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~~~~'''~~Arithmetic rational~~''' '''~~divisions~~ of octave''' ~~~~

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.

~~~~'''~~ARDO~~''' ~~(which is simplified as '~~''[~~http://sites~~.~~google~~.~~com/site/240edo/arithmeticrationaldivisionsofoctave ADO])~~''' is ~~an intervallic system considered as </span~~>

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).

~~~~[~~http://www~~.~~richland~~.~~edu/james/lecture/m116/sequences/arithmetic.html arithmetic sequence] with divisions of system as terms of sequence~~. ~~~~

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.

~~If the first division~~ is ~~'''R1'''~~ (~~wich is ratio~~ of ~~C/C~~) ~~and~~ the ~~last~~ , ~~'''Rn''' (wich~~ is ~~ratio~~ of ~~2C/C~~)~~, with common difference~~ of ~~'''d'''~~

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.

~~(which is~~ '''~~1/C~~'''~~), we have : ~~

== Formula ==

Within each period of any ''n''-afdo system, the [[frequency ratio]] ''r'' of the ''m''-th degree is

<~~span style="display: block; text-align: left;">~~'''R2~~ = R1+~~d''' <~~/~~span>~~</~~span>

<math>\displaystyle r = (n + m)/n</math>

~~~~'''R3= ~~R1+2d~~''' ~~~~

Alternatively, with common frequency difference ''d'' = 1/''n'', we have:

<~~span style="display: block; text-align: left;"~~>~~'''R4 = R1~~+~~3d '''~~</~~span~~>

<math>

r = 1 + md

</math>

~~~~'''~~………~~'''~~~~

In particular, when ''m'' = 0, ''r'' = 1, and when ''m'' = ''n'', ''r'' = 2.

<~~span style~~=~~"display: block~~; ~~text-align: left~~;"><~~span style="font-family: Times New Roman~~;">'''<~~span style="color: black; font-family: arial~~,~~sans-serif~~; ~~font-size: 15px~~;">~~Rn = R1+(n-1)d~~</~~span~~>'''<~~/span~~></~~span~~>

== Relation to string lengths ==

If the first division has ratio of ''r''1 and length of ''l''1 and the last, ''r''''n'' and ''l''''n'' , we have: ''l''''n'' = 1/''r''''n'' and if ''r''''n'' > … > ''r''3 > ''r''2 > ''r''1, then ''l''1 > ''l''2 > ''l''3 > … > ''l''''n''

~~Each consequent divisions like '''R4''' and '''R3''' have a difference of '''d''' with each other.The concept of division here is a bit different from '''EDO''' and other systems (which is the difference of cents of two consequent degree). In '''ADO~~''', a division is frequency-related and is the ratio of each degree due to the first degree.For example ratio of 1.5 is the size of 3/2 in 12~~-~~ADO system~~.~~

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

~~For any '''C-ADO'''~~ system ~~with~~ [~~http~~://~~www~~.~~tonalsoft~~.com/~~enc~~/c/cardinality.aspx **cardinality**] ~~of '''C''', we have ratios related to different degrees of '''m''' as~~ : ~~~~

These lengths are related to the inverse of ratios in the system. The above picture shows the differences between divisions of length in 12ado system. On the contrary, we have equal divisions of length in [[EDL]] systems (→ [https://sites.google.com/site/240edo/equaldivisionsoflength(edl) EDL system]):

