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An '''AFDO''' ('''arithmetic frequency division of the octave''') or '''ODO''' ('''otonal division of the octave''') is a [[period]]ic [[tuning system]] which divides the [[octave]] according to an [[Wikipedia: Arithmetic progression|arithmetic progression]] of frequency.  
An '''AFDO''' ('''arithmetic frequency division of the octave''') or '''ODO''' ('''otonal division of the octave''') is a [[period]]ic [[tuning system]] which divides the [[octave]] according to an [[Wikipedia: Arithmetic progression|arithmetic progression]] of frequency.  


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]]. AFDOs with more divisors such as [[Highly composite equal division|highly composite]] AFDOs generally have more useful just intervals.  
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.  
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 ==
Line 10: Line 12:
<math>\displaystyle r = (n + m)/n</math>
<math>\displaystyle r = (n + m)/n</math>


If the first division is ''r''<sub>0</sub> (which is ratio of (''n'' + 0)/''n'' = 1) and the last, ''r''<sub>''n''</sub> (which is ratio of (''n'' + ''n'')/''n'' = 2), with common difference of ''d'' (which is 1/''n''), we have:  
Alternatively, with common frequency difference ''d'' = 1/''n'', we have:  


<math>
<math>
r_1 = r_0 + d \\
r = 1 + md
r_2 = r_0 + 2d \\
r_3 = r_0 + 3d \\
\vdots \\
r_m = r_0 + md
</math>
</math>
In particular, when ''m'' = 0, ''r'' = 1, and when ''m'' = ''n'', ''r'' = 2.


== Relation to string lengths ==
== Relation to string lengths ==
If the first division has ratio of ''r''<sub>1</sub> and length of ''l''<sub>1</sub> and the last, ''r''<sub>''n''</sub> and L<sub>''n''</sub> , we have: ''l''<sub>''n''</sub> = 1/''r''<sub>''n''</sub> and if ''r''<sub>''n''</sub> &gt; … &gt; ''r''<sub>3</sub> &gt; ''r''<sub>2</sub> &gt; ''r''<sub>1</sub>, then ''l''<sub>1</sub> &gt; ''l''<sub>2</sub> &gt; ''l''<sub>3</sub> &gt; … &gt; ''l''<sub>''n''</sub>
If the first division has ratio of ''r''<sub>1</sub> and length of ''l''<sub>1</sub> and the last, ''r''<sub>''n''</sub> and ''l''<sub>''n''</sub> , we have: ''l''<sub>''n''</sub> = 1/''r''<sub>''n''</sub> and if ''r''<sub>''n''</sub> &gt; … &gt; ''r''<sub>3</sub> &gt; ''r''<sub>2</sub> &gt; ''r''<sub>1</sub>, then ''l''<sub>1</sub> &gt; ''l''<sub>2</sub> &gt; ''l''<sub>3</sub> &gt; … &gt; ''l''<sub>''n''</sub>


[[File:ADO-4.jpg|350px|center]]
[[File:ADO-4.jpg|350px|center]]
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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.  


== Individual pages for AFDOs ==
== Properties ==
* <nowiki>*</nowiki>1afdo – identical to [[1edo]]
* ''n''-afdo has [[maximum variety]] ''n''.
* [[2afdo]]
* Except for 1afdo and 2afdo, AFDOs are [[chiral]]. The inverse of ''n''-afdo is ''n''-ifdo.
* [[3afdo]]
** 1afdo is equivalent to 1ifdo and 1edo;
* [[4afdo]]
** 2afdo is equivalent to 2ifdo.
* [[5afdo]]
 
