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Encyclopedia of Microtonal Music Theory

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[Joe Monzo]

At its most basic, modern Western musical notation employs a staff utilizing 5 horizontal lines and the spaces between those lines, with a clef at the left side of the staff indicating the range of pitches (definition 1) available on that staff. There is often much more additional material in the notation of a musical composition, including indications for rhythm (durations), dynamics (volume), phrasing (various means of articulation for the notes, i.e., whether they are to be played connected or detached), and text specifying any number of other aspects of performance, including tempo (speed) and specific methods of playing the instrument (sul ponticello ["on the bridge"], etc.). This page will trace the history of the development of Western musical notation.

Much of our current Western music-theory is traceable in a direct line from the time of the Frankish Empire under Charlemagne, with their first books written around 800 AD. The Franks, who spoke a Germanic language, struggled to understand Latin and to find a way to notate their music, basing a lot of their theory on Boethius's Latin translation c.505 of ancient Greek books on music. So by studying Boethius they were aware of some of the Greek music-theory, but ultimately discarded some of it (such as Greek descriptions and measurements of microtonality) as apparently irrelevant to their own music, settling on using only pythagorean tuning of the diatonic scale as described by Boethius. The Greek system was based on equivalence at the perfect-4th, not on octave-equivalence as our system is today, and it used a complicated naming scheme where the names of both the tetrachord and the note had to be specified, and the note-names were based on the physical distance of lyre strings from the player's body as he held the lyre. The Greeks also had two types of notation for practical purposes, one instrumental and one vocal, which used letters and symbols derived from letters. Boethius gives an extensive explanation of these Greek notations, and he was in fact also the first author who used letters of the Roman alphabet to describe notes, but his use of them was in geometrical diagrams where they represent "point A", "point B", etc., extending well beyond "Z" by doubling letters after that, and they were not presented as being for practical use of musicians.

European musical notation started during the 700s with neumes, which were squiggly lines written above the words, and which were intended to sort of sketch out the shape of the melody on single syllables (which might have a lot of notes if sustained a long time). The neumes apparently developed from accent symbols that would be written over Greek/Byzantine text. They spread into Europe during the 800s-900s -- the Carolingian era when the Franks were the greatest European power.

But as described above, the ancient Greeks also had a musical notation, two different ones in fact, one called "vocal" and one called "instrumental". The "vocal" notation used letters of the Greek alphabet to indicate notes. The "instrumental" notation used rotations of symbols, some of which are Greek letters but the others of which seem to me to be derived from the earlier Phoenician alphabet, which is where the Greeks got the idea for their alphabet anyway.

(The Greek name for those people, "Phonike", refers to the sounds of speech -- cf. the modern English word "phonics". The Akkadian/Babylonian name for these people was "Kaininu", which is a Semitic word related to the Hebrew word which means "Canaanites". So they were apparently the people who invented the modern type of alphabet where each letter symbol represents approximately one phoneme. Older "alphabets" were really syllabaries, where one sound represented an entire syllable -- a more modern example is the Indian languages derived from Sanskrit, or the scripts of the Dravidian languages of southern India.)

The gamut of pitches was basically simply a holdover from the ancient Greek theory. The Greek version had an ascending interval structure which would be represented today by A2 B2 C3 D3 E3 F3 G3 A3 Bb3 B3 C4 D4 E4 F4 G4 A4, and whose tetrachords have interval pattern tone-semitone (definition 3)-tone.

Because of the nature of the Frankish chant used in the church Mass, and the fact that D was viewed as the most important starting pitch, the Franks realigned the ancient Greek tetrachord system known as the Perfect Immutable System, and added a new note ("gamma" γ, definition 3) at the bottom, so that the whole system was shifted a "tone" (definition 1) downward, G2 ... G4. Thus, the intervallic (definition 2) structure between degrees (defintion 1) of the tetrachords was different from the ancient Greek: tone-tone-semitone (defnition 3). This additional lower note "gamma" γ is where the word "gamut" originated. Hucbald (840-930 AD) was the theorist primarily responsible for this reorientation of the tetrachords, c.880.

The neumes up to this time are called "campo aperto", which means "open field", because the squiggly lines were written on otherwise blank parchment (i.e., with no lines), and showed a relative movement of melody but without any reference pitch. One simply had to learn a chant by rote, and the neumes were just there to help.

