User:CompactStar/Overtone scale: Difference between revisions

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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:


{| class="wikitable"
|-
| harmonic
| 5
| 6
| 7
| 8
| 9
| 10
|-
| JI ratio
| [[1/1]]
| [[6/5]]
| [[7/5]]
| [[8/5]]
| [[9/5]]
| [[2/1]]
|}
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.
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
(which is 1/C), we have :
<math>
R_2 = R_1 + d \\
R_3= R_1 + 2d \\
R_4 = R_1 + 3d \\
\vdots \\
R_n = R_1 + (n-1)d
</math>
== Lengths ==
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 :
L1 &gt; L2 &gt; L3 &gt; …… &gt; Ln
[[File:ADO-4.jpg|350px|center]]
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]**:
[[File:ADO-5.jpg|346px|center]]
== Relation to superparticular ratios ==
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]].
== Relation to otonality & harmonic series ==
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.
== Individual pages for ADOs == 
* [[2ado]]
* [[3ado]]
* [[4ado]]
* [[5ado]]
* [[6ado]]
* [[7ado]]
* [[8ado]]
* [[9ado]]
* [[10ado]]
* [[11ado]]
* [[12ado]]
* [[13ado]]
* [[14ado]]
* [[15ado]]
* [[20ado]]
* [[30ado]]
* [[60ado]]
* [[120ado]]
== See also ==
* [[Arithmetic temperament]]
* [[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]
* [[OD|OD, or otonal division]]: An n-ADO is equivalent to an n-ODO.
[[Category:ADO]]

Revision as of 04:20, 2 March 2023

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