View source for Enharmonic scale

{{Short description|Ancient Greek musical scale}}
{{for|enharmonic keys|Enharmonic}}
[[Image:Enharmonic scale segment on C.png|thumb|Enharmonic scale [segment] on C.<ref name="Moore">{{cite CEM |url=https://archive.org/stream/completeencyclop00moor#page/n286/mode/1up |year=1875 |title=Enharmonic scale |page=281}}. Moore cites Greek use of quarter tones until the time of Alexander the Great.</ref><ref name=Callcott-1833/> {{audio|Enharmonic scale segment on C.mid|Play}}<ref name=Callcott-1833/> Note that in this depiction C{{music|#}} and D{{music|b}} are distinct rather than equivalent as in modern notation.]]
[[Image:Enharmonic scale on C.png|thumb|Enharmonic scale on C.<ref name=Elson-1905>
{{cite book
 |first=Louis Charles |last=Elson 
 |year=1905
 |title=Elson's Music Dictionary
 |page=100
 |publisher=O. Ditson Company
}}
</ref>]]
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In music theory, an '''enharmonic scale''' is a ''very'' [[musical system of ancient Greece|ancient Greek musical scale]] which contains four notes tuned to approximately [[quarter tone]] pitches, bracketed (as pairs) between four fixed pitches.<ref name=ML-West-1992/> For example, in modern [[microtonal]] notation, one of the several '''enharmonic scales''' aligned with the conventional key of [[C major]] would be
: {{sc|'''C'''}} (0[[musical cents|&nbsp;¢]]), {{sc|'''D'''}}{{music|##}} (400[[musical cents|&nbsp;¢]]), {{sc|'''E'''}}{{music|t}} (450[[musical cents|&nbsp;¢]]), {{sc|'''F'''}} (500[[musical cents|&nbsp;¢]]),
: {{sc|'''G'''}} (700[[musical cents|&nbsp;¢]]), {{sc|'''A'''}}{{music|##}} (1000[[musical cents|&nbsp;¢]]), {{sc|'''B'''}}{{music|t}}(1150&nbsp;¢), {{sc|'''c′'''}} (1200[[musical cents|&nbsp;¢]]).

The symbol {{music|t}} in this example represents a [[half-sharp]], or sharpening by a [[quartertone]] (50[[musical cents|&nbsp;cents]]), although raising pitch by exactly 50&nbsp;cents is not at all required, nor even usual among the different Greek enharmonic tunings, which tended instead to have the movable, inner notes (here, {{sc|'''D'''}} & {{sc|'''E'''}}; {{sc|'''A'''}} & {{sc|'''B'''}}) variably spaced, with about 20~30&nbsp;cents between each other, and likewise spaced from their closest fixed note (for this example those are {{sc|'''C'''}}, {{sc|'''F'''}}, {{sc|'''G'''}}, and {{sc|'''c′'''}}).<ref name=ML-West-1992/>

==Bracketing tetrachords==
Four of the scale notes – the [[tonic (music)|tonic]] ({{sc|'''C'''}} in the example), [[subdominant]] ({{sc|'''F'''}}), [[dominant (music)|dominant]] ({{sc|'''G'''}} ), and [[octave]] ({{sc|'''c′<nowiki/>'''}}) – are all fixed: They are nearly exactly the same [[relative pitch]]es in all three categories of ancient Greek scales ('''''enharmonic''''', [[chromatic genus|''chromatic'']], and [[diatonic genus|''diatonic'']]),<ref name=ML-West-1992/> and in ancient Greek music, the fixed tones [[relative pitch]]es were very nearly the same as the corresponding notes in the modern [[12 equal temperament|conventional scale]]. On the other hand, the four notes contained between the brackets, from the example {{sc|'''D'''}} and {{sc|'''E'''}} (between {{sc|'''C'''}} and {{sc|'''F'''}}); and {{sc|'''A'''}} and {{sc|'''B'''}} (between {{sc|'''G'''}} and {{sc|'''c′<nowiki/>'''}}) are the two pairs of bracketed, variable notes; they can have nearly any pitch. After pitches chosen for them, if the interval between a movable note and any other note is about a quarter tone or less, the scale is called "enharmonic". The small, or "[[microtonal]]" interval can be between either of the bracketing fixed notes, or from the other movable note, inside the bracket.

