Editing Just intonation (section)

==Twelve-tone scale==
{{anchor|Chromatic scale}}
There are several ways to create a just tuning of the twelve-tone scale.

===Pythagorean tuning===
{{Main|Pythagorean tuning}}

[[Pythagorean tuning]] can produce a twelve-tone scale, but it does so by involving ratios of very large numbers, corresponding to natural harmonics very high in the harmonic series that do not occur widely in physical phenomena. This tuning uses ratios involving only powers of 3 and 2, creating a sequence of just [[perfect fifth|fifths]] or [[perfect fourth|fourth]]s, as follows:

{| class="wikitable" style="text-align: center"
|-
! style="width:5em;" | Note
! style="width:5em;" | G{{music|flat}}
! style="width:5em;" | D{{music|flat}}
! style="width:5em;" | A{{music|flat}}
! style="width:5em;" | E{{music|flat}}
! style="width:5em;" | B{{music|flat}}
! style="width:5em;" | F
! style="width:5em;" | C
! style="width:5em;" | G
! style="width:5em;" | D
! style="width:5em;" | A
! style="width:5em;" | E
! style="width:5em;" | B
! style="width:5em;" | F{{music|sharp}}
|-
! Ratio
| 1024:729
| 256:243
| 128:81
| 32:27
| 16:9
| 4:3
| 1:1
| 3:2
| 9:8
| 27:16
| 81:64
| 243:128
| 729:512
|-
! Cents
|588
|90
|792
|294
|996
|498
|0
|702
|204
|906
|408
|1110
|612
|}

The ratios are computed with respect to C (the ''base note''). Starting from C, they are obtained by moving six steps (around the [[circle of fifths]]) to the left and six to the right. Each step consists of a multiplication of the previous pitch by {{frac|2|3}} (descending fifth), {{frac|3|2}} (ascending fifth), or their [[inversion (music)|inversions]] ({{frac|3|4}} or {{frac|4|3}}).

Between the [[enharmonic]] notes at both ends of this sequence is a [[pitch (music)|pitch]] ratio of {{nowrap|{{sfrac|3{{sup|12}}|2{{sup|19}}}} {{=}} {{sfrac|531441|524288}}}}, or about 23 [[Cent (music)|cents]], known as the [[Pythagorean comma]]. To produce a twelve-tone scale, one of them is arbitrarily discarded. The twelve remaining notes are repeated by increasing or decreasing their frequencies by a power of 2 (the size of one or more [[octave]]s) to build scales with multiple octaves (such as the keyboard of a piano). A drawback of Pythagorean tuning is that one of the twelve fifths in this scale is badly tuned and hence unusable (the [[wolf fifth]], either F{{music|sharp}}–D{{music|flat}} if G{{music|flat}} is discarded, or B–G{{music|flat}} if F{{music|sharp}} is discarded). This twelve-tone scale is fairly close to [[equal temperament]], but it does not offer much advantage for [[tonality|tonal]] harmony because only the perfect intervals (fourth, fifth, and octave) are simple enough to sound pure. Major thirds, for instance, receive the rather unstable interval of 81:64, sharp of the preferred 5:4 by an 81:80 ratio.<ref name="autogenerated1968">{{Cite book|last=Daniélou|first=Alain|author-link=Alain Daniélou| title=The Ragas of Northern Indian Music | year=1968 | publisher=Barrie & Rockliff |location=London | isbn = 0-214-15689-3}}</ref> The primary reason for its use is that it is extremely easy to tune, as its building block, the perfect fifth, is the simplest and consequently the most [[consonance and dissonance|consonant]] interval after the octave and unison.

Pythagorean tuning may be regarded as a "three-limit" tuning system, because the ratios can be expressed as a product of integer powers of only whole numbers less than or equal to 3.

===Five-limit tuning===
{{Main|Five-limit tuning}}

A twelve-tone scale can also be created by compounding harmonics up to the fifth: namely, by multiplying the frequency of a given reference note (the base note) by powers of 2, 3, or 5, or a combination of them. This method is called five-limit tuning.

To build such a twelve-tone scale (using C as the base note), we may start by constructing a table containing fifteen pitches:

:{| class="wikitable" style="text-align:center"
|-
! Factor
!style="width:4em;"| {{sfrac| 1 | 9 }}
!style="width:4em;"| {{sfrac| 1 | 3 }}
!style="width:4em;"| 1
!style="width:4em;"| 3
!style="width:4em;"| 9
!
|-
!rowspan="3"| 5
|style="background:coral;"| '''D'''
| '''A'''
| '''E'''
| '''B'''
|style="background:springgreen;"| '''F'''{{sup|{{music|#}}}}
! note
|-
|style="background:coral;"| 10:9
| 5:3
| 5:4
| 15:8
|style="background:springgreen;"| 45:32
! ratio
|-
|style="background:coral;"| 182&nbsp;¢
| 884&nbsp;¢
| 386&nbsp;¢
| 1088&nbsp;¢
|style="background:springgreen;"| 590&nbsp;¢
! cents
|-
!rowspan="3"| 1
|style="background:gold;"| '''B'''{{sup|{{music|b}}}}
| '''F'''
| {{red|'''{{big|C}}'''}}
| '''G'''
|style="background:coral;"| '''D'''
! note
|-
|style="background:gold;"| 16:9
| 4:3
| 1:1
| 3:2
|style="background:coral;"| 9:8
! ratio
|-
|style="background:gold;"| 996&nbsp;¢
| 498&nbsp;¢
| 0&nbsp;¢
| 702&nbsp;¢
|style="background:coral;"| 204&nbsp;¢
! cents
|-
! rowspan="3" | {{sfrac| 1 | 5 }}
|style="background:springgreen;"| '''G'''{{sup|{{music|b}}}}
| '''D'''{{sup|{{music|b}}}}
| '''A'''{{sup|{{music|b}}}}
| '''E'''{{sup|{{music|b}}}}
| style="background:gold;" | '''B'''{{sup|{{music|b}}}}
! note
|-
|style="background:springgreen;"| 64:45
| 16:15
| 8:5
| 6:5
|style="background:gold;"| 9:5
! ratio
|-
|style="background:springgreen;"| 610&nbsp;¢
| 112&nbsp;¢
| 814&nbsp;¢
| 316&nbsp;¢
|style="background:gold;"| 1018&nbsp;¢
! cents
|}