The treatise musica enchiriadis, which has been dated variously from c 860 - 950 AD, but which Monzo believes is an earlier work, has examples of "diastematic" notation, that is, a system of parallel horizontal lines which represent specific pitches, with T (for "tone") and S ("semitone") written in the margins next to the lines, and with the text written in the spaces.

example of diastematic notation from music enchiriadis

The musica enchiriadis also presents, as its first main topic, the so-called "daesian" notation, which was an adaptation of Greek letters which would be rotated for different tetrachords, and can be seen in the left margin of the image above. But Monzo believes that the daesian notation means something different from what all other commentators have written. (His unpublished paper A New Look at the musica enchiriadis, part of which appears in that link, examines his interpretation of this fascinating treatise.)

The type of vertical graphic thinking in the musica enchiriadis eventually resulted in a reference line being drawn across the "campo aperto" to represent a reference pitch. It could be any note, but was generally F (for the adult male singers), G (for the boys), or C (for voices in between those two).

Eventually this practice proved very useful, and scribes began to use a line for all three notes, with letters at the beginning to indicate which was which. This is the origin of our modern clefs: the letter F evolved into the bass clef and the letter G evolved into the treble clef. C became the C-clefs which used to be used on every one of the 5 lines, and the two still in use are still moveable today (alto and tenor clefs), but its evolution from the letter C is not as obvious as the bass and treble clefs.

These lines were also often given different colors, as a further visual aid to help the different singers stay on their proper part. So a score would have three lines on it with a lot of space in between each line, since the intervals are:

        treble G
"5th" <
        middle C
"5th" <
        bass F

This is precisely the reason why "middle-C" is referred to as such, and only coincidentally because it is near the middle of the full-size piano keyboard. It is exactly between bass F and treble G, and is the only diatonic note which falls between both the treble staff and the bass staff, touching neither.

The Dialogus of pseudo-Odo, written approximately the middle-900s, was the first music-theory treatise to name the pitches with the letters of the Roman alphabet in modulo-7 form, with A B C D E F G representing the diatonic pitches of the ascending scale, and repeating the letters in each octave (definition 1), thereby also employing octave-equivalence in practice for the first time in writing.

It was Guido d'Arezzo, around 1050 or so, who hit on the idea of using the diastematic notation of the musica enchiriadis for notating all the chants he needed to teach to his students, and he also developed the solfege syllables "ut - re - mi - fa - so - la" to represent the pitches in each hexachord (definition 2), which was his new system of pitch organization.

Guido used a 4-line staff, which became the standard (and still is) for notating Greogorian chant. Composers began adding other lines (up to 12), but eventually settled on the 5-line staff as a standard.

Even tho musicians thought in terms of hexachords (as explained here), the basic gamut of pitches used at that time was 2 octaves of the heptatonic (7-tone) diatonic scale plus the Bb below middle-C.

The hexachord system was based on those 3 reference pitches (which is exactly why they became reference pitches), and thus hexachords came in 3 versions which all had the same symmetrical intervallic structure:

hard    G - A - B - C  - D - E
natural C - D - E - F  - G - A
soft    F - G - A - Bb - C - D
"steps"   1   1  1/2   1   1
          t   t   s    t   t

The process of moving between different types of hexachords by employing common-tones was called "mutation", and in fact was another hold-over from ancient Greek practice.

As far back as pseudo-Odo, i.e., from the very beginning of using the alphabet to represent musical nominals, there two written symbols for the two different types of "B": round-b (or "soft-b") and square-b (or "hard-b"). The German words "mol/molle" (soft) and "dur" (hard) are still used today in many European languages to describe the difference between "major" and "minor".

Musicians eventually (c.1300) began to modulate more widely and thus employed the idea of mutation to travel to different tonal centers in addition to these 3.

Essentially, mutation employed the "soft-b" to indicate in a relative way a modulation into the "soft" hexachord, whatever its actual pitch might be. So the "soft-b" at this time always indicated a relative and not an absolute pitch.

The "soft-b" gradually evolved into our modern "flat" symbol flat - music notation accidental symbol, and the "hard-b" evolved into both the "sharp" sharp - music notation accidental symbol and, later, the "natural natural - music notation accidental symbol.

During the 1300s -- the "manneristic" era when music was composed with a rhythmic complexity that would not be seen again until the 1900s -- modulation became very popular, and by the end of the century the full chromatic scale had evolved, as the flat and sharp symbols were used more and more often to indicate absolute pitches.

Around 1400, there was a sharp decline in chromaticism, and over the next couple of centuries the notation largely settled down into the system in general use today.

We have only addressed the notation of pitch here. Of course, there's a whole other aspect of musical notation which is an entire study in itself: the notation of rhythm.

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