Approaches to microtonal notation
There are three broad theoretical approaches to microtonal music notation:  pragmatic;  structural;  schismatic.
The pragmatic approach
The first approach stresses the importance of continuity. According to this philosophy, the most important criterion for any microtonal notation is that it must ease (insofar as is possible) the transition from a standard 19th century European 12-tone equal tempered music notation to a new microtonal notation. Microtonalists who advocate this pragmatic approach typically compose music for traditional 19th century European instruments with pitch inflections added to standard notes. Microtonalists who advocate this approach to xenharmonic notation include Ezra Sims, Johnny Reinhard, the Catler Brothers, the Netherlands Huyghens-Fokker 31-tone group, virtually all quartertone composers from the 1920s-1940s (viz., Ivan Wyschnegradsky, Alois Haba, Julian Carrillo). All of these xenharmonists typically propose variants of traditional notation using 7 standard western note-names on a standard western 5-line staff with modified or added accidentals (or other signs; + or -, for instance, or up-arrow, or down-arrow) to indicate non-12 pitches.
A subgroup of these "pragmatic" composers advocate using unmodified traditional western notation with a new meaning for each symbol. One of the most popular 19-tone equal notations, for instance, uses completely traditional notation but with new meanings for "sharp" and "flat:" in this case, D# is a different pitch from Eb, giving 19 discrete different pitches all told (Cb is identical to B#, and Fb is identical to E#--all other sharp/flat pairs which would be considered enharmonic in 12 tone equal temperament are considered different pitches in this notation for 19 equal). Ivor Darreg, M. Joel Mandelbaum and other composers advocated this type of notation. The other "new use" for traditional notation can be found in tablature, as typically used for digital keyboard instruments. According to this form of microtonal notation, the unmodified standard notation now indicates only the keys to be depressed by the performer--the pitches sounded will typically be radically different.
Composers who advocate this type of "pragmatic" microtonal notation include Enrique Moreno, Warren Burt, Gary Morrison, and Harry Partch in his Chromelodeon notation.
The structural approach
The second group of microtonalists stress the pre-eminent importance of reflecting the deep harmonic/melodic structure of an intonation in its notation. According to this approach, it is less important that a microtonal notation be compatible with current practice than that the notation produce results which do not violate group theoretic or harmonic properties of the tuning. Microtonalists who advocate this approach to notation typically compose music on paper, with the realization of the composition regarded as secondary. Such microtonalists include Rudolf Rasch, Siemen Terpstra, Easley Blackwood and Paul Rapoport. These microtonalists consider it important that different tunings with similar properties be notated in a similar way. As a result, this group typically favors adding new accidentals and using them in new ways. For example, Easley Blackwood's notation for 15 tone equal temperament includes the implicit rule that E and F sound identical pitches, as do B and C. This brings out the essential pentatonic structure buried within the 15 tone equal temperament, although it could prove confusing for someone unfamiliar with the notation.
One of the greatest problems faced by the structuralists is that many tunings exist for which no modification of standard 12-tone notation is sufficient to adequately bring out the deep structure of the tuning. Paul Rapoport has pointed out that 13 and 14 tone equal temperament tend to require drastic notational solutions which generally do violence to traditional concepts of musical notation: these two tunings are notated by using every pitch out of 26 and 28 tone equal temperament, respectively--yet this creates a problem, since 26 and 13 tone equal temperaments sound completely different and share almost nothing in common from a musical standpoint. Ditto 14 and 28 tone. 50 and 33 tone equal temperaments are also difficult to deal with given the confines of traditional notation.
The schismatic approach
Advocates of this mode of microtonal notation stress that non-12 tunings often produce musical results incompatible with traditional western music. Intervals much smaller than the semitone, as well as exotic scales in which higher members of the harmonic series are exactly represented, demand that traditional 12-tone equal tempered notation be thrown out (according to this group). Accordingly, advocates of the schismatic approach to microtonal notation typically design entirely new musical staves, or do away with the musical staff entirely. They will often use new note-shapes, new types of note-heads, or more than 7 note-names: microtonalists who have taken this approach to notating non-12 music typically invent new instruments for which traditional notation would be useless.
Such microtonalists include Leo deVries (with his "Twinline" notation), Joseph Yasser (with his keys X, Y and Z), Ted Mook (with his differently-shaped note-heads embodying Harry Partch's just ratios) and Karlheinz Stockhausen (with his graphic notation for the 25th root of 5 in the 1956 composition "Studie II").
Four different usages
The three general theoretical approaches mentioned above stress abstract considerations. The intent: to bring out one or another generalized property of an intonational system. Down in the trenches, however, real-world microtonal performers generally do not worry so about whether a microtonal notation reflects some arcane group theoretic or harmonic-series property of a tuning. Real microtonal performers in the real world are apt to worry more about whether the notation fits their performance technique and their compositional/improvisational methodology. Thus, in addition to the three highly abstract approaches to xenharmonic notation taken by microtonal theorists, there exist 4 lower-level practical strategies for notating microtonal music. These strategies are:  diatonic-ornamental;  computer-numerical;  gestural;  procedural.
Diatonic with microtonal inflections
Some microtonal music consists of primarily diatonic music using either 5 or 7 relatively large intervals with a constellation of much smaller microtonal intervals used as inflections of the primary pitches. Sundanese Jaipongan and Japanese koto and gagaku music, along with many Indian ragas can be viewed this way; some of higher divisions of the octave are also used this way, viz., 53 and 41 and 31 tone equal temperament (in the hands of some composers). Most important, however, is the issue of the instrument used to perform microtonal music. Unfretted string instruments, double reed instruments, valveless brass and voice encourage this kind of microtonal performance: indeed, almost all of the world's vocal music even when nominally in the 12 tone equal temperament uses microtonal inflections to add "life" and "verve" to the performance.