~~(C+m/C)~~

[[File:ADO-5.jpg|346px|center]]

~~~~For example , ~~in '''12-ADO''' the ratio related to the first degree is 13~~/~~12 .<~~/~~span><~~/~~span>~~

== Relation to superparticular ratios ==

An AFDO has step sizes of [[superparticular ratio]]s with increasing numerators. For example, [[5ado]] has step sizes of [[6/5]], [[7/6]], [[8/7]], and [[9/8]].

~~'''12-ADO''' can be shown as~~ series like: '''12:13'''''':14:15:16:17:18:19:20:21:22:23:24''' or '''12 13 '''14 15 16 17 18 19 20 21 22 23 24 '''.'''

== Relation to otonality & harmonic series ==

~~For~~ an ~~'''ADO''' intervallic~~ system ~~with '''n''' divisions we have unequal divisions of length ~~by ~~dividing string length~~ to~~'''n''' unequal divisions based on each degree ratios.If~~ the ~~first division has ratio~~ of ~~'''R1''' and length~~ of ~~'''L1<~~/~~span>'''~~ and ~~the last~~, ~~'''Rn''' and '''Ln<~~/~~span>'''~~ , ~~we have: '''Ln = 1~~/~~Rn''' and if '''Rn >.....~~...~~> R3 > R2 > R1''' so : ~~

We can consider an AFDO system as an [[otonal]] system. Otonality is a term introduced by [[Harry Partch]] to describe chords whose notes are the overtones (multiples) of a given fixed tone. Considering AFDO, an otonality is a collection of pitches which can be expressed in ratios that have equal denominators. For example, 1/1, 5/4, and 3/2 form an otonality because they can be written as 4/4, 5/4, 6/4. Every otonality is therefore part of the harmonic series. An otonality corresponds to an arithmetic series of frequencies or a harmonic series of wavelengths or distances on a string instrument.

<~~span style="display~~: ~~block; text~~-~~align: left;~~">'''~~L1 > L2 > L3 > …… > Ln~~'''~~~~

== History ==

In the earliest materials, the AFDO was known as the ADO, for ''arithmetic division of the octave''. The term was proposed by [[Shaahin Mohajeri]] in 2006, along with the term [[EDL]] (equal division of length)<ref>[https://yahootuninggroupsultimatebackup.github.io/makemicromusic/topicId_13427.html#13427 Yahoo! Tuning Group | ''for c.m.bryan , about ado and edl'' ]</ref>. Previously, the set of pitch materials equivalent to ''n''-ado's had been known as "mode ''n'' of the harmonic series", "over-''n'' scales", and ''n''-edl's had been known as "aliquot-''n''" scales, as the distinction between tunings and scales were not made. Neither of Shaahin's two new concepts were systematic extensions of the term [[EDO]] (equal division of the octave), and no one else used these two terms besides Shaahin himself.

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

In 2021, a team consisting of [[Douglas Blumeyer]], [[Billy Stiltner]], and [[Paul Erlich]] developed the first systematic extension of EDO from equal divisions of pitch to equal divisions of frequency and length, including special terms for divisions of rational intervals such as the octave; under this system, an ''n''-ADO would be an ''n''-ODO.

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"

|+ style=white-space:nowrap | 0…99

| [[0afdo|0]]

| [[1edo|1]]

| [[2afdo|2]]

| [[3afdo|3]]

| [[4afdo|4]]

| [[5afdo|5]]

| [[6afdo|6]]

| [[7afdo|7]]

| [[8afdo|8]]

| [[9afdo|9]]

|-

| [[10afdo|10]]

| [[11afdo|11]]

| [[12afdo|12]]

| [[13afdo|13]]

| [[14afdo|14]]

| [[15afdo|15]]

| [[16afdo|16]]

| [[17afdo|17]]

| [[18afdo|18]]

| [[19afdo|19]]

|-

| [[20afdo|20]]

| [[21afdo|21]]

| [[22afdo|22]]

| [[23afdo|23]]

| [[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}}

~~This lengths are related to reverse of ratios in system.The above picture shows the differences between divisions of length in~~ 12~~-ADO system . On the contrary~~ , ~~we have equal divisios of length in [http://sites.google.com/site/240edo/equaldivisionsoflength(edl) **EDL system**]:~~

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

~~[[File:ADO~~-5~~.jpg~~|~~346px|center]]~~

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

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

~~[[File~~:~~ADO-3.jpg|604px~~|~~center]]~~

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

