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| An '''ADO''' (arithmetic divisions of the octave) or '''overtone scale''' is a tuning system which divides the octave arithmetically rather than logarithmically. This is equivalent to taking an octave-long subset of the harmonic series and make it repeat at the octave. For any C-ADO system, the m-th degree is equal to the ratio (C+m)/C. For example, the [[5ado]] system consists of the 5th to 10th harmonics:
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| {| class="wikitable"
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| |-
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| | harmonic
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| | 5
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| | 6
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| | 7
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| | 8
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| | 9
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| | 10
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| | JI ratio
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| | [[1/1]]
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| | [[6/5]]
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| | [[7/5]]
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| | [[8/5]]
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| | [[9/5]]
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| | [[2/1]]
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| |}
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| For an ADO system, the distance between interval ratios is equal, rather than the distance between their logarithms like in EDO systems. 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.
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| 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
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| (which is 1/C), we have :
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| <math>
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| R_2 = R_1 + d \\
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| R_3= R_1 + 2d \\
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| R_4 = R_1 + 3d \\
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| \vdots \\
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| R_n = R_1 + (n-1)d
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| </math>
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| == Lengths ==
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| 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 >........> R3 > R2 > R1 so :
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| L1 > L2 > L3 > …… > Ln
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| [[File:ADO-4.jpg|350px|center]]
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| 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]**:
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| [[File:ADO-5.jpg|346px|center]]
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| == Relation to superparticular ratios ==
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| 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]].
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| == Relation to otonality & harmonic series ==
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| 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. An otonality corresponds to an arithmetic series of frequencies or a harmonic series of wavelengths or distances on a string instrument.
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| == Individual pages for ADOs ==
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| * [[2ado]]
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| * [[3ado]]
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| * [[4ado]]
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| * [[5ado]]
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| * [[6ado]]
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| * [[7ado]]
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| * [[8ado]]
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| * [[9ado]]
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| * [[10ado]]
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| * [[11ado]]
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| * [[12ado]]
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| * [[13ado]]
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| * [[14ado]]
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| * [[15ado]]
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| * [[20ado]]
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| * [[30ado]]
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| * [[60ado]]
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| * [[120ado]]
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| == See also ==
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| * [[Arithmetic temperament]]
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| * [[Arithmetic MOS scale]]
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| * [http://240edo.googlepages.com/ADO-EDL.XLS Fret position calculator] (excel sheet) based on EDL system and string length
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| * How to approximate EDO and ADO systems with each other? [https://sites.google.com/site/240edo/ADOandEDO.xls Download this file]
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| * [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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| * [[OD|OD, or otonal division]]: An n-ADO is equivalent to an n-ODO.
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| [[Category:ADO]]
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