AFDO: Difference between revisions

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For example, in [[12afdo]] the first degree is [[13/12]], the second is 14/12 ([[7/6]]), and so on. For an AFDO system, the ''difference'' between interval ratios is equal (they form an arithmetic progression), rather than their ''ratios'' between interval ratios being equal as in [[EDO]] systems (a geometric progression). All AFDOs are subsets of [[just intonation]], and up to transposition, any AFDO is a superset of a smaller AFDO and a subset of a larger AFDO (i.e. ''n''-afdo is a superset of (''n'' - 1)-afdo and a subset of (''n'' + 1)-afdo for any integer ''n'' > 1).

When treated as a scale, the AFDO is equivalent to the [[overtone scale]]. An AFDO is equivalent to an ODO ([[otonal division]] of the octave). It may also be called an EFDO ([[equal frequency division]] of the octave), however, this more general acronym is typically reserved for divisions of irrational intervals (unlike the octave) which are therefore not subsets of just intonation.

When treated as a scale, the AFDO is equivalent to the [[overtone scale]]. However, an overtone scale often has an assumption of a tonic whereas an AFDO simply describes all the theoretically available pitch relations. Therefore, a passage built on 12::22 could be said to be in Mode 12, but is technically covered by 11afdo.

An AFDO is equivalent to an ODO ([[otonal division]] of the octave). It may also be called an EFDO ([[equal frequency division]] of the octave), however, this more general acronym is typically reserved for divisions of irrational intervals (unlike the octave) which are therefore not subsets of just intonation.

== Formula ==

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In 2023, [[Flora Canou]] revived the old term, reinterpreting the word "arithmetic" as a reference to the [[Wikipedia: Arithmetic mean|arithmetic mean]] in addition to arithmetic progressions, then extended it through the other [[Pythagorean means]], and later through all [[power mean]]s. As it was shown that ''arithmetic'' alone was insufficient to define the object since frequency could not be assumed, the term was eventually changed to AFDO, showing both the type of power mean and the sampled resource.

== Properties ==

* ''n''-afdo has [[maximum variety]] ''n''.

* Except for 1afdo and 2afdo, AFDOs are [[chiral]]. The inverse of ''n''-afdo is ''n''-ifdo.

** 1afdo is equivalent to 1ifdo and 1edo;

** 2afdo is equivalent to 2ifdo.

== Individual pages for AFDOs ==

=== By size ===

{| class="wikitable center-all"

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=== By prime family ===

'''Over-2''': {{AFDOs|2, 4, 8, 16, 32, 64, 128, 256, 512}}

'''Over-3''': {{AFDOs|3, 6, 12, 24, 48, 96, 192, 384}}

'''Over-5''': {{AFDOs|5, 10, 20, 40, 80, 160, 320}}

'''Over-7''': {{AFDOs|7, 14, 28, 56, 112, 224, 448}}

'''Over-11''': {{AFDOs|11, 22, 44, 88, 176, 352}}

'''Over-13''': {{AFDOs|13, 26, 52, 104, 208, 416}}

'''Over-17''': {{AFDOs|17, 34, 68, 136, 272, 544}}

'''Over-19''': {{AFDOs|19, 38, 76, 152, 304}}

'''Over-23''': {{AFDOs|23, 46, 92, 184, 368}}

'''Over-29''': {{AFDOs|29, 58, 116, 232, 464}}

'''Over-31''': {{AFDOs|31, 62, 124, 248, 496}}

=== By other properties ===

'''Prime''': {{AFDOs|2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271}}

'''Semiprime''': {{AFDOs|4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 169, 177, 178, 183, 185, 187}}

'''Odd squarefree semiprime''': {{AFDOs|15, 21, 33, 35, 39, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, 95, 111, 115, 119, 123, 129, 133, 141, 143, 145, 155, 159, 161, 177, 183, 185, 187, 201, 203, 205, 209, 213, 215, 217, 219, 221, 235, 237, 247, 249, 253, 259, 265, 267, 287, 291, 295, 299, 301, 303}}

'''Nonprime prime power''': {{AFDOs|4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1024, 1331, 1369, 1681, 1849, 2048}}