Despite the music of [[India]] and the [[Middle East]] still using similar intervals in traditional and classical scales, even the idea of the very small pitch intervals used in the enharmonic scale has lain outside the competence of musicians trained in occidental music at least since the time of the early Roman Empire.<ref name=ML-West-1992/>

== Difference in meaning of "enharmonic" between the classical-era and now ==
The ancient Greek meaning of '''''enharmonic''''' is that the scale contains at least one very narrow interval. (The spacing of each pair notes between their bracketing fixed notes is usually either approximately or exactly the same, so when there is one narrow interval in one bracket there is almost always another one inside the other bracket.)<ref name=ML-West-1992/> 
Modern musical vocabulary has re-used the word ''"enharmonic"'' altered to have the most extreme possible meaning of its ancient sense, to mean two differently-named notes which happen to actually have the same pitch. In [[musical system of ancient Greece|ancient Greek music]] from which ''enharmonic scales'' come, the meaning of ''enharmonic'' not so extreme: It means that the notes are ''not'' actually the same, but do only differ in pitch by a very slight amount, and had a similar connotation to "[[microtonal]]" in modern musical vocabulary.

Since an enharmonic scale uses (approximately) [[quarter tones]], or more technically [[diesis|dieses]] (divisions) which do not occur on standard modern keyboards,<ref name=Callcott-1833>
{{cite book
 |first=John Wall |last=Callcott
 |year=1833
 |title=A Musical Grammar in Four Parts
 |page=109
 |publisher=James Loring
}}
</ref>
nor were even used in the preceding western tuning systems, such as [[quarter comma meantone|¼&nbsp;comma temperament]] (the predominant tuning about 200&nbsp;years ago) or [[well temperament]] (finally went out of use as conventional tuning about 140~150&nbsp;years ago) the pitches and intervals in the several ancient Greek enharmonic scales are foreign to nearly any modern-trained musician, and generally outside the scope of musical competence of modern occidental musicians: People playing modern fixed-pitch instruments have no opportunity to experiment with musical scales containing these notes, since piano keyboards only have provisions for [[half tone]]s, as do frets on [[guitar]]s and [[mandolin]]s, fingering holes on [[woodwind]]s, and valves on [[brass instrument]]s. This has been the situation for more than 150&nbsp;years for fixed-pitch occidental instruments.

Even among [[Ancient Greece|Hellenic]] musicians, enharmonic scales appear to have gone out of style around {{nobr|{{gaps|2|500}} years}} ago, and only persisted as a perfunctory part of normal musical training; enharmonic scales seem to have been oddities even to the Greek writers in the [[Roman Empire]], whose works on music theory we still have.<ref name=ML-West-1992/> So the idea of such very small pitch intervals used in the enharmonic scale has lain outside of the scope of musicians' training for occidental music, despite music of [[India]] and the [[Middle East]] still using similar intervals traditional and classical scales.

== Unfamiliar, variable-size quarter tones ==
An otherwise well regarded 19th&nbsp;century musicologist once wrote the rather blatantly false definition in his 1905 musical dictionary, that the '''enharmonic scale''' is

: ... "an [imaginary] gradual progression by [[quarter tone]]s" or any "[[scale (music)|[musical] scale]] proceeding by [[quarter tone]]s". — Elson (1905)<ref name=Elson-1905/>{{anchor|Elsons_stupid_remark_anchor}}

However, enharmonic tuning does seem "imaginary" to many modern western musicians because of the intentional limitations placed into [[12 tone equal temperament|conventional tuning]], and deficient musical training which only prepares modern students to deal with a single tuning system, even though many others were in use in the west in the recent past, and still more are in current use in other parts of the world. Even well-educated musicologists have little or no understanding of [[ancient Greek music]]al scales (among whom sits Elson<ref name=Elson-1905/>) nor even relatively recently disused tuning systems, such as the [[quarter comma meantone|¼&nbsp;comma meantone temperament]] predominantly used up to the time of [[Johan Sebastian Bach|Bach]], and the later unequal [[well temperament]]s based on it.

