AFDO: Difference between revisions

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]], 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]]. 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>

r = 1 + md

</math>

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''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'' &gt; … &gt; ''r''₃ &gt; ''r''₂ &gt; ''r''₁, then ''l''₁ &gt; ''l''₂ &gt; ''l''₃ &gt; … &gt; ''l''_''n''

 

[[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]].  

 

== Relation to otonality & harmonic series ==

 

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.

 

== 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.

 

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.

 

=== By size ===

{| class="wikitable center-all"

|+ style=white-space:nowrap | 0…99

|-

|-

|-

|-

|-

|-

|-

|-

{| class="wikitable center-all mw-collapsible mw-collapsed"

|+ style=white-space:nowrap | 100…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}}

 

 

== See also ==

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

* [[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 ==

 

[[Category:AFDO| ]] <!-- main article -->

[[Category:Lists of scales]]

[[Category:Just intonation]]