Editing Just intonation (section)

===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.