The factors listed in the first row and column are powers of 3 and 5, respectively (e.g., {{nobr| {{sfrac| 1 | 9 }} {{=}} 3{{sup|−2}} ).}} Colors indicate couples of [[enharmonic]] notes with almost identical pitch. The ratios are all expressed relative to C in the centre of this diagram (the base note for this scale). They are computed in two steps:

# For each cell of the table, a ''base ratio'' is obtained by multiplying the corresponding factors. For instance, the base ratio for the lower-left cell is {{nobr| {{sfrac| 1 | 9 }} × {{sfrac| 1 | 5 }} {{=}} {{sfrac| 1 | 45 }} .}}
# The base ratio is then multiplied by a negative or positive power of 2, as large as needed to bring it within the range of the octave starting from C (from 1:1 to 2:1). For instance, the base ratio for the lower left cell ({{sfrac| 1 | 45 }}) is multiplied by 2{{sup|6}}, and the resulting ratio is 64:45, which is a number between 1:1 and 2:1.

Note that the powers of 2 used in the second step may be interpreted as ascending or descending [[octave]]s. For instance, multiplying the frequency of a note by 2{{sup|6}} means increasing it by 6 octaves. Moreover, each row of the table may be considered to be a sequence of fifths (ascending to the right), and each column a sequence of major thirds (ascending upward). For instance, in the first row of the table, there is an ascending fifth from D and A, and another one (followed by a descending octave) from A to E. This suggests an alternative but equivalent method for computing the same ratios. For instance, one can obtain A, starting from C, by moving one cell to the left and one upward in the table, which means descending by a fifth and ascending by a major third:

: {{sfrac|2|3}} × {{sfrac|5|4}} {{=}} {{sfrac|10|12}} = {{sfrac|5|6}} .

Since this is below C, one needs to move up by an octave to end up within the desired range of ratios (from 1:1 to 2:1):

: {{sfrac|5|6}} × {{sfrac|2|1}} {{=}} {{sfrac|10|6}} {{=}} {{sfrac|5|3}} .

A 12&nbsp;tone scale is obtained by removing one note for each couple of enharmonic notes. This can be done in four ways that have in common the removal of G{{music|b}}, according to a convention which was valid even for C-based Pythagorean and [[quarter-comma meantone]] scales. Note that it is a [[diminished fifth]], close to half an octave, above the tonic C, which is a discordant interval; also its ratio has the largest values in its numerator and denominator of all tones in the scale, which make it least harmonious: All are reasons to avoid it.

The following chart shows one way to obtain a 12&nbsp;tone scale by removing one note for each pair of enharmonic notes.  In this method one discards the first column of the table (labeled "{{sfrac| 1 | 9 }}").

:{| class="wikitable" style="text-align: center"
|-
!colspan="7"| Asymmetric scale
|-
! Factor
!style="width: 3em"| {{sfrac| 1 | 3 }}
!style="width: 3em"| 1
!style="width: 3em"| 3
!style="width: 3em"| 9
|-
!rowspan="2"| 5
| '''A'''
| '''E'''
| '''B'''
|style="background:springgreen;"| '''F'''{{sup|{{music|#}}}}
|-
| 5:3
| 5:4
| 15:8
|style="background:springgreen;"| 45:32
|-
!rowspan="2"| 1
| '''F'''
| {{red|'''{{big|C}}'''}}
| '''G'''
|style="background:coral;"| '''D'''
|-
| 4:3
| 1:1
| 3:2
|style="background:coral;"| 9:8
|-
! rowspan="2" | {{sfrac| 1 | 5 }}
| '''D'''{{sup|{{music|b}}}}
| '''A'''{{sup|{{music|b}}}}
| '''E'''{{sup|{{music|b}}}}
|style="background:gold;"| '''B'''{{sup|{{music|b}}}}
|- style="text-align:center;"
| 16:15
| 8:5
| 6:5
|style="background:gold;"| 9:5
|}

This scale is "asymmetric" in the sense that going up from the tonic two semitones we multiply the frequency by {{sfrac| 9 | 8 }}, while going down from the tonic two semitones we do not divide the frequency by {{sfrac| 9 | 8 }}. For two methods that give "symmetric" scales, see [[Five-limit tuning#Twelve-tone scale|Five-limit tuning: Twelve-tone scale]].

===Extension of the twelve-tone scale===

The table above uses only low powers of 3 and 5 to build the base ratios. However, it can be easily extended by using higher positive and negative powers of the same numbers, such as 5{{sup|2}} = 25, 5{{sup|−2}} = {{frac|1|25}}, 3{{sup|3}} = 27, or 3{{sup|−3}} = {{frac|1|27}}. A scale with 25, 35 or even more pitches can be obtained by combining these base ratios.