In music theory, anenharmonic scale is averyancient Greek musical scale which contains four notes tuned to approximatelyquarter tone pitches, bracketed (as pairs) between four fixed pitches.[4] For example, in modernmicrotonal notation, one of the severalenharmonic scales aligned with the conventional key ofC major would be
The symbol
in this example represents ahalf-sharp, or sharpening by aquartertone (50 cents), although raising pitch by exactly 50 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,D &E;A &B) variably spaced, with about 20~30 cents between each other, and likewise spaced from their closest fixed note (for this example those areC,F,G, andC′).[4]
Four of the scale notes – thetonic (C in the example),subdominant (F),dominant (G ), andoctave (C′) – are all fixed: They are nearly exactly the samerelative pitches in all three categories of ancient Greek scales (enharmonic,chromatic, anddiatonic),[4] and in ancient Greek music, the fixed tonesrelative pitches were very nearly the same as the corresponding notes in the modernconventional scale. On the other hand, the four notes contained between the brackets, from the exampleD andE (betweenC andF); andA andB (betweenG andC′) 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 ofIndia and theMiddle 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.[4]
The ancient Greek meaning ofenharmonic 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.)[4] 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. Inancient Greek music from whichenharmonic scales come, the meaning ofenharmonic not so extreme: It means that the notes arenot 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 technicallydieses (divisions) which do not occur on standard modern keyboards,[2]nor were even used in the preceding western tuning systems, such as¼ comma temperament (the predominant tuning about 200 years ago) orwell temperament (finally went out of use as conventional tuning about 140~150 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 forhalf tones, as do frets onguitars andmandolins, fingering holes onwoodwinds, and valves onbrass instruments. This has been the situation for more than 150 years for fixed-pitch occidental instruments.
Even amongHellenic musicians, enharmonic scales appear to have gone out of style around2500 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 theRoman Empire, whose works on music theory we still have.[4] 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 ofIndia and theMiddle East still using similar intervals traditional and classical scales.
An otherwise well regarded 19th century musicologist once wrote the rather blatantly false definition in his 1905 musical dictionary, that theenharmonic scale is
However, enharmonic tuning does seem "imaginary" to many modern western musicians because of the intentional limitations placed intoconventional 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 ofancient Greek musical scales (among whom sits Elson[3]) nor even relatively recently disused tuning systems, such as the¼ comma meantone temperament predominantly used up to the time ofBach, and the later unequalwell temperaments 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[4]) into theRoman Imperial era. Although still taught as a perfunctory part ofHellenistic education, the enharmonic scale was only rarely – if ever – used during the period of 180~400 CE when the Greek musical theory books which still survive were written.[5][4]
The enharmonic scale usesdieses (divisions) which are not tuned in any pitch present on standard modern keyboards,[2]since modern, standard keyboards only have provisions forhalf-tone steps. The two different notations used for vocal and instrumental notes inancient 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.[4] Despite the pitches being unknown to naïve occidentally-trained musicians, all theancient 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 fixedtonic andsubdominant (orfourth) (relative toCMaj, the notes betweenC andF), and the other two movable notes between fixeddominant /fifth and theoctave (betweenG andC′). When expressing notes with modern letter notation, it is conventional to use some elaborately sharpened or flattened version of the notesD,E,A, andB, representing not their precise pitches, but merely to follow the modern standard of giving every distinct pitch in a scale its own, separate letter.[4]
Since theancient 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'sremark calling them "imaginary". The only difficulty is retuning the strings (on an acoustic piano or harpsichord) or convincing an electronicsound module (for a modernelectronic keyboard) to produce the bizarre pitches required for enharmonic scaleD,E,A, andB notes; the fixed notes (C,F,G, andC′) may also need comparatively slight adjustments, but in enharmonic scales they are all very nearly (or even exactly) tuned to the samerelative pitches they have in theconventional modern scale.[4]
For example, in modernmicrotonal notation, and standard-pitchquarter tones (approximately 50 ¢ up =, down =
), a simplified version of one of the enharmonic scales is
None of the pitches used in any standard enharmonic scale would actually be rounded to the nearest 50 ¢, but the approximate positions would be within about ±20 ¢ 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
The symbol in this instance represents ahalf-sharp, or sharpening by aquartertone, however the actual pitches forancient Greek music the half sharp () and double sharp (
) pitches were allowed to be anything between around = 30~70 cents, and = 130~240 cents, depending on the aesthetics of the musician tuning the instrument.[4]
Note that the modern sharp (♯), flat (♭), half-sharp (), and half-flat () symbols donot (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 theblues, that usepitch 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 isenharmonically related to, such as in the quarter tone scale. As an example, F♯ and G♭ are equivalent in achromatic scale (the same sound is spelled differently), but they are different sounds in an enharmonic scale (as well as nearly every known musical tuningexcept for the modern12-tone E.T. scale). (See:musical tuning for a more complete introduction to the many non-12-tone E.T. tuning systems.)
Musical keyboards which distinguish between enharmonic notes are called by some modern scholarsenharmonic keyboards, and more genericallymicrotonal keyboards. (Theenharmonic genus, a tetrachord with roots in early Greek music, is only loosely related to enharmonic scales.)
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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 somekitharodes 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 differentkitharas during a performance, with each tuned to a different, but tonally interlocking enharmonic scale.
Consider a scale constructed throughPythagorean tuning: A Pythagorean scale can be constructed "upwards" by wrapping a chain ofperfect fifths around anoctave, but it can also be constructed "downwards" by wrapping a chain ofperfect fourths 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:
| Note | Ratio | Decimal | Cents | Difference (cents) |
|---|---|---|---|---|
| C | 1:1 | 1 | 0 | |
| D♭ | 256:243 | 1.05350 | 90.225 | 23.460 |
| C♯ | 2187:2048 | 1.06787 | 113.685 | |
| D | 9:8 | 1.125 | 203.910 | |
| E♭ | 32:27 | 1.18519 | 294.135 | 23.460 |
| D♯ | 19683:16384 | 1.20135 | 317.595 | |
| E | 81:64 | 1.26563 | 407.820 | |
| F | 4:3 | 1.33333 | 498.045 | |
| G♭ | 1024:729 | 1.40466 | 588.270 | 23.460 |
| F♯ | 729:512 | 1.42383 | 611.730 | |
| G | 3:2 | 1.5 | 701.955 | |
| A♭ | 128:81 | 1.58025 | 792.180 | 23.460 |
| G♯ | 6561:4096 | 1.60181 | 815.640 | |
| A | 27:16 | 1.6875 | 905.865 | |
| B♭ | 16:9 | 1.77778 | 996.090 | 23.460 |
| A♯ | 59049:32768 | 1.80203 | 1019.550 | |
| B | 243:128 | 1.89844 | 1109.775 | |
| C′ | 2:1 | 2 | 1200 |
In the above scale the following pairs of notes are said to be enharmonic:
In this example, natural notes are sharpened by multiplying its frequency ratio by 256 / 243 (called alimma), and a natural note is flattened by multiplying its ratio by 243 / 256 . A pair of enharmonic notes are separated by aPythagorean comma, which is equal to531441/524288 (about 23.46cents).