~~~~'''~~Relation between harmonics and ADO system~~'''~~~~

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

~~~~'''~~ADO'~~'' ~~(like~~ '''EDL)''' is based on [http://en.wikipedia.org/wiki/Superparticular_number **Superparticular ratios**] and [http://en.wikipedia.org/wiki/Harmonic_series_%28music%29 **harmonic series**]. Have a look at 12-ADO in this picture:~~~~

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

~~[[File~~:~~ADO-2.jpg|378px~~|~~center]]~~

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

~~The above picture shows that~~ '''~~ADO~~''' ~~system is classified as~~ :~~~~

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

- System with unequal [http://tonalsoft.com/enc/e/epimorios.aspx **epimorios**] '''('''~~[http~~:~~//en.wikipedia.org/wiki/Superparticular_number **Superparticular**]''')''' divisions.~~

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

~~- System based on ascending series of superparticular ratios with descending sizes.~~

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

- System which covers superparticular ratios between harmonic of number C (in this example 12)to harmonic of Number 2C(in this example 24).

=== By other properties ===

~~~~'''~~- [http://sites.google.com/site/240edo/ADO-EDL.XLS An spreadsheet showing relation between harmonics~~ , ~~superparticular ratios and ADO system]'''~~

'''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}}

~~~~'''-''' ~~[http://www.music.sc.edu/fs/bain/software/BainTheOvertoneSeries.pdf The Overtone Series]~~

'''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}}

~~~~'''~~Relation between Otonality and ADO system~~'''~~~~

'''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}}

~~We can consider ~~'''~~ADO~~'''~~ system as [http://en.wikipedia.org/wiki/Otonal **Otonal system**] .'''Otonality''' is a term introduced by [http://en.wikipedia.org/wiki/Harry_Partch **Harry Partch**] to describe chords whose notes are the overtones (multiples) of a given fixed tone.Considering ADO , ~~an Otonality is a collection of pitches which can be expressed in ratios that have equal denominators. For example~~, ~~1/1~~, ~~5/4~~, ~~and 3/2 form an Otonality because they can be written as 4/4~~, ~~5/4, 6/4. Every Otonality is therefore part of the [http://en.wikipedia.org/wiki/Harmonic_series_%28music%29 **harmonic series**]. nominator here is called "[http://tonalsoft.com/enc/n/nexus.aspx **Numerary nexus**]".An Otonality corresponds to an [http://en.wikipedia.org/wiki/Arithmetic_series **arithmetic series**] of frequencies or a [http://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29 **harmonic series**] of wavelengths or distances on a [http://en.wikipedia.org/wiki/String_instrument **string instrument**].

'''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}}

~~'''- [http://240edo.googlepages.com/ADO~~-~~EDL.XLS Fret position calculator (excel sheet ) based on EDL system and string length~~]~~'''~~

== See also ==

* [[AFS]] (arithmetic frequency sequence)

* [[IFDO]] (inverse-arithmetic frequency division of the octave)

* [[Frequency temperament]]

