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'''Relative cent''' ('''rct''', '''r¢''') is a logarithmic [[Interval_size_measure|interval size measure]] based on a given [[Equal|equal]]-stepped tonal system. Its size is 1 percent of the distance between adjacent pitches.
The '''relative cent''' (symbol: '''r¢''', '''rct''', '''¢<sub>EDO</sub>''') is a [[unit of interval size]] based on a given [[equal]]-stepped tonal system (especially [[EDO]] systems). In being the size of 1 percent of the distance between adjacent pitches, it is the generalization of the [[common cent]] (¢).


Given any N [[EDO|EDO]], the size of an interval in ''relative cents'' is N/12 times its size in [[cent|cents]]; or equivalently, 100 N times its logarithm base 2. Hence in [[7edo|7edo]], the octave is 700 relative cents, in [[53edo|53edo]], 5300 relative cents and so forth.
The term ''centidegree'' was suggested as an alias, but this seems to be used already as a unit for temperature.


An existing example is the [[turkish_cent|turkish cent]], which is the relative cent of [[106edo|106edo]]. The iota, the relative cent for [[17edo|17edo]], has been proposed by [[George_Secor|George Secor]] and [[Margo_Schulter|Margo Schulter]] for use with 17edo, and [[Tútim_Dennsuul_Wafiil|Tútim Dennsuul]] has advocated the [[Purdal|purdal]], which divides the octave into [[9900edo|9900]] parts. The [[millioctave|millioctave]] is another such measure, as it can be viewed as the relative cent measure for [[10edo|10edo]].
Given any ''N'' [[EDO]], the size of an interval in ''relative cents'' is ''N''/12 times its size in [[cent]]s; or equivalently, 100''N'' times its logarithm base 2. Hence in [[7edo|7EDO]], the octave is 700 relative cents, in [[53edo|53EDO]], 5300 relative cents and so forth.


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|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.
An existing example is the [[turkish cent]], which is the relative cent of [[106edo|106EDO]]. The iota, the relative cent for [[17edo|17EDO]], has been proposed by [[George Secor]] and [[Margo Schulter]] for use with 17EDO, and [[Tútim Dennsuul Wafiil|Tútim Dennsuul]] has advocated the [[purdal]], which divides the octave into 9900 parts (being relative cents of [[99edo|99EDO]]). The [[millioctave]] is another such measure, as it can be viewed as the relative cent measure for [[10edo|10EDO]]. The śat, for use with Armodue, divides the octave into 1600 parts (Armodue being a theory of [[16edo|16EDO]]).


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:
== Relative error ==
{{Main| Relative interval error }}


