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Encyclopedia of Microtonal Music Theory

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[Joe Monzo]

A symbol placed before a note-head in musical notation, or after a letter-name in a text description, which indicates some intonational change or pitch adjustment relative to the tuning of the plain note or letter-name. The name "accidental" is derived from the fact that these notes are felt to lie outside the regular diatonic scale.

The standard musical accidentals are:

symbol   name           effect

  #   sharp         raise 1 semitone
  b   flat          lower 1 semitone
  x   double-sharp  raise 2 semitones (sometimes written ##)
  bb  double-flat   lower 2 semitones

There is also a "natural", which cancels the effect of a previously given accidental, whether that accidental occurs in a particular instance or in the key-signature.

In the standard 12-edo scale, the double-sharps and double-flats have the effect of altering the pitch by a whole-tone and are often simply written as the equivalent natural with a change of letter-name (i.e., Cx = D, Bbb = A), but this is not necessarily the circumstance in other tunings. The exact amount of alteration by all of these standard accidentals depends on the size of the chromatic semitone in the given tuning.

In addition to these four standard accidentals, many others have been devised by microtonalists to specify various intonational adjustments. My own proposal is dubbed HEWM, which uses these additional symbols:

- + < > v ^

and comes in two varieties, one based on extended just-intonation and the other (simpler) version based on 72-edo. The latter eliminates the need for multiple instances of a particular symbol, because ++ equals > and >+ equals ^, etc, and bb and x are written as the equivalent natural with a change of letter-name, as with 12-edo.

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Below is an explanation of the origin of the standard Western accidentals.

hi Lorenzo, Johnny, and Haresh,

--- In tuning@yahoogroups.com, "lorenzofrizzera" wrote:

In this site http://www.uk-piano.org/history/compass.html, I've found this:

The first sharp to be added to the keyboard was probably the F sharp, according to academic research.

The interval of an augmented fourth, between the notes we would call F and B, was considered discordant, so the B was often lowered, bringing in an extra note, B flat, shorter and narrower, between the A and the B. After the B flat probably came the E flat, then C sharp and finally G sharp.

The author wrote me that those sections were taken from his notes at history and tuning letures back in the 1970's.

Are these informations correct?


it may be true that F# was the first accidental added to the *keyboard*, i'm not sure ... but Bb came before any other accidental in both theory and non-keyboard practice.

it arose as a result of the ancient Greek Perfect Immutable System (PIS), which included both the Greater Perfect System (GPS) and the Lesser Perfect System (LPS).

all of these scales were based on "similar tetrachords", i.e., tetrachords which all had the same interval structure.

the GPS had 2 pairs of tetrachords conjunct (connected together by the same note serving simultaneously as the bottom note of one tetrachord and the top note of the one below it: the hyperbolaion-diezeugmenon pair, and the meson-hypaton pair), but also a "tone of disjunction" (9:8 ratio) between the main reference note mese ("middle", the top note of tetrachord meson) and the bottom note of the tetrachord above it (tetrachord diezeugmenon).

the LPS used 3 tetrachords which were all conjunct: synemmenon-meson-hypaton.

(and both the LPS and GPS had the "added tone" proslambanomenos at the bottom.)

when medieval theorists hit on the idea of naming the notes with Roman letters, and then tried to give their scales cachet by making analogies with ancient Greek theory, they named mese "a" and used the other letters b, c, d, e, f, g in ascending order for the rest of the distinct notes in the PIS, then repeated the letters for notes which were an octave apart, which was an early recognition of "octave"-equivalence.

(this had actually already been done with the Greek-letter notation attested by Alypius, Aristides Quintilianus, and Boethius ... but all three of these authors were well after the absorption of Greece into the Roman Empire. my own guess is that the Greeks themselves never recognized "octave"-equivalence in their notation, because for them everything was based on tetrachords.)

the whole PIS spanned 2 octaves, one each below and above mese "a", so capital letters were used for the notes in the lower "octave" below mese "a", and lower-case for the notes in the upper "octave" above mese "a".

however, the "b" in diezeugmenon was a whole-tone (9:8, i.e., the "tone of disjunction") above "a", whereas the "b" in synemmenon followed the tetrachord pattern of having the semitone at the bottom of the tetrachord, and was thus only a semitone above "a". so there were two different "b"s, one a semitone higher than the other.

thus a means of distinguishing between them was necessary, and so the regular round "b" ("soft-b") was used for the lower "b", and a "b" whose round part was written with straight lines so as to make a square ("hard-b") was used for the higher "b".

the "soft-b" eventually evolved into our modern flat symbol (quite obviously), and the "hard-b" evolved into both the natural symbol and the sharp symbol, which for quite a long time were used interchangeably -- this is because the practice of mutation and musica ficta, based on solfege syllables, preceded our modern practice of having specific pitch alteration meanings for the accidentals.

i've simplified the story a bit ... one big thing i left out was that the boundaries of all the tetrachords were shifted a tone lower in pitch, and the reference tetrachord shifted down by one, to the one which contained D, E, F, G. this was because the Frankish theorists wanted to use what they could of the ancient Greek theory, but base their system on the finals of the church modes which were then being used in *their* music, the actual Greek music having become obsolete long before then.

you should be able to piece together all of the details from my relevant Encyclopaedia pages ... here are a few, which should be consulted in this order:


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