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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).  
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 ==
== 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.  
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 ==  
== Individual pages for AFDOs ==  
=== By size ===
=== By size ===
{| class="wikitable center-all"
{| class="wikitable center-all"
Line 268: Line 275:
| [[199afdo|199]]
| [[199afdo|199]]
|}
|}
'''200 and above'''
*[[Over the Hedge|3746579]]


=== By prime family ===
=== By prime family ===


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


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


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


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


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


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


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


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


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


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


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


=== By other properties ===
=== 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}}
'''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}}
'''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}}
'''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}}
'''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}}
== Using AFDOs to make EDO interval tables ==
To make a quick and dirty table of intervals in an edo, you can go to [[Scale Workshop]] and do the following:
# Create a new Excel spreadsheet
# Make cell A1 = 1, A2 = 2 and so on. Flood-fill down to the size of your EDO
# Make cell B1 = (1200/EDO size)*A1 and flood-fill this formula all the way down column B
# Copy-paste column B into Scale Workshop
# Modify > approximate by harmonics > an afdo selected according to the table below (or choose your own)
# Modify > reduce
# Paste the result into the third (C) column in Excel
# Insert a row at the top of the spreadsheet to use as a header row
# Title the first column "Edo step", the second column “cents” and the third column "JI ratios"
Your table is now complete!
Do note that this is not a replacement for a manually curated table of intervals, nor is it even a replacement for a more cleverly computed table of intervals such as those generated by [[Template:Interval table]] or [[Template:Huge table]]. This method is very crude compared to those.
But it is very computationally light, so it can be helpful for really, really big edos that are difficult to document any other way.
Here is the table recommending what afdo to choose (i.e. what number to put for "approximate by harmonics"). The afdos recommended are the superior highly composite numbers (OEIS A002201):
{| class="wikitable"
|+
!EDO size
!Ideal AFDO to map to
!Included denominators
!Excluded denominators
|-
| 1edo to 20edo
| [[60afdo]]
| 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30
| 7, 8, 9, 11, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32
|-
| 21edo to 40edo
| [[120afdo]]
| 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30
| 7, 9, 11, 13, 14, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32
|-
| 41edo to 120edo
| [[360afdo]]
| 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30
| 7, 11, 13, 14, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32
|-
| 121edo to 840edo
| [[2520afdo]]
| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30
| 11, 13, 16, 17, 19, 22, 23, 25, 26, 27, 29, 31, 32
|-
| 841edo to 1680edo
| [[5040afdo]]
| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30
| 11, 13, 17, 19, 22, 23, 25, 26, 27, 29, 31, 32
|-
| 1681edo to 18480edo
| [[55440afdo]]
| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 28, 30
| 13, 17, 19, 23, 25, 26, 27, 29, 31, 32
|-
| 18481edo to 240240edo
| [[720720afdo]]
| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30
| 17, 19, 23, 25, 27, 29, 31, 32
|-
| 240241edo to 480480edo
| 1441440afdo
| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32
| 17, 19, 23, 29, 31
|-
| 480480edo to 1,441,440edo
| 4324320afdo
| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32
| 17, 19, 23, 29, 31
|-
| 1,441,441edo to 7,207,200edo
| 21621600afdo
| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32
| 17, 19, 23, 29, 31
|-
| 7,207,201edo to 122,522,400edo
| 367567200afdo
| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32
| 19, 23, 29, 31
|-
| 122,522,401edo to 2,327,925,600edo
| 6983776800afdo
| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32
| 23, 29, 31
|}


== See also ==
== See also ==
Retrieved from "https://en.xen.wiki/w/AFDO"