Examples: Sinead O'Conner's nominally 12-tone equal but inflexionally microtonal performance of "Nothing Compares To You," the microtonal inflexions in The Doors' song "The End," Odetta and Louis Armstrong's rendition of "St. James Infirmary," etc. Microtonalists who use this strategy of microtonal notation include trombonist George Lewis, guitarist Dan Stearns, and many others.
Computer and MIDI formats
The year 1986 marks a sharp break with all previous musical history. Prior to that year, a microtonalist who wanted to compose and perform music outside the 12 tone equal temperament had to build special-purpose one of a kind instruments and train players in exotic performance techniques. This changed in 1983 with the advent of MIDI. In 1986 the first commercial affordable retunable synthesizers had appeared, and since that time retunability has always been available on at least some MIDI synthesizers. This means that since 1986 microtonality has been standardized to some degree on electronic instruments, and brought within the budget of the average musician. The standard protocol for microtonal electronic music is called MIDI (an acronym which stands for "Musical Instrument Digital Interface"). This standard protocol allows any MIDI synthesizer or controller to send notes to any other MIDI synthesizer or controller. The MIDI protocol is revolutionary because it does not embody pitch information: a MIDI "note" is merely an instruction which tells a synthesizer to look at one of 128 locations in a numerical table and sound the pitch which resides there. Once MIDI notes are stored in a file, a composition can be reproduced exactly on another computer provided that an identically-tuned synthesizer is available. Thus MIDI is a musical notation, but a singularly oblique one: in this notation, there are 16 channels, each with 128 "slots" to which any pitch can be assigned. By itself, a MIDI note betokens nothing: the notation only makes sense when combined with the values of the numeric pitch table stored in the synthesizer's memory.
The oblique nature of MIDI's reference to pitch incurs some disadvantages--for example, it is not possible to tell by looking at a printout of a MIDI note file which pitches the synthesizer is actually playing, since the "notes" are merely abstract numbers from 0 to 127 indicating arbitrary sound frequencies. It can also be confusing, when performing a composition which uses several pitch tables spread over several channels: which pitch sounds on which MIDI "note" on which channel? Yet another potential drawback derives from the fact that all MIDI synthesizers typically require that a pitch be specified to the nearest 1/768 of an octave, or the nearest 1/1024 of an octave. This accuracy may not be sufficient for some musical purposes, particularly long-held microtonal just intonation chords. However the MIDI protocol's uniquely generality also confers some advantages. For one, MIDI allows any conceivable tuning to be played--as long as it can be represented within a gamut of 128 abstract numbers. For another, a MIDI controller need not be a physical keyboard: it can, for instance, be an entirely abstract software program which generates MIDI "notes" as a result of purely mathematical procedures.
A related protocol is the Csound note format. This protocol represents notes as either Hertz or 12-tone equal tempered chromatic pitches with a numerical microtonal offset. When Hertz are used, this format specifies microtonal pitches unambiguously and also places no restrictions on the pitch granularity (rather than 1/768 of an octave, Csound typically allows pitch to be specified to within 1/10,000 of a Herz or so) or on the rhythmic complexity.
Composers and performers who use the MIDI and/or some modification of the Csound numerical format for notating microtonal music typically use computers and/or syntehsizers as their primary musical instruments. Such microtonalists include Richard Boulanger, William Schottsteadt (PLA and Common Music, formats similar to the Csound numerical pitch format), James Dashow, John Chowning, Buzz Kimball, Elaine Mullen, Mathew Puzan, most members of the San Francisco-based Just Intonation Network, and others.
Microtonal acoustic or electronic performances need not consist of specifically defined discrete notes; microtonal tone-complexes and clusters or clouds of notes can be produced by gestures. For example, many of Tom Nunn's and Bart Hopkins' and Trimpin's and Q.R. Ghazala's microtonal instruments generate sweeps of sound in response to continuous movements, pressures or manipulations with a bow or stick or hand. Notations for these instruments are often gestural in nature. Similar gestures repeated on the same section of an electroacoustic or acoustic microtonal instrument whose components resonate or interact electronically will often produce distinctly microtonal and reproducibly similar sets of pitches. Such notations typically place more emphasis on the mode of excitation, the object used, and the exact spot on the instrument stroked or bowed, than on the pitch produced.
Microtonalists who use such notations typically invent new instruments based on resonant or gestural principles. Such microtonalists include Jacques Dudon (who employs sequencer disks in his photoelectric synthesizer to control the operation of the other discs) Bart Hopkins, Jonathan Glasier, Tom Nunn, Q.R. Ghazala, Trimpin, Iannis Xenakis (in his UPIC system), Donald Buchla (in his Lightning series of digital sensors) and others.
In many cases, it is unnecessary to denote pitch because performers work with microtonal instruments in which the pitch is rigidly fixed and unchangeable--for instance, metallophones, lightly-struck zithers, xylophones, etc. Microtonal music played on groups of such instruments is often notated in terms of groups of actions repeated en bloc: in effect, a meta-notation indicating repeated sets of phrases (which are typically visualized by performers as repeated locations to be periodically struck or plucked). Such notations represent a high level of musical abstraction and typically require a good deal of additional explanation.
Microtonalists who use such notations typically work in microtonal ensembles where tight-knit coordination is of prime importance. Such microtonalists include Sundanese, Javanese and Balinese gamelan performers or American gamelan performers/composers who use gamelan notation, classical Korean musicians who use a kind of matrix indicating both rhythm and pitch; William Wesley, who also uses a similar pitch-rhythm matrix; Kraig Grady, who uses a block-sectioned graph paper notation indicating repeated phrases and rhythms; and so on.
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