The enharmonic scale was a very real tuning system that survived from pre-classical Greek music (when it seems to have been put to more use<ref name=ML-West-1992/>) into the [[Roman Empire|Roman Imperial era]]. Although still taught as a perfunctory part of [[Hellenistic culture|Hellenistic education]], the enharmonic scale was only rarely – if ever – used during the period of 180~400&nbsp;[[Common Era|CE]] when the Greek musical theory books which still survive were written.<ref>See the articles on [[Claudius Ptolemy]] (''Harmonics''), and [[Boethius]].</ref><ref name=ML-West-1992>
{{cite book
 |last=West |first=Martin Litchfield |author-link=Martin Litchfield West
 |year=1992
 |title=Ancient Greek Music
 |place=Oxford, UK
 |publisher=[[Oxford University Press]]
 |isbn=0-19-814975-1
}}
</ref>

The enharmonic scale uses [[diesis|dieses]] (divisions) which are not tuned in any pitch present on standard modern keyboards,<ref name=Callcott-1833/>
since modern, standard keyboards only have provisions for [[semitone|half-tone]] steps. The two different notations used for vocal and instrumental notes in [[ancient Greek Musical Notation|ancient Greek music notation]] are more tonally versatile, since they are based on quarter-tones = half-sharps, with step sizes that could be altered from a strict quarter tone step.<ref name=ML-West-1992/> Despite the pitches being unknown to naïve occidentally-trained musicians, all the [[musical system of ancient Greece|ancient Greek tuning]] systems only require seven distinct pitches in a completed octave, and only the four of those pitches, the two that lie between the fixed [[tonic (music)|tonic]] and [[subdominant]] (or [[perfect fourth|fourth]]) (relative to [[C major|C{{sup|Maj}}]], the notes between {{sc|'''C'''}} and {{sc|'''F'''}}), and the other two movable notes between fixed [[Dominant (music)|dominant]] / [[perfect fifth|fifth]] and the [[octave]] (between {{sc|'''G'''}} and {{sc|'''c′'''}}). When expressing notes with modern letter notation, it is conventional to use some elaborately sharpened or flattened version of the notes {{sc|'''D'''}}, {{sc|'''E'''}}, {{sc|'''A'''}}, and {{sc|'''B'''}}, representing not their precise pitches, but merely to follow the modern standard of giving every distinct pitch in a scale its own, separate letter.<ref name=ML-West-1992/>

Since the [[musical system of ancient Greece|ancient Greek pitch systems]] only require eight different notes in a completed octave, and a modern keyboard has twelve, there actually are more than enough keys on any keyboard to implement one of the several enharmonic scales, contrary to Elson's [[#Elsons_stupid_remark_anchor|remark calling them "imaginary"]]. The only difficulty is retuning the strings (on an acoustic piano or harpsichord) or convincing an electronic [[sound module]] (for a modern [[MIDI keyboard|electronic keyboard]]) to produce the bizarre pitches required for enharmonic scale {{sc|'''D'''}}, {{sc|'''E'''}}, {{sc|'''A'''}}, and {{sc|'''B'''}} notes; the fixed notes ({{sc|'''C'''}}, {{sc|'''F'''}}, {{sc|'''G'''}}, and {{sc|'''c′'''}}) may also need comparatively slight adjustments, but in enharmonic scales they are all very nearly (or even exactly) tuned to the same [[relative pitch]]es they have in the [[12 equal temperament|conventional modern scale]].<ref name=ML-West-1992/>

For example, in modern [[microtone (music)|microtonal notation]], and standard-pitch [[quarter tone]]s (approximately 50[[musical cents|&nbsp;¢]] up = {{music|t}}, down = {{music|d}}), a simplified version of one of the enharmonic scales is
: {{sc|'''C'''}} (0[[musical cents|&nbsp;¢]]), {{sc|'''D'''}}{{music|d}} (50[[musical cents|&nbsp;¢]]), {{sc|'''E'''}}{{music|bb}} (100[[musical cents|&nbsp;¢]]), {{sc|'''F'''}} (500[[musical cents|&nbsp;¢]]),
: {{sc|'''G'''}} (700[[musical cents|&nbsp;¢]]), {{sc|'''A'''}}{{music|d}} (750[[musical cents|&nbsp;¢]]), {{sc|'''B'''}}{{music|bb}} (800&nbsp;¢), {{sc|'''c′'''}} (1200[[musical cents|&nbsp;¢]]).

