Editing Pythagorean tuning (section)

==Method==

12-tone Pythagorean temperament is based on a sequence of perfect fifths, each tuned in the ratio 3:2, the next simplest ratio after 2:1 (the octave). Starting from D for example (''D-based'' tuning), six other notes are produced by moving six times a ratio 3:2 up, and the remaining ones by moving the same ratio down:

:E♭&ndash;B♭&ndash;F&ndash;C&ndash;G&ndash;'''D'''&ndash;A&ndash;E&ndash;B&ndash;F♯&ndash;C♯&ndash;G♯

This succession of eleven 3:2 intervals spans across a wide range of [[frequency]] (on a [[piano keyboard]], it encompasses 77 keys). Since notes differing in frequency by a factor of 2 are perceived as similar and given the same name ([[octave equivalence]]), it is customary to divide or multiply the frequencies of some of these notes by 2 or by a power of 2. The purpose of this adjustment is to move the 12 notes within a smaller range of frequency, namely within the interval between the '''base note''' D and the D above it (a note with twice its frequency). This interval is typically called the '''basic octave''' (on a piano keyboard, an [[octave]] has only 12 keys). This dates to antiquity: in Ancient Mesopotamia, rather than stacking fifths, tuning was based on alternating ascending fifths and descending fourths (equal to an ascending fifth followed by a descending octave), resulting in the notes of a pentatonic or heptatonic scale falling within an octave.

:{| class="wikitable" style="text-align: center"
! class="unsortable"|Note
! class="unsortable"|Interval from D
! class="unsortable"|Formula
!=
! class="unsortable"|=
! class="unsortable"|Frequency<br />ratio
! Size<br />(cents)
! 12-TET-dif<br />(cents)
|-
| D
| [[unison]]
| <math>\frac{1}{1}</math>
|<math>3^{0} \times 2^{0}</math>
| <math>\frac{3^0}{2^0}</math>
|<math>\frac{1}{1}</math>
|style="text-align: right"| 0.00
|style="text-align: right"| 0.00
|-
| E{{music|b}}
| [[minor second]]
| <math>\left( \frac{2}{3} \right)^5 \times 2^3</math>
|<math>3^{-5} \times 2^{8}</math>
| <math>\frac{2^8}{3^5}</math>
|<math>\frac{256}{243}</math>
|style="text-align: right"| 90.22
|style="text-align: right"| −9.78
|-
| E
| [[major second]]
| <math>\left( \frac{3}{2} \right)^2 \times \frac{1}{2}</math>
|<math>3^{2} \times 2^{-3}</math>
| <math>\frac{3^2}{2^3}</math>
|<math>\frac{9}{8}</math>
|style="text-align: right"| 203.91
|style="text-align: right"| 3.91
|-
| F
| [[minor third]]
| <math>\left( \frac{2}{3} \right)^3 \times 2^2</math>
|<math>3^{-3} \times 2^{5}</math>
| <math>\frac{2^5}{3^3}</math>
|<math>\frac{32}{27}</math>
|style="text-align: right"| 294.13
|style="text-align: right"| −5.87
|-
| F{{music|#}}
| [[major third]]
| <math>\left( \frac{3}{2} \right)^4 \times \left( \frac{1}{2} \right)^2</math>
|<math>3^{4} \times 2^{-6}</math>
| <math>\frac{3^4}{2^6}</math>
|<math>\frac{81}{64}</math>
|style="text-align: right"| 407.82
|style="text-align: right"| 7.82
|-
| G
| [[perfect fourth]]
| <math>\frac{2}{3} \times 2</math>
|<math>3^{-1} \times 2^{2}</math>
| <math>\frac{2^2}{3^1}</math>
|<math>\frac{4}{3}</math>
|style="text-align: right"| 498.04
|style="text-align: right"| −1.96
|-
| A{{music|b}}
| [[diminished fifth]]
| <math>\left( \frac{2}{3} \right)^6 \times 2^4</math>
|<math>3^{-6} \times 2^{10}</math>
| <math>\frac{2^{10}}{3^6}</math>
|<math>\frac{1024}{729}</math>
|style="text-align: right"| 588.27
|style="text-align: right"| −11.73
|-
| G{{music|#}}
| [[augmented fourth]]
| <math>\left( \frac{3}{2} \right)^6 \times \left( \frac{1}{2} \right)^3</math>
|<math>3^{6} \times 2^{-9}</math>
| <math>\frac{3^6}{2^9}</math>
|<math>\frac{729}{512}</math>
|style="text-align: right"| 611.73
|style="text-align: right"| 11.73
|-
| A
| [[perfect fifth]]
| <math>\frac{3}{2}</math>
|<math>3^{1} \times 2^{-1}</math>
| <math>\frac{3^1}{2^1}</math>
|<math>\frac{3}{2}</math>
|style="text-align: right"| 701.96
|style="text-align: right"| 1.96
|-
| B{{music|b}}
| [[minor sixth]]
| <math>\left( \frac{2}{3} \right)^4 \times 2^3</math>
|<math>3^{-4} \times 2^{7}</math>
| <math>\frac{2^7}{3^4}</math>
|<math>\frac{128}{81}</math>
|style="text-align: right"| 792.18
|style="text-align: right"| −7.82
|-
| B
| [[major sixth]]
| <math>\left( \frac{3}{2} \right)^3 \times \frac{1}{2}</math>
|<math>3^{3} \times 2^{-4}</math>
| <math>\frac{3^3}{2^4}</math>
|<math>\frac{27}{16}</math>
|style="text-align: right"| 905.87
|style="text-align: right"| 5.87
|-
| C
| [[minor seventh]]
| <math>\left( \frac{2}{3} \right)^2 \times 2^2</math>
|<math>3^{-2} \times 2^{4}</math>
| <math>\frac{2^4}{3^2}</math>
|<math>\frac{16}{9}</math>
|style="text-align: right"| 996.09
|style="text-align: right"| −3.91
|-
| C{{music|#}}
| [[major seventh]]
| <math>\left( \frac{3}{2} \right)^5 \times \left( \frac{1}{2} \right)^2</math>
|<math>3^{5} \times 2^{-7}</math>
| <math>\frac{3^5}{2^7}</math>
|<math>\frac{243}{128}</math>
|style="text-align: right"| 1109.78
|style="text-align: right"| 9.78
|-
| D
| [[octave]]
| <math>\frac{2}{1}</math>
|<math>3^{0} \times 2^{1}</math>
| <math>\frac{2^1}{3^0}</math>
|<math>\frac{2}{1}</math>
|style="text-align: right"| 1200.00
|style="text-align: right"| 0.00
|}

