Editing Enharmonic scale (section)

== Example of a modern, multi-tone enharmonic scale ==
As opposed to ancient Greek enharmonic scales, which only employed seven notes in an octave, modern musicians have expanded the idea of an "enharmonic scale" to include most of the pitches which ancient Greek tuning might select from to create a seven pitch octave. This gives the modern musician options for in-effect modulating between multiple different ancient Greek scales. This creates musical options that, as far as we now understand, was never possible for ancient Greeks musicians. Although note that some [[kitharode]]s were musically experimental and inventive, and sought musical novelty, so they might well have imagined alternating between different enharmonic scales. They might even accomplished it, by one musician switching between several different [[kithara]]s during a performance, with each tuned to a different, but tonally interlocking enharmonic scale.

Consider a scale constructed through [[Pythagorean tuning]]: A Pythagorean scale can be constructed "upwards" by wrapping a chain of [[perfect fifth]]s around an [[octave]], but it can also be constructed "downwards" by wrapping a chain of [[perfect fourth]]s around the same octave. By juxtaposing these two slightly different scales, it is possible to create an enharmonic scale.

The following Pythagorean scale is enharmonic:

:{| class="wikitable"
! Note
! Ratio
! Decimal
! [[Cent (music)|Cents]]
! Difference<br>(cents)
|-
| C || {{0|0000}}1:1 || 1 || {{0|000}}0 || style="background: darkgrey" |
|-
| D{{music|flat}} || {{0|00}}256:243 || 1.05350 || {{0|00}}90.225 || rowspan="2" | 23.460
|-
| C{{music|sharp}} || {{0}}2187:2048 || 1.06787 || {{0}}113.685
|-
| D || {{0|0000}}9:8 || 1.125 || {{0}}203.910 || style="background: darkgrey" |
|-
| E{{music|flat}} || {{0|000}}32:27 || 1.18519 || {{0}}294.135 || rowspan="2" | 23.460
|-
| D{{music|sharp}} || 19683:16384 || 1.20135 || {{0}}317.595
|-
| E || {{0|000}}81:64 || 1.26563 || {{0}}407.820 || rowspan="2" style="background: darkgrey" |
|-
| F || {{0|0000}}4:3 || 1.33333 || {{0}}498.045
|-
| G{{music|flat}} || {{0}}1024:729 || 1.40466 || {{0}}588.270 || rowspan="2" | 23.460
|-
| F{{music|sharp}} || {{0|00}}729:512 || 1.42383 || {{0}}611.730
|-
| G || {{0|0000}}3:2 || 1.5 || {{0}}701.955 || style="background: darkgrey" |
|-
| A{{music|flat}} || {{0|00}}128:81 || 1.58025 || {{0}}792.180 || rowspan="2" | 23.460
|-
| G{{music|sharp}} || {{0|0}}6561:4096 || 1.60181 || {{0}}815.640
|-
| A || {{0|000}}27:16 || 1.6875 || {{0}}905.865 || style="background: darkgrey" |
|-
| B{{music|flat}} || {{0|000}}16:9 || 1.77778 || {{0}}996.090 || rowspan="2" | 23.460
|-
| A{{music|sharp}} || 59049:32768 || 1.80203 || 1019.550
|-
| B || {{0|00}}243:128 || 1.89844 || 1109.775 || rowspan="2" style="background: darkgrey" |
|-
| C′ || {{0|0000}}2:1 || 2 || 1200
|}

In the above scale the following pairs of notes are said to be enharmonic:
* C{{music|sharp}} and D{{music|flat}}
* D{{music|sharp}} and E{{music|flat}}
* F{{music|sharp}} and G{{music|flat}}
* G{{music|sharp}} and A{{music|flat}}
* A{{music|sharp}} and B{{music|flat}}

In this example, natural notes are sharpened by multiplying its frequency ratio by {{sfrac| 256 | 243 }} (called a [[Pythagorean limma|limma]]), and a natural note is flattened by multiplying its ratio by {{sfrac| 243 | 256 }} . A pair of enharmonic notes are separated by a [[Pythagorean comma]], which is equal to {{sfrac| {{gaps|531|441}} | {{gaps|524|288}} }} (about 23.46 [[cent (music)|cents]]).