* [[5- to 10-tone scales from the modes of the harmonic series]]

~~~~[~~http~~://sites.google.com/site/240edo/ADOandEDO.xls ~~- How to approximate EDand ADO~~ systems with each other~~?Download this file~~]</~~span><~~/~~span><~~/~~span>~~

== 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]]

~~'''Related to ADO'''~~

== Notes ==

<~~span style="display: block; text~~-~~align: center;">[~~http~~://www.soundtransformations.btinternet.co.uk/Danerhudyarthemarmonicseries.htm **Magic of Tone and the Art of Music by the late Dane Rhudyar**]~~~~ [[Category:~~ADO]]

[[Category:AFDO| ]]

[[Category:~~todo:cleanup~~]]

[[Category:Acronyms]]

[[Category:Lists of scales]]

[[Category:Just intonation]]

@@ Line 1: / Line 1: @@
-<span style="color: #000000; font-family: arial,sans-serif; font-size: 140%;">'''Arithmetic rational''' '''divisions of octave''' </span>
+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.
-<span style="display: block; text-align: left;"><span style="color: black; font-family: arial; font-size: 13px;">'''ARDO''' (which is simplified as '''[http://sites.google.com/site/240edo/arithmeticrationaldivisionsofoctave ADO])''' is an intervallic system <span style="color: black; font-family: arial,sans-serif; font-size: 15px;">considered as </span></span></span>
+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).
-<span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">[http://www.richland.edu/james/lecture/m116/sequences/arithmetic.html arithmetic sequence] with divisions of system as <span style="color: black; font-family: arial,sans-serif; font-size: 15px;">terms of sequence. </span></span></span>
+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.
-<span style="display: block; text-align: left;"><span style="font-family: Times New Roman;"><span style="font-family: arial,sans-serif;">If the first division is <u>'''R1'''</u> (wich is ratio of C/C) and the last , <u>'''Rn'''</u> </span><span style="color: black; font-size: 15px;">(wich is ratio of 2C/C), with common difference of </span><u><span style="color: black; font-size: 15px;">'''d'''</span></u></span></span>
+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.
-<span style="display: block; text-align: center;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">(which is '''1/C'''), we have : </span></span>
+== Formula ==
+Within each period of any ''n''-afdo system, the [[frequency ratio]] ''r'' of the ''m''-th degree is
-<span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">'''R2 = R1+d''' </span></span>
+<math>\displaystyle r = (n + m)/n</math>
-<span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">'''R3= R1+2d''' </span></span>
+Alternatively, with common frequency difference ''d'' = 1/''n'', we have:
-<span style="display: block; text-align: left;"><span style="font-family: arial,sans-serif;">'''<span style="color: black; font-size: 15px;">R4 = R1+3d </span>'''</span></span>
+<math>
+r = 1 + md
+</math>
-<span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">'''………'''</span></span>
+In particular, when ''m'' = 0, ''r'' = 1, and when ''m'' = ''n'', ''r'' = 2.
-<span style="display: block; text-align: left;"><span style="font-family: Times New Roman;">'''<span style="color: black; font-family: arial,sans-serif; font-size: 15px;">Rn = R1+(n-1)d</span>'''</span></span>
+== 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>
-<span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">Each consequent divisions like '''R4''' and '''R3''' have a difference of '''d''' with each other.The concept of division here is a bit different from '''EDO''' and other systems (which is the difference of cents of two consequent degree). In '''ADO''', a division is frequency-related and is the ratio of each degree due to the first degree.For example ratio of 1.5 is the size of 3/2 in 12-ADO system.</span></span>
+[[File:ADO-4.jpg|350px|center]]
-<span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">For any '''C-ADO''' system with [http://www.tonalsoft.com/enc/c/cardinality.aspx **cardinality**] of '''C''', we have ratios related to different degrees of '''m''' as : </span></span>
