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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.  For any C-ADO system, the m-th degree is equal to the ratio (C+m)/C. 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 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.
An '''overtone scale''' is an octave-long subset of the [[harmonic series]] repeating at the octave. It is also known as an '''AFDO''' (arithmetic frequency divisions of the octave) due to an overtone scale being an arithmetically equal division of the octave, and '''ODO''' ([[otonal division]]s of the octave).


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
An overtone scale with n notes maybe referred to as mode n of the [[harmonic series]] or n-AFDO. For example, [[Mode 5]] is a pentatonic scale with the intervals [[1/1]]-[[6/5]]-[[7/5]]-[[8/5]]-[[9/5]]-[[2/1]] or 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]]
|}
 
A "mode" in other musical contexts is usually a different rotation of the same intervals. In the case of different harmonic modes, that's not exactly the case. However, in some sense it's a reasonable comparison, because as you slide the subset of harmonics around, you're essentially sampling different segments of integers whose prime factorizations follow simple, constant patterns (every 2nd number has a 2, every 3rd number has a 3, every 5th number has a 5) and therefore the full internal interval set (all dyads, triads, tetrads, etc.) from one mode to the next is more alike than it is different.
 
For a Mode C system, the m-th degree is equal to the ratio (C+m)/C. 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 :  
(which is 1/C), we have :  
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R_n = R_1 + (n-1)d
R_n = R_1 + (n-1)d
</math>
</math>
== Over-n scales ==
Another way to describe Mode 5 is that it is an example of an "Over-5 Scale." As 5 is octave-redundant with 10, 20, 40, 80 etc, any scale with one of those (the form is technically 2<sup>n</sup>*5, where n is any integer greater than or equal to zero) in the denominator of every tone could be called an Over-5 Scale.


== Lengths ==
== Lengths ==
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[[File:ADO-4.jpg|350px|center]]
[[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]**:
This lengths are related to reverse of ratios in system.The above picture shows the differences between divisions of length in Mode 12 system . On the contrary , we have equal divisions of length in **[https://sites.google.com/site/240edo/equaldivisionsoflength(edl) EDL system]**:
[[File:ADO-5.jpg|346px|center]]
[[File:ADO-5.jpg|346px|center]]


== Relation to superparticular ratios ==
== 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]].  
An overtone scale has step sizes of [[superparticular ratio]]s with increasing numerators. For example, [[Mode 5]] has step sizes of [[6/5]], [[7/6]], [[8/7]], and [[9/8]].


== Relation to otonality & harmonic series ==  
== Relation to otonality ==  


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.
We can consider an overtone scale 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 ==
== Primodality ==
* [[2ado]]
* [[3ado]]
* [[4ado]]
* [[5ado]]
* [[6ado]]
* [[7ado]]
* [[8ado]]
* [[9ado]]
* [[10ado]]
* [[11ado]]
* [[12ado]]
* [[13ado]]
* [[14ado]]
* [[15ado]]
* [[20ado]]
* [[30ado]]
* [[60ado]]
* [[120ado]]


