AFDO: Difference between revisions

An '''ADO''' ('''arithmetic divisions 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 [[12ado]] the first degree is [[13/12]], the second is 14/12 ([[7/6]]), and so on. For an ADO 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 ADOs are subsets of [[just intonation]]. ADOs with more divisors such as [[Highly_composite_equal_division|highly composite]] ADOs generally have more useful just intervals.  

When treated as a scale, the ado is equivalent to the [[overtone scale]]. An ADO 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.  

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

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

r_1 = r_0 + d \\

r_m = r_0 + md

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''

An ADO 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]].  

We can consider ADO 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 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. An otonality corresponds to an arithmetic series of frequencies or a harmonic series of wavelengths or distances on a string instrument.

The term ADO 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</ref>. Previously, scales/tunings 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. 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 term ADO by leveraging the ambiguity in the word "arithmetic", repurposing it as a reference to the [[Wikipedia:Arithmetic_mean|arithmetic mean]] rather than to arithmetic progressions, then extended this interpretation to the term IDO, for "inverse-arithmetic division of the octave" by coining "inverse-harmonic mean".

* [[Arithmetic interval chain]]

* [[Arithmetic MOS scale]]

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

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

* [http://www.soundtransformations.btinternet.co.uk/Danerhudyarthemarmonicseries.htm Magic of Tone and the Art of Music by the late Dane Rhudyar]

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