* [[6afdo]]
== Individual pages for AFDOs ==  
* [[7afdo]]
=== By size ===
* [[8afdo]]
{| class="wikitable center-all"
* [[9afdo]]
|+ style=white-space:nowrap | 0…99
* [[10afdo]]
| [[0afdo|0]]
* [[11afdo]]
| [[1edo|1]]
* [[12afdo]]
| [[2afdo|2]]
* [[13afdo]]
| [[3afdo|3]]
* [[14afdo]]
| [[4afdo|4]]
* [[15afdo]]
| [[5afdo|5]]
* [[16afdo]]
| [[6afdo|6]]
* [[17afdo]]
| [[7afdo|7]]
* [[18afdo]]
| [[8afdo|8]]
* [[19afdo]]
| [[9afdo|9]]
* [[20afdo]]
|-
* [[21afdo]]
| [[10afdo|10]]
* [[22afdo]]
| [[11afdo|11]]
* [[23afdo]]
| [[12afdo|12]]
* [[24afdo]]
| [[13afdo|13]]
* [[25afdo]]
| [[14afdo|14]]
* [[26afdo]]
| [[15afdo|15]]
* [[27afdo]]
| [[16afdo|16]]
* [[28afdo]]
| [[17afdo|17]]
* [[29afdo]]
| [[18afdo|18]]
* [[30afdo]]
| [[19afdo|19]]
* [[31afdo]]
|-
* [[32afdo]]
| [[20afdo|20]]
* [[33afdo]]
| [[21afdo|21]]
* [[34afdo]]
| [[22afdo|22]]
* [[35afdo]]
| [[23afdo|23]]
* [[36afdo]]
| [[24afdo|24]]
| [[25afdo|25]]
| [[26afdo|26]]
| [[27afdo|27]]
| [[28afdo|28]]
| [[29afdo|29]]
|-
| [[30afdo|30]]
| [[31afdo|31]]
| [[32afdo|32]]
| [[33afdo|33]]
| [[34afdo|34]]
| [[35afdo|35]]
| [[36afdo|36]]
| [[37afdo|37]]
| [[38afdo|38]]
| [[39afdo|39]]
|-
| [[40afdo|40]]
| [[41afdo|41]]
| [[42afdo|42]]
| [[43afdo|43]]
| [[44afdo|44]]
| [[45afdo|45]]
| [[46afdo|46]]
| [[47afdo|47]]
| [[48afdo|48]]
| [[49afdo|49]]
|-
| [[50afdo|50]]
| [[51afdo|51]]
| [[52afdo|52]]
| [[53afdo|53]]
| [[54afdo|54]]
| [[55afdo|55]]
| [[56afdo|56]]
| [[57afdo|57]]
| [[58afdo|58]]
| [[59afdo|59]]
|-
| [[60afdo|60]]
| [[61afdo|61]]
| [[62afdo|62]]
| [[63afdo|63]]
| [[64afdo|64]]
| [[65afdo|65]]
| [[66afdo|66]]
| [[67afdo|67]]
| [[68afdo|68]]
| [[69afdo|69]]
|-
| [[70afdo|70]]
| [[71afdo|71]]
| [[72afdo|72]]
| [[73afdo|73]]
| [[74afdo|74]]
| [[75afdo|75]]
| [[76afdo|76]]
| [[77afdo|77]]
| [[78afdo|78]]
| [[79afdo|79]]
|-
| [[80afdo|80]]
| [[81afdo|81]]
| [[82afdo|82]]
| [[83afdo|83]]
| [[84afdo|84]]
| [[85afdo|85]]
| [[86afdo|86]]
| [[87afdo|87]]
| [[88afdo|88]]
| [[89afdo|89]]
|-
| [[90afdo|90]]
| [[91afdo|91]]
| [[92afdo|92]]
| [[93afdo|93]]
| [[94afdo|94]]
| [[95afdo|95]]
| [[96afdo|96]]
| [[97afdo|97]]
| [[98afdo|98]]
| [[99afdo|99]]
|}
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | 100…199
| [[100afdo|100]]
| [[101afdo|101]]
| [[102afdo|102]]
| [[103afdo|103]]
| [[104afdo|104]]
| [[105afdo|105]]
| [[106afdo|106]]
| [[107afdo|107]]
| [[108afdo|108]]
| [[109afdo|109]]
|-
| [[110afdo|110]]
| [[111afdo|111]]
| [[112afdo|112]]
| [[113afdo|113]]
| [[114afdo|114]]
| [[115afdo|115]]
| [[116afdo|116]]
| [[117afdo|117]]
| [[118afdo|118]]
| [[119afdo|119]]
|-
| [[120afdo|120]]
| [[121afdo|121]]
| [[122afdo|122]]
