463 (472.66B14)
127 1524 74.29 891.483
128 1536 (60016) 7mu 74.875 (4A.E0116) 898.502 (382.8116) 7mu
129 1548 75.46 905.522
130 1560 ρ¢ 76.045 912.5415 ρ¢
131 1572 76.63 919.561
132 1584 (E0012) 77.215 (65.2712) 926.581 (652.6E712)
133 1596 77.8 933.6
134 1608 78.385 940.62
135 1620 78.97 947.639
136 1632 ₮ 79.555 954.659 ₮
137 1644 80.14 961.678
138 1656 80.725 968.698
139 1668 81.31 975.7175
140 1680 (88014) 81.895 (5B.C7514) 982.737 (502.A4614)
141 1692 82.48 989.757
142 1704 83.065 996.776
143 1716 83.65 1003.796
144 1728 (100012) ħ 84.235 (70.29912) 1010.82 (702.99512) ħ

In other words, only each 12th divisor of the next 28 after 100 has the same problem as 100 does for use as the base of a relative cent. Also, the last three have the problem that they put 3/2 in the middle of a cent.

Application for quantifying approximation

If you want to quantify the approximation of a given JI interval in a given equal-stepped tonal system, you can consider the absolute distance of 50 r¢ as the worst possible and 0 r¢ as the best possible. For example, 5edo has a relatively good approximated natural seventh with the ratio 7/4: the absolute distance of 4\5 in 5edo is 8.826 ¢ or 3.677 r¢ flat of 7/4. But the approximations of its multiple edos 10edo (7.355 r¢), 15edo (11.032 r¢) ... become progressively worse (in a relative sense). So in 65edo, there is the 7/4 situated halfway between two adjacent pitches (off by at least 47.807 r¢), but its absolute distance from this interval in cents is still the same as for 5edo: 8.826 ¢ flat. See Pepper ambiguity for a mathematical approach to quantify the approximations for sets of intervals.


...also the term centidegree was suggested, but this seems to be used already as a unit for temperature.