[[Primodality]] involves the use of large prime modes of the harmonic series.
== A solfege system ==
[[Andrew Heathwaite]] proposes a solfege system for harmonics 16-32 (Mode 16):
{| class="wikitable"
|-
| harmonic
| 16
| 17
| 18
| 19
| 20
| 21
| 22
| 23
| 24
| 25
| 26
| 27
| 28
| 29
| 30
| 31
| 32
|-
| JI ratio
| 1/1
| 17/16
| 9/8
| 19/16
| 5/4
| 21/16
| 11/8
| 23/16
| 3/2
| 25/16
| 13/8
| 27/16
| 7/4
| 29/16
| 15/8
| 31/16
| 2/1
|-
| solfege
| '''do'''
| '''ra'''
| '''re'''
| '''me'''
| '''mi'''
| '''fe'''
| '''fu'''
| '''su'''
| '''sol'''
| '''le'''
| '''lu'''
| '''la'''
| '''ta'''
| '''tu'''
| '''ti'''
| '''da'''
| '''do'''
|}
Thus, the pentatonic scale in the example at the top (Mode 5) could be sung: '''mi sol ta do re mi'''
== Twelve scales ==
For those interested in learning to sing and hear just intervals, here are twelve of the simplest otonal scales to try. I leave it up to the curious learner to decide the value, beauty, or usefulness of these particular scales for their compositional purposes.
{| class="wikitable"
|-
|
|
| 1
| 2
| 3
| 4
| 5
| 6
| 7
| 8
| 9
| 10
| 11
| 12
| 13
| 14
| 15
| 16
| 17
| 18
| 19
| 20
| 21
| 22
| 23
| 24
|-
| Mode 1
| 1-note
| '''do'''
| '''do'''
|
|
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|
|
|
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|
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|-
| Mode 2
| 2-note
|
| '''do'''
| '''sol'''
| '''do'''
|
|
|
|
|
|
|
|
|
|
|
|
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|-
| Mode 3
| 3-note
|
|
| '''sol'''
| '''do'''
| '''mi'''
| '''sol'''
|
|
|
|
|
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|-
| Mode 4
| 4-note
|
|
|
| '''do'''
| '''mi'''
| '''sol'''
| '''ta'''
| '''do'''
|
|
|
|
|
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|-
| Mode 5
| 5-note
|
|
|
|
| '''mi'''
| '''sol'''
| '''ta'''
| '''do'''
| '''re'''
| '''mi'''
|
|
|
|
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|-
| Mode 6
| 6-note
|
|
|
|
|
| '''sol'''
| '''ta'''
| '''do'''
| '''re'''
| '''mi'''
| '''fu'''
| '''sol'''
|
|
|
|
|
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|-
| Mode 7
| 7-note
|
|
|
|
|
|
| '''ta'''
| '''do'''
| '''re'''
| '''mi'''
| '''fu'''
| '''sol'''
| '''lu'''
| '''ta'''
|
|
|
|
|
|
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|-
| Mode 8
| 8-note
|
|
|
|
|
|
|
| '''do'''
| '''re'''
| '''mi'''
| '''fu'''
| '''sol'''
| '''lu'''
| '''ta'''
| '''ti'''
| '''do'''
|
|
|
|
|
|
|
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|-
| Mode 9
| 9-note
|
|
|
|
|
|
|
|
| '''re'''
| '''mi'''
| '''fu'''
| '''sol'''
| '''lu'''
| '''ta'''
| '''ti'''
| '''do'''
| '''ra'''
| '''re'''
|
|
|
|
|
|
|-
| Mode 10
| 10-note
|
|
|
|
|
|
|
|
|
| '''mi'''
| '''fu'''
| '''sol'''
| '''lu'''
| '''ta'''
| '''ti'''
| '''do'''
| '''ra'''
| '''re'''
| '''me'''
| '''mi'''
|
|
|
|
|-
| Mode 11
| 11-note
|
|
|
|
|
|
|
|
|
|
| '''fu'''
| '''sol'''
| '''lu'''
| '''ta'''
| '''ti'''
| '''do'''
| '''ra'''
| '''re'''
| '''me'''
| '''mi'''
| '''fe'''
| '''fu'''
|
|
|-
| Mode 12
| 12-note
|
|
|
|
|
|
|
|
|
|
|
| '''sol'''
| '''lu'''
| '''ta'''
| '''ti'''
| '''do'''
| '''ra'''
| '''re'''
| '''me'''
| '''mi'''
| '''fe'''
| '''fu'''
| '''su'''
| '''sol'''
|}
== Individual pages for overtone scales ==
* [[Mode 2]]
* [[Mode 3]]
* [[Mode 4]]
* [[Mode 5]]
* [[Mode 6]]
* [[Mode 7]]
* [[Mode 8]]
* [[Mode 9]]
* [[Mode 10]]
* [[Mode 11]]
* [[Mode 12]]
* [[Mode 13]]
* [[Mode 14]]
* [[Mode 15]]
* [[Mode 16]]
* [[Mode 17]]
* [[Mode 18]]
* [[Mode 19]]
* [[Mode 20]]
* [[Mode 21]]
* [[Mode 22]]
* [[Mode 23]]
* [[Mode 24]]
* [[Mode 25]]
* [[Mode 26]]
* [[Mode 27]]
* [[Mode 28]]
* [[Mode 29]]
* [[Mode 30]]
* [[Mode 31]]
* [[Mode 32]]
* [[Mode 60]]
* [[Mode 120]]
== See also ==
== See also ==
* [[Arithmetic temperament]]
* [[Arithmetic temperament]]
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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]
* 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]
* [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.
* [[Primodality]]
 
* [[8th Octave Overtone Tuning]]
[[Category:ADO]]
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