+These lengths are related to the inverse of ratios in the system. The above picture shows the differences between divisions of length in 12ado system. On the contrary, we have equal divisions of length in [[EDL]] systems (→ [https://sites.google.com/site/240edo/equaldivisionsoflength(edl) EDL system]):
-<span style="display: block; text-align: center;">(C+m/C)</span>
+[[File:ADO-5.jpg|346px|center]]
-<span style="display: block; text-align: left;"><span style="font-family: arial,sans-serif;">For example , in '''12-ADO''' the ratio related to the first degree is 13/12 .</span></span>
+== Relation to superparticular ratios ==
+An AFDO has step sizes of [[superparticular ratio]]s with increasing numerators. For example, [[5ado]] has step sizes of [[6/5]], [[7/6]], [[8/7]], and [[9/8]].
-<span style="display: block; text-align: left;"><span style="font-family: arial,sans-serif;">'''12-ADO''' can be shown as series like: </span>'''<span style="color: black; font-family: Arial; font-size: 13px;">12:13</span>''''''<span style="color: black; font-family: Arial; font-size: 13px;">:14:15:16:17:18:19:20:21:22:23:24</span>'''<span style="color: black; font-family: arial; font-size: 13px;"> or </span>'''<span style="color: black; font-family: arial; font-size: 13px;">12 13 </span>'''<span style="color: black; font-family: Arial; font-size: 13px;">14 15 16 17 18 19 20 21 22 23 24</span> '''<span style="color: black; font-family: Arial; font-size: 13px;">.</span>'''</span>
+== Relation to otonality & harmonic series ==
-<span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">For an '''ADO''' intervallic system with '''n''' divisions we have <span style="font-family: arial,sans-serif;">unequal divisions of length </span>by dividing string length to'''<span style="color: black; font-family: Arial; font-size: 13px;">n</span>''' unequal divisions based on each degree ratios.If the first division has ratio of '''R1''' and length of '''<span style="color: black; font-family: Arial; font-size: 13px;">L1</span>''' and the last, '''Rn''' and '''<span style="color: black; font-family: Arial; font-size: 13px;">Ln</span>''' , we have: '''Ln = 1/Rn''' and if '''Rn &gt;........&gt; R3 &gt; R2 &gt; R1''' so : </span></span>
+We can consider an AFDO system as an [[otonal]] system. Otonality is a term introduced by [[Harry Partch]] to describe chords whose notes are the overtones (multiples) of a given fixed tone. Considering AFDO, an otonality is a collection of pitches which can be expressed in ratios that have equal denominators. For example, 1/1, 5/4, and 3/2 form an otonality because they can be written as 4/4, 5/4, 6/4. Every otonality is therefore part of the harmonic series. An otonality corresponds to an arithmetic series of frequencies or a harmonic series of wavelengths or distances on a string instrument.
-<span style="display: block; text-align: left;">'''<span style="color: black; font-family: Arial; font-size: 13px;">L1 &gt; L2 &gt; L3 &gt; …… &gt; Ln</span>'''</span>
+== History ==
+In the earliest materials, the AFDO was known as the ADO, for ''arithmetic division of the octave''. The term was proposed by [[Shaahin Mohajeri]] in 2006, along with the term [[EDL]] (equal division of length)<ref>[https://yahootuninggroupsultimatebackup.github.io/makemicromusic/topicId_13427.html#13427 Yahoo! Tuning Group | ''for c.m.bryan , about ado and edl'' ]</ref>. Previously, the set of pitch materials equivalent to ''n''-ado's had been known as "mode ''n'' of the harmonic series", "over-''n'' scales", and ''n''-edl's had been known as "aliquot-''n''" scales, as the distinction between tunings and scales were not made. Neither of Shaahin's two new concepts were systematic extensions of the term [[EDO]] (equal division of the octave), and no one else used these two terms besides Shaahin himself.
-[[File:ADO-4.jpg|350px|center]]