| [[123afdo|123]]
| [[124afdo|124]]
| [[125afdo|125]]
| [[126afdo|126]]
| [[127afdo|127]]
| [[128afdo|128]]
| [[129afdo|129]]
|-
| [[130afdo|130]]
| [[131afdo|131]]
| [[132afdo|132]]
| [[133afdo|133]]
| [[134afdo|134]]
| [[135afdo|135]]
| [[136afdo|136]]
| [[137afdo|137]]
| [[138afdo|138]]
| [[139afdo|139]]
|-
| [[140afdo|140]]
| [[141afdo|141]]
| [[142afdo|142]]
| [[143afdo|143]]
| [[144afdo|144]]
| [[145afdo|145]]
| [[146afdo|146]]
| [[147afdo|147]]
| [[148afdo|148]]
| [[149afdo|149]]
|-
| [[150afdo|150]]
| [[151afdo|151]]
| [[152afdo|152]]
| [[153afdo|153]]
| [[154afdo|154]]
| [[155afdo|155]]
| [[156afdo|156]]
| [[157afdo|157]]
| [[158afdo|158]]
| [[159afdo|159]]
|-
| [[160afdo|160]]
| [[161afdo|161]]
| [[162afdo|162]]
| [[163afdo|163]]
| [[164afdo|164]]
| [[165afdo|165]]
| [[166afdo|166]]
| [[167afdo|167]]
| [[168afdo|168]]
| [[169afdo|169]]
|-
| [[170afdo|170]]
| [[171afdo|171]]
| [[172afdo|172]]
| [[173afdo|173]]
| [[174afdo|174]]
| [[175afdo|175]]
| [[176afdo|176]]
| [[177afdo|177]]
| [[178afdo|178]]
| [[179afdo|179]]
|-
| [[180afdo|180]]
| [[181afdo|181]]
| [[182afdo|182]]
| [[183afdo|183]]
| [[184afdo|184]]
| [[185afdo|185]]
| [[186afdo|186]]
| [[187afdo|187]]
| [[188afdo|188]]
| [[189afdo|189]]
|-
| [[190afdo|190]]
| [[191afdo|191]]
| [[192afdo|192]]
| [[193afdo|193]]
| [[194afdo|194]]
| [[195afdo|195]]
| [[196afdo|196]]
| [[197afdo|197]]
| [[198afdo|198]]
| [[199afdo|199]]
|}
 
=== 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}}


== See also ==
== See also ==
* [http://240edo.googlepages.com/ADO-EDL.XLS Fret position calculator] (excel sheet) based on EDL system and string length
* [[AFS]] (arithmetic frequency sequence)
* How to approximate EDO and AFDO systems with each other? [https://sites.google.com/site/240edo/ADOandEDO.xls Download this file]{{dead link}}
* [[IFDO]] (inverse-arithmetic frequency division of the octave)
* [http://www.soundtransformations.btinternet.co.uk/Danerhudyarthemarmonicseries.htm Magic of Tone and the Art of Music by the late Dane Rhudyar]
* [[AFS]]
* [[Frequency temperament]]
* [[Frequency temperament]]
* [[5- to 10-tone scales from the modes of the harmonic series]]
== External links ==
* [https://sites.google.com/site/240edo/ADOandEDO.xls Approximate EDO and AFDO systems with each other (Excel sheet)]{{dead link}}
* [http://240edo.googlepages.com/ADO-EDL.XLS Fret position calculator (Excel sheet)]
* [http://www.soundtransformations.btinternet.co.uk/Danerhudyarthemarmonicseries.htm Magic of Tone and the Art of Music] by the late [[Dane Rhudyar]]
== Notes ==
== Notes ==


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