{| class="wikitable"
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 1.955{{c}} flat, or −1.955 cents sharp, which is therefore also its error in relative cents. The fifth of [[41edo|41EDO]] is 1.654 relative cents sharp. Thus for {{nowrap|53 {{=}} 41 + 12}}, the fifth is {{nowrap|−1.955 + 1.654 {{=}} −0.301}} relative cents sharp, and hence {{nowrap|(−0.301) × (12/53) {{=}} −0.068{{c}}}} sharp, which is to say 0.068{{c}} flat.
|+
!Divisor
!Octave
!3/2
!*12
|-
|101
|1212 f¢
|59.081°
|708.975 f¢
|-
|102
|1224
|59.666
|715.994
|-
|103
|1236
|60.251
|723.014
|-
|104
|1248
|60.836
|730.033
|-
|105
|1260
|61.421
|737.053
|-
|106
|'''1272'''
|'''62.006'''
|'''744.072'''
|-
|107
|1284
|62.591
|751.092
|-
|108
|1296 (900<sub>12</sub>)
|63.176 (53.214<sub>12</sub>)
|758.111 (532.141<sub>12</sub>)
|-
|109
|1308
|63.761
|765.131
|-
|110
|1320
|64.346
|772.1505
|-
|111
|1332
|64.931
|779.17
|-
|112
|1344 (800<sub>14</sub>)
|''65.516 (49.731<sub>14</sub>)''
|''786.19 (402.292<sub>14</sub>)''
|-
|113
|1356
|66.101
|793.209
|-
|114
|1368
|66.686
|800.229
|-
|115
|1380
|67.271
|807.248
|-
|116
|1392
|67.856
|814.268
|-
|117
|1404
|68.441
|821.287
|-
|118
|1416
|69.026
|828.307
|-
|119
|1428
|69.6105
|835.3265
|-
|120
|1440 (X00<sub>12</sub>)
|70.1955 (5X.242<sub>12</sub>)
|842.346 (5X2.41X<sub>12</sub>)
|-
|121
|1452
|70.7805
|849.366
|-
|122
|1464
|71.365
|856.385
|-
|123
|1476
|71.95
|863.405
|-
|124
|1488
|''72.535''
|''870.424''
|-
|125
|1500
|73.12
|877.444
|-
|126
|1512 (900<sub>14</sub>)
|73.705 (53.9C3<sub>14</sub>)
|''884.463 (472.66B<sub>14</sub>)''
|-
|127
|1524
|74.29
|''891.483''
|-
|128
|1536 (600<sub>16</sub>)
|74.875 (4A.E01<sub>16</sub>)
|''898.502 (382.81<sub>16</sub>)''
|}
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 ==
== Application for quantifying approximation ==
If you want to quantify the approximation of a given [[JI|JI]] interval in a given [[Equal|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|5edo]] has a relatively good approximated [[Natural_seventh|natural seventh]] with the ratio [[7/4|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|10edo]] (7.355 r¢), [[15edo|15edo]] (11.032 r¢) ... become progressively worse (in a relative sense). So in [[65edo|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|Pepper ambiguity]] for a mathematical approach to quantify the approximations for sets of intervals.
If you want to quantify the approximation of a given [[JI]] interval in a given [[Equal|equal-stepped]] tonal system, you can consider the absolute distance of 50&#x202F;r¢ as the worst possible and 0&#x202F;r¢ as the best possible. For example, [[5edo|5EDO]] has a relatively good approximated [[natural seventh]] with the ratio [[7/4]]: the absolute distance of 4\5 in 5EDO is 8.82{{c}} or 3.677&#x202F;r¢ flat of 7/4. But the approximations of its multiple EDOs [[10edo|10EDO]] (7.355&#x202F;r¢), [[15edo|15EDO]] (11.032&#x202F;r¢) … become progressively worse (in a relative sense). So in [[65edo|65EDO]], there is the 7/4 situated halfway between two adjacent pitches (off by at least 47.807&#x202F;r¢), but its absolute distance from this interval in cents is still the same as for 5EDO: 8.826{{c}} flat. See [[Pepper ambiguity]] for a mathematical approach to quantify the approximations for sets of intervals.


-----
== See also ==
''...also the term [[centidegree|centidegree]] was suggested, but this seems to be used already as a unit for temperature.''     
* [[Relative error]] − a measure for mapping quality
[[Category:approximation]]
 
[[Category:interval_measure]]
[[Category:Approximation]]
[[Category:practical_help]]
[[Category:Interval size measures]]
[[Category:relative_measure]]
[[Category:Elementary math]]
[[Category:term]]
[[Category:Relative measures]]
[[Category:unit]]
[[Category:Terms]]

Latest revision as of 13:21, 5 March 2025

The relative cent (symbol: r¢, rct, ¢EDO) is a unit of interval size based on a given equal-stepped tonal system (especially EDO systems). In being the size of 1 percent of the distance between adjacent pitches, it is the generalization of the common cent (¢).

The term centidegree was suggested as an alias, but this seems to be used already as a unit for temperature.

Given any N EDO, the size of an interval in relative cents is N/12 times its size in cents; or equivalently, 100N 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 (being relative cents of 99EDO). The millioctave is another such measure, as it can be viewed as the relative cent measure for 10EDO. The śat, for use with Armodue, divides the octave into 1600 parts (Armodue being a theory of 16EDO).

Relative error

Main article: Relative interval error

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 1.955 ¢ 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 ¢ sharp, which is to say 0.068 ¢ flat.

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 8.82 ¢ 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.

See also

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