Relative cent
Relative cent (rct, r¢) 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 ¢^{t} | 61.421 | 737.053 ¢^{t} |
106 | 1272 π^{¢} | 62.006 | 744.072 π^{¢} |
107 | 1284 | 62.591 | 751.092 |
108 | 1296 (900_{12}) ¢^{2.5} | 63.176 (53.214_{12}) | 758.111 (532.141_{12}) ¢^{2.5} |
109 | 1308 | 63.761 | 765.131 |
110 | 1320 | 64.346 | 772.1505 |
111 | 1332 | 64.931 | 779.17 |
112 | 1344 (6C0_{14}) lb | 65.516 (49.731_{14}) | 786.19 (402.292_{14}) 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 (X00_{12}) dF | 70.1955 (5X.242_{12}) | 842.346 (5X2.41X_{12}) 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 (7A0_{14}) | 73.705 (53.9C3_{14}) | 884.463 (472.66B_{14}) |
127 | 1524 | 74.29 | 891.483 |
128 | 1536 (600_{16}) 7mu | 74.875 (4A.E01_{16}) | 898.502 (382.81_{16}) 7mu |
129 | 1548 | 75.46 | 905.522 |
130 | 1560 ρ^{¢} | 76.045 | 912.5415 ρ^{¢} |
131 | 1572 | 76.63 | 919.561 |
132 | 1584 (E00_{12}) | 77.215 (65.27_{12}) | 926.581 (652.6E7_{12}) |
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 (880_{14}) | 81.895 (5B.C75_{14}) | 982.737 (502.A46_{14}) |
141 | 1692 ¢^{p} | 82.48 | 989.757 |
142 | 1704 | 83.065 | 996.776 |
143 | 1716 | 83.65 | 1003.796 |
144 | 1728 (1000_{12}) ħ | 84.235 (70.299_{12}) | 1010.82 (702.995_{12}) 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.