+In 2021, a team consisting of [[Douglas Blumeyer]], [[Billy Stiltner]], and [[Paul Erlich]] developed the first systematic extension of EDO from equal divisions of pitch to equal divisions of frequency and length, including special terms for divisions of rational intervals such as the octave; under this system, an ''n''-ADO would be an ''n''-ODO.
+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"
+|+ style=white-space:nowrap | 0…99
+| [[0afdo|0]]
+| [[1edo|1]]
+| [[2afdo|2]]
+| [[3afdo|3]]
+| [[4afdo|4]]
+| [[5afdo|5]]
+| [[6afdo|6]]
+| [[7afdo|7]]
+| [[8afdo|8]]
+| [[9afdo|9]]
+|-
+| [[10afdo|10]]
+| [[11afdo|11]]
+| [[12afdo|12]]
+| [[13afdo|13]]
+| [[14afdo|14]]
+| [[15afdo|15]]
+| [[16afdo|16]]
+| [[17afdo|17]]
+| [[18afdo|18]]
+| [[19afdo|19]]
+|-
+| [[20afdo|20]]
+| [[21afdo|21]]
+| [[22afdo|22]]
+| [[23afdo|23]]
+| [[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}}
-<span style="display: block; text-align: left;"><span style="color: black; font-family: arial; font-size: 13px;">This lengths are related to reverse of ratios in system.The above picture shows the differences between divisions of length in 12-ADO system . On the contrary , we have equal divisios of length in [http://sites.google.com/site/240edo/equaldivisionsoflength(edl) **EDL system**]:</span></span>
+'''Over-3''': {{AFDOs|3, 6, 12, 24, 48, 96, 192, 384}}
-[[File:ADO-5.jpg|346px|center]]
+'''Over-5''': {{AFDOs|5, 10, 20, 40, 80, 160, 320}}
+'''Over-7''': {{AFDOs|7, 14, 28, 56, 112, 224, 448}}
-[[File:ADO-3.jpg|604px|center]]
+'''Over-11''': {{AFDOs|11, 22, 44, 88, 176, 352}}
-<span style="display: block; text-align: center;">'''<span style="color: black; font-family: Arial; font-size: 13px;"><u>Relation between harmonics and ADO system</u></span>'''</span>
+'''Over-13''': {{AFDOs|13, 26, 52, 104, 208, 416}}
-<span style="display: block; text-align: left;"><span style="color: black; font-family: arial; font-size: 13px;">'''ADO''' (like '''EDL)''' is based on [http://en.wikipedia.org/wiki/Superparticular_number **Superparticular ratios**] and [http://en.wikipedia.org/wiki/Harmonic_series_%28music%29 **harmonic series**]. Have a look at 12-ADO in this picture:</span></span>
+'''Over-17''': {{AFDOs|17, 34, 68, 136, 272, 544}}
-[[File:ADO-2.jpg|378px|center]]
+'''Over-19''': {{AFDOs|19, 38, 76, 152, 304}}
-<span style="color: black; font-family: arial; font-size: 13px;">The above picture shows that '''ADO''' system is classified as :</span>
+'''Over-23''': {{AFDOs|23, 46, 92, 184, 368}}
-<span style="display: block; text-align: left;"><span style="font-family: Times New Roman;"><span style="color: black; font-family: arial; font-size: 13px;">- System with unequal </span><span style="color: blue; font-family: arial; font-size: 13px;">[http://tonalsoft.com/enc/e/epimorios.aspx **epimorios**]</span><span style="color: black; font-family: arial; font-size: 13px;"> '''('''[http://en.wikipedia.org/wiki/Superparticular_number **Superparticular**]''')''' divisions.</span></span></span>
+'''Over-29''': {{AFDOs|29, 58, 116, 232, 464}}
-<span style="display: block; text-align: left;"><span style="color: black; font-family: arial; font-size: 13px;">- System based on ascending series of superparticular ratios with descending sizes.</span></span>
+'''Over-31''': {{AFDOs|31, 62, 124, 248, 496}}
-<span style="display: block; text-align: left;"><span style="color: black; font-family: arial; font-size: 13px;">- System which covers superparticular ratios between harmonic of number C (in this example 12)to harmonic of Number 2C(in this example 24).</span></span>
+=== By other properties ===
-<span style="display: block; text-align: left;"><span style="color: black; font-family: arial,sans-serif; font-size: 15px;">'''- <span style="font-family: arial,sans-serif;">[http://sites.google.com/site/240edo/ADO-EDL.XLS An spreadsheet showing relation between harmonics , superparticular ratios and ADO system]</span>'''</span></span>
+'''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}}