None of the  pitches used in any standard enharmonic scale would actually be rounded to the nearest 50[[musical cents|&nbsp;¢]], but the approximate positions would be within about ±20[[musical cents|&nbsp;¢]] of those shown. It is also not necessary for the movable pitches to all lean toward their lower-bound fixed note; a somewhat more realistic example would be
: {{sc|'''C'''}} (0[[musical cents|&nbsp;¢]]), {{sc|'''D'''}}{{music|##}} (380[[musical cents|&nbsp;¢]]), {{sc|'''E'''}}{{music|t}} (420[[musical cents|&nbsp;¢]]), {{sc|'''F'''}} (500[[musical cents|&nbsp;¢]]),
: {{sc|'''G'''}} (700[[musical cents|&nbsp;¢]]), {{sc|'''A'''}}{{music|##}} (970[[musical cents|&nbsp;¢]]), {{sc|'''B'''}}{{music|t}}(1130&nbsp;¢), {{sc|'''c′'''}} (1200[[musical cents|&nbsp;¢]]).<ref name=ML-West-1992/>

The symbol {{music|t}} in this instance represents a [[half-sharp]], or sharpening by a [[quartertone]], however the actual pitches for [[music of ancient Greece|ancient Greek music]] the half sharp ({{music|t}}) and double sharp ({{music|##}}) pitches were allowed to be anything between around {{music|t}} = 30~70[[musical cents|&nbsp;cents]], and {{music|##}} = 130~240[[musical cents|&nbsp;cents]], depending on the aesthetics of the musician tuning the instrument.<ref name=ML-West-1992/>

Note that the modern sharp ({{music|#}}), flat ({{music|b}}), half-sharp ({{music|t}}), and half-flat ({{music|d}}) symbols do ''not'' (usually) represent fixed pitch changes when used to annotate ancient Greek notes, but instead only the approximate location of the actual pitches used in the Greek scale.

Although the movable notes are highly variable when a scale is devised, after the choice is made, all the notes are stuck in their respective positions until the end of a musical piece. So their use is not like modern musical forms, like the [[blues]], that use [[pitch bend]] on notes played on pitch elsewhere, and for those modern styles that use sliding pitch, at least in principle, any note might be bent during performance. As far as now known, the only form of "pitch bend" used by the ancient Greeks was in the initial tuning, with a bent pitch remaining bent until the instrument was retuned for the next piece of music.

More broadly, an enharmonic scale is a scale in which (using standard notation) there is no exact equivalence between a sharpened note and the flattened note it is [[enharmonic]]ally related to, such as in the quarter tone scale. As an example, F{{Music|sharp}} and G{{Music|flat}} are equivalent in a [[chromatic scale]] (the same sound is spelled differently), but they are different sounds in an enharmonic scale (as well as nearly every known musical tuning ''except'' for the modern [[12 equal temperament|12-tone E.T.]] scale). (''See'': [[musical tuning]] for a more complete introduction to the many non-12-tone E.T. tuning systems.)

[[Musical keyboard]]s which distinguish between enharmonic notes are called by some modern scholars [[enharmonic keyboard]]s, and more generically [[microtonal]] keyboards. (The [[enharmonic genus]], a tetrachord with roots in early Greek music, is only loosely related to enharmonic scales.)

[[Image:Lesser diesis (difference m2-A1).PNG|thumb|right|440px|Diesis defined in [[quarter-comma meantone]] as a [[diminished second]] {{nobr|( {{sub|min}}2nd − {{sup|Aug}}1st ≈ 117.1 − 76.0 ≈ 41.1 [[musical cents|cents]]),}} or an interval between two [[enharmonic|enharmonically equivalent]] notes (from D{{Music|b}} to C{{Music|#}}). {{audio|Enharmonic scale segment on C.mid|Play}}]]

== 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]]).

==References==
{{reflist|25em}}

==External links==
* {{cite book
 |last=Barbieri |first=Patrizio
 |year=2008
 |title=Enharmonic instruments and music, 1470–1900
 |location=Latina
 |publisher=Il Levante Libreria Editrice
 |url=http://www.patriziobarbieri.it/1.htm
 |access-date=2008-12-17   |url-status=dead
 |archive-url=https://web.archive.org/web/20090215045859/http://www.patriziobarbieri.it/1.htm
 |archive-date=2009-02-15
}}

{{scales}}

[[Category:Musical scales]]
[[Category:Musical tuning]]