@@ Line 3: / Line 3: @@
 For example, in [[12afdo]] the first degree is [[13/12]], the second is 14/12 ([[7/6]]), and so on. For an AFDO system, the ''difference'' between interval ratios is equal (they form an arithmetic progression), rather than their ''ratios'' between interval ratios being equal as in [[EDO]] systems (a geometric progression). All AFDOs are subsets of [[just intonation]], and up to transposition, any AFDO is a superset of a smaller AFDO and a subset of a larger AFDO (i.e. ''n''-afdo is a superset of (''n'' - 1)-afdo and a subset of (''n'' + 1)-afdo for any integer ''n'' > 1).
-When treated as a scale, the AFDO is equivalent to the [[overtone scale]]. An AFDO is equivalent to an ODO ([[otonal division]] of the octave). It may also be called an EFDO ([[equal frequency division]] of the octave), however, this more general acronym is typically reserved for divisions of irrational intervals (unlike the octave) which are therefore not subsets of just intonation.
+When treated as a scale, the AFDO is equivalent to the [[overtone scale]]. However, an overtone scale often has an assumption of a tonic whereas an AFDO simply describes all the theoretically available pitch relations. Therefore, a passage built on 12::22 could be said to be in Mode 12, but is technically covered by 11afdo.
+An AFDO is equivalent to an ODO ([[otonal division]] of the octave). It may also be called an EFDO ([[equal frequency division]] of the octave), however, this more general acronym is typically reserved for divisions of irrational intervals (unlike the octave) which are therefore not subsets of just intonation.
 == Formula ==
@@ Line 40: / Line 42: @@
 In 2023, [[Flora Canou]] revived the old term, reinterpreting the word "arithmetic" as a reference to the [[Wikipedia: Arithmetic mean|arithmetic mean]] in addition to arithmetic progressions, then extended it through the other [[Pythagorean means]], and later through all [[power mean]]s. As it was shown that ''arithmetic'' alone was insufficient to define the object since frequency could not be assumed, the term was eventually changed to AFDO, showing both the type of power mean and the sampled resource.
+== Properties ==
+* ''n''-afdo has [[maximum variety]] ''n''.
+* Except for 1afdo and 2afdo, AFDOs are [[chiral]]. The inverse of ''n''-afdo is ''n''-ifdo.
+** 1afdo is equivalent to 1ifdo and 1edo;
+** 2afdo is equivalent to 2ifdo.
 == Individual pages for AFDOs ==
 === By size ===
 {| class="wikitable center-all"
@@ Line 270: / Line 277: @@
 === By prime family ===
 '''Over-2''': {{AFDOs|2, 4, 8, 16, 32, 64, 128, 256, 512}}
 '''Over-3''': {{AFDOs|3, 6, 12, 24, 48, 96, 192, 384}}
 '''Over-5''': {{AFDOs|5, 10, 20, 40, 80, 160, 320}}
 '''Over-7''': {{AFDOs|7, 14, 28, 56, 112, 224, 448}}
 '''Over-11''': {{AFDOs|11, 22, 44, 88, 176, 352}}
 '''Over-13''': {{AFDOs|13, 26, 52, 104, 208, 416}}
 '''Over-17''': {{AFDOs|17, 34, 68, 136, 272, 544}}
 '''Over-19''': {{AFDOs|19, 38, 76, 152, 304}}
 '''Over-23''': {{AFDOs|23, 46, 92, 184, 368}}
 '''Over-29''': {{AFDOs|29, 58, 116, 232, 464}}
 '''Over-31''': {{AFDOs|31, 62, 124, 248, 496}}
 === By other properties ===
 '''Prime''': {{AFDOs|2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271}}
 '''Semiprime''': {{AFDOs|4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 169, 177, 178, 183, 185, 187}}
 '''Odd squarefree semiprime''': {{AFDOs|15, 21, 33, 35, 39, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, 95, 111, 115, 119, 123, 129, 133, 141, 143, 145, 155, 159, 161, 177, 183, 185, 187, 201, 203, 205, 209, 213, 215, 217, 219, 221, 235, 237, 247, 249, 253, 259, 265, 267, 287, 291, 295, 299, 301, 303}}
 '''Nonprime prime power''': {{AFDOs|4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1024, 1331, 1369, 1681, 1849, 2048}}