-<span style="display: block; text-align: left;"><span style="color: black; font-family: arial; font-size: 15px;">'''-''' <span style="font-family: arial,sans-serif;">[http://www.music.sc.edu/fs/bain/software/BainTheOvertoneSeries.pdf The Overtone Series]</span></span></span>
+'''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}}
-<span style="display: block; text-align: center;"><span style="color: black; font-family: arial; font-size: 15px;">'''<span style="color: black; font-family: Arial; font-size: 13px;"><u>Relation between Otonality and ADO system</u></span>'''</span></span>
+'''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}}
-<span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">We can consider </span>'''<span style="color: black; font-family: Arial; font-size: 13px;">ADO</span>'''<span style="color: black; font-family: Arial; font-size: 13px;"> system as </span><span style="color: blue; font-family: Arial; font-size: 13px;">[http://en.wikipedia.org/wiki/Otonal **Otonal system**]</span><span style="color: black; font-family: Arial; font-size: 13px;"> .'''Otonality''' is a term introduced by </span><span style="color: blue; font-family: Arial; font-size: 13px;">[http://en.wikipedia.org/wiki/Harry_Partch **Harry Partch**]</span><span style="color: black; font-family: Arial; font-size: 13px;"> to describe chords whose notes are the overtones (multiples) of a given fixed tone.Considering ADO , an Otonality is a collection of pitches which can be expressed in ratios that have equal denominators. For example, 1/1, 5/4, and 3/2 form an Otonality because they can be written as 4/4, 5/4, 6/4. Every Otonality is therefore part of the [http://en.wikipedia.org/wiki/Harmonic_series_%28music%29 **harmonic series**]. nominator here is called "</span><span style="color: blue; font-family: Arial; font-size: 13px;">[http://tonalsoft.com/enc/n/nexus.aspx **Numerary nexus**]</span><span style="color: black; font-family: Arial; font-size: 13px;">".An Otonality corresponds to an [http://en.wikipedia.org/wiki/Arithmetic_series **arithmetic series**] of frequencies or a [http://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29 **harmonic series**] of wavelengths or distances on a [http://en.wikipedia.org/wiki/String_instrument **string instrument**].</span></span>
+'''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}}
-<span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;">'''<span style="color: black; font-family: 'Times New Roman'; font-size: 13px;">- </span><u><span style="color: windowtext; font-family: 'times new roman'; font-size: 16px;">[http://240edo.googlepages.com/ADO-EDL.XLS Fret position calculator (excel sheet ) based on EDL system and string length]</span></u>'''</span></span>
+== See also ==
+* [[AFS]] (arithmetic frequency sequence)
+* [[IFDO]] (inverse-arithmetic frequency division of the octave)
+* [[Frequency temperament]]
+* [[5- to 10-tone scales from the modes of the harmonic series]]
-<span style="display: block; text-align: left;"><span style="color: black; font-family: Arial; font-size: 13px;"><span style="color: #0000ff; font-family: arial,sans-serif; font-size: 16px;">[http://sites.google.com/site/240edo/ADOandEDO.xls - How to approximate EDand ADO systems with each other?Download this file]</span></span></span>
+== 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]]
-<span style="display: block; text-align: center;"><span style="color: black; font-family: Arial; font-size: 13px;">'''<u><span style="color: windowtext; font-family: arial,sans-serif; font-size: 16px;">Related to ADO</span></u>'''</span></span>
+== Notes ==
-<span style="display: block; text-align: center;"><span style="color: black; font-family: arial; font-size: 24px;">[http://www.soundtransformations.btinternet.co.uk/Danerhudyarthemarmonicseries.htm **Magic of Tone and the Art of Music by the late Dane Rhudyar**]</span></span>      [[Category:ADO]]
+[[Category:AFDO| ]] <!-- main article -->
-[[Category:todo:cleanup]]
+[[Category:Acronyms]]
+[[Category:Lists of scales]]
+[[Category:Just intonation]]