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R_n = R_1 + (n-1)d
R_n = R_1 + (n-1)d
</math>
</math>
== Relation to superparticular ratios ==
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 ==
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.
== Primodality ==
[[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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|
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|-
| Mode 3
| 3-note
|
|
| '''sol'''
| '''do'''
| '''mi'''
| '''sol'''
|
|
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|
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|-
| Mode 4
| 4-note
|
|
|
| '''do'''
| '''mi'''
| '''sol'''
| '''ta'''
| '''do'''
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|-
| Mode 5
| 5-note
|
|
|
|
| '''mi'''
| '''sol'''
| '''ta'''
| '''do'''
| '''re'''
| '''mi'''
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|-
| Mode 6
| 6-note
|
|
|
|
|
| '''sol'''
| '''ta'''
| '''do'''
| '''re'''
| '''mi'''
| '''fu'''
| '''sol'''
|
|
|
|
|
|
|
|
|
|
|
|
|-
| Mode 7
| 7-note
|
|
|
|
|
|
| '''ta'''
| '''do'''
| '''re'''
| '''mi'''
| '''fu'''
| '''sol'''
| '''lu'''
| '''ta'''
|
|
|
|
|
|
|
|
|
|
|-
| Mode 8
| 8-note
|
|
|
|
|
|
|
| '''do'''
| '''re'''
| '''mi'''
| '''fu'''
| '''sol'''
| '''lu'''
| '''ta'''
| '''ti'''
| '''do'''
|
|
|
|
|
|
|
|
|-
| 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'''
|}
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