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'''ARDO''' (which is simplified as ADO) refers to Arithmetic Rational Divisions of the Octave. it is an intervallic system considered as an arithmetic sequence with divisions of the system as terms of a sequence.  
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.  


If the first division is <math>R_1</math> (which is ratio of C/C) and the last , <math>R_n</math> (which is ratio of 2C/C), with common difference of d
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).


(which is 1/C), we have :  
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 ==
Within each period of any ''n''-afdo system, the [[frequency ratio]] ''r'' of the ''m''-th degree is
 
<math>\displaystyle r = (n + m)/n</math>
 
Alternatively, with common frequency difference ''d'' = 1/''n'', we have:  


<math>
<math>
R_2 = R_1 + d \\
r = 1 + md
R_3= R_1 + 2d \\
R_4 = R_1 + 3d \\
\vdots \\
R_n = R_1 + (n-1)d
</math>
</math>


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.
In particular, when ''m'' = 0, ''r'' = 1, and when ''m'' = ''n'', ''r'' = 2.
 
== 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>
 
[[File:ADO-4.jpg|350px|center]]
 
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]):
 
[[File:ADO-5.jpg|346px|center]]
 
== 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]].  


For any C-ADO system with **cardinality** of C, we have ratios related to different degrees of m as :
== Relation to otonality & harmonic series ==


(C+m/C)
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.


For example , in 12-ADO the ratio related to the first degree is 13/12 .
== 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.  


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 .
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.  


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


L1 &gt; L2 &gt; L3 &gt; …… &gt; Ln
== 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.


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


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 **[https://sites.google.com/site/240edo/equaldivisionsoflength(edl) EDL system]**:
'''Over-11''': {{AFDOs|11, 22, 44, 88, 176, 352}}


[[File:ADO-5.jpg|346px|center]]
'''Over-13''': {{AFDOs|13, 26, 52, 104, 208, 416}}


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


[[File:ADO-3.jpg|604px|center]]
'''Over-19''': {{AFDOs|19, 38, 76, 152, 304}}


Relation between harmonics and ADO system
'''Over-23''': {{AFDOs|23, 46, 92, 184, 368}}


ADO (like EDL) is based on **Superparticular ratios** and **harmonic series**. Have a look at 12-ADO in this picture:
'''Over-29''': {{AFDOs|29, 58, 116, 232, 464}}


[[File:ADO-2.jpg|378px|center]]
'''Over-31''': {{AFDOs|31, 62, 124, 248, 496}}


The above picture shows that ADO system is classified as :
=== By other properties ===


*System with unequal **epimorios** (**Superparticular**) divisions.
'''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}}
*System based on ascending series of superparticular ratios with descending sizes.
*System which covers superparticular ratios between harmonic of number C (in this example 12)to harmonic of Number 2C(in this example 24).
*[https://sites.google.com/site/240edo/ADO-EDL.XLS An spreadsheet] showing relation between harmonics , superparticular ratios and ADO system


Relation between Otonality and ADO system
'''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}}


We can consider ADO system as **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 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 **harmonic series**. nominator here is called "**Numerary nexus**".An Otonality corresponds to an **arithmetic series** of frequencies or a **harmonic series** of wavelengths or distances on a **string instrument**.
'''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}}


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


- How to approximate EDO and ADO systems with each other?[https://sites.google.com/site/240edo/ADOandEDO.xls Download this file]
== 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]]


Related to ADO
== 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]]


<nowiki>**</nowiki>[http://www.soundtransformations.btinternet.co.uk/Danerhudyarthemarmonicseries.htm Magic of Tone and the Art of Music by the late Dane Rhudyar]**     
== Notes ==
[[Category:ADO]]
[[Category:todo:cleanup]]
[[Category:todo:Change table images to wikitables]]


[[OD, or otonal division]]: An n-ADO is equivalent to an n-ODO.
[[Category:AFDO| ]] <!-- main article -->
[[Category:Acronyms]]
[[Category:Lists of scales]]
[[Category:Just intonation]]
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