In the formulas, the ratios 3:2 or 2:3 represent an ascending or descending perfect fifth (i.e. an increase or decrease in frequency by a perfect fifth, while 2:1 or 1:2 represent a rising or lowering octave). The formulas can also be expressed in terms of powers of the third and the second [[Harmonic series (music)|harmonics]].

The [[major scale]] based on C, obtained from this tuning is:<ref>Asiatic Society of Japan (1879). ''[https://books.google.com/books?id=DX0uAAAAYAAJ&dq=pythagorean+interval&pg=PA82 Transactions of the Asiatic Society of Japan], Volume 7'', p.&nbsp;82. Asiatic Society of Japan.</ref>

:{| class="wikitable" style="text-align:center"
!Note
!colspan="2" | '''C'''
!colspan="2" | '''D'''
!colspan="2" | '''E'''
!colspan="2" | '''F'''
!colspan="2" | '''G'''
!colspan="2" | '''A'''
!colspan="2" | '''B'''
!colspan="2" | '''C'''
|-
!Ratio
|colspan="2" |{{frac|1|1}}
|colspan="2" |{{frac|9|8}}
|colspan="2" |{{frac|81|64}}
|colspan="2" |{{frac|4|3}}
|colspan="2" |{{frac|3|2}}
|colspan="2" |{{frac|27|16}}
|colspan="2" |{{frac|243|128}}
|colspan="2" |{{frac|2|1}}
|-
!Step
|colspan="1" | —
|colspan="2" |{{frac|9|8}}
|colspan="2" |{{frac|9|8}}
|colspan="2" |{{frac|256|243}}
|colspan="2" |{{frac|9|8}}
|colspan="2" |{{frac|9|8}}
|colspan="2" |{{frac|9|8}}
|colspan="2" |{{frac|256|243}}
|colspan="1" | —
|}

In equal temperament, pairs of [[enharmonic]] notes such as A{{music|b}} and G{{music|#}} are thought of as being exactly the same note&mdash;however, as the above table indicates, in Pythagorean tuning they have different ratios with respect to D, which means they are at a different frequency. This discrepancy, of about 23.46 cents, or nearly one quarter of a semitone, is known as a ''[[Pythagorean comma]]''.

To get around this problem, Pythagorean tuning constructs only twelve notes as above, with eleven fifths between them. For example, one may use only the 12 notes from E{{music|b}} to G{{music|#}}. This, as shown above, implies that only eleven just fifths are used to build the entire chromatic scale. The remaining interval (the diminished sixth from G{{music|#}} to E{{music|b}}) is left badly out-of-tune, meaning that any music which combines those two notes is unplayable in this tuning. A very out-of-tune interval such as this one is known as a ''[[wolf interval]]''. In the case of Pythagorean tuning, all the fifths are 701.96 cents wide, in the exact ratio 3:2, except the wolf fifth, which is only 678.49 cents wide, nearly a quarter of a [[semitone]] flatter.

<!-- Image with unknown copyright status removed: [[media:Wolf fifth.ogg|Wolf fifth.ogg]]  (33.1KB) is a sound file demonstrating this out of tune fifth. The first two fifths are perfectly tuned in the ratio 3:2, the third is the D{{Music|#}}&ndash;E{{Music|b}} wolf fifth. It may be useful to compare this to [[media:Et fifths.ogg|Et fifths.ogg]] (38.2KB), which is the same three fifths tuned in [[equal temperament]], each of them tolerably well in tune.
-->
If the notes G{{music|sharp}} and E{{music|flat}} need to be sounded together, the position of the wolf fifth can be changed. For example, a C-based Pythagorean tuning would produce a stack of fifths running from D{{music|flat}} to F{{music|sharp}}, making F{{music|sharp}}-D{{music|flat}} the wolf interval. However, there will always be one wolf fifth in Pythagorean tuning, making it impossible to play in all [[key (music)|keys]] in tune.