Notation
There are many different forms of musical notation in xenharmony. This page will serve as an introduction to some of the concepts found in xenharmonic notation, and some of the more common types. Note that this does not cover interval naming systems with no notational counterpart. Notation systems are usually designed for consistent tuning systems such as regular temperaments, or approximately consistent tuning systems.
In general, a notation system on this wiki allows for the transcription of musical information such as pitches, note durations, rhythms, etc. Most notation systems here act as extensions or modifications of standard staff notation; however, as this wiki focuses on xenharmony (that is, non-standard intervals), any way of naming and notating pitches (and optionally chords) is generally sufficient to be considered a notation system. A notation system may also include an interval naming system.
Introduction
Staff notation
Staff
A staff is a set of lines (or other reference) that symbols representing notes are placed in relation to. In most notation systems, notes are placed on or in between the lines, resulting in (2n+1) staff positions for n staff lines (without needing additional lines, called ledger lines). Different staff positions represent different pitches. In standard diatonic notation, there are five lines per staff, and therefore 11 distinct staff positions without adding ledger lines.
Clef
A clef is a symbol placed on a staff (usually at the very start) to indicate what note a particular staff line represents. The other notes can then be determined from the clef and a characteristic scale.
Characteristic scale
The characteristic scale is the "default" set of notes that serve as the basis for a system being notated. The notes of the characteristic scale are represented in text by nominals (written symbols representing notes, usually alphabetic, numerical, or drawn from some other extramusical sequence) and on the staff by distinct staff positions. For standard notation, the characteristic scale is mosdiatonic in the mode LsLLsLL on A = 440Hz. The nominals are ABCDEFG, and then A an octave up.
Note that the actual order of the alphabet may be of secondary importance to some musicians. While retaining it (as favored in KISS and diamond-mos notation) may aid intuition for people getting into music theory through xenharmony, some experienced musicians may prefer to treat other parts of their relationships as more fundamental in nonstandard notation schemes. The letters are thus in those cases arranged in a non-alphabetical order based on pitch or chain position, more letters may be added or existing letters removed, and the series may be otherwise altered to support the intuition of the arranger.
Accidentals
Accidentals are the term for any symbol representing an alteration to a nominal. In standard notation, the # accidental raises a note by a chromatic semitone, and the b accidental lowers by a chromatic semitone. Microtonal notation systems usually add new accidentals, rather than changing the set of nominals. For example, the ^ and v accidentals (arrows) raise and lower by an edostep. Note that accidentals other than the standard # and b (and derivatives) may instead be called inflections in some sources, wherein the category as a whole are alterations. Accidentals generally continue to apply to the same staff position until the next barline is reached.
Note that the words "sharp" and "flat" in alterations (and the symbols "#" and "b") are more specific than the adjectives sharp and flat as used in regards to tuning: the former refer to a fixed alteration by the chroma of mosdiatonic, which often generalizes outside diatonic (e.g. Bohlen-Pierce, KISS) to the moschroma in general, while the latter just mean "higher in pitch/distance" and "lower in pitch/distance", functioning as a generalization from the alignment of # and b with a single edostep in 12edo. More at Notation/Generalizations of sharp and flat.
Etymology
"Accident" originally referred to something non-essential or incidental, so an "accidental" represented a note that was non-essential to the key being played in.
Key signatures
A key signature is a set of accidentals placed at the beginning of the sheet music that apply to the whole piece and serve as the "default" accidentals to apply to staff positions. These accidentals also apply to notes separated by octaves from their position, which is not true of other accidentals. In general, there are only about 15 key signatures in use (based on the circle of fifths) representing the various transpositions of the diatonic scale. However, key signatures are theoretically possible for any set of defined alterations from the base notes of the characteristic scale, so that (for instance) one could specify a non-MOS diatonic scale with alterations to the standard version. (Note that otherwise, accidentals work normally, so that if the key signature specifies C#, placing a b accidental before a C note will read as Cb, not C (which is what the natural accidental is used for).) However, it is important to note that people usually only read the number of sharps or flats in the key signature to determine the scale, so scales other than diatonic modes are somewhat awkward to write and read as key signatures.
Etymology
Keys in Western music are usually each associated with a mode and transposition of the diatonic scale, the latter of which the key signature specifies.
Note symbols
While not directly pitch- or interval-related, an important consideration in standard notation is that note duration is represented by differences in symbol shape, rather than by physical size, which allows the precise reading of durations.
Annotations
Figured bass is a prominent form of notation accompanying a standard score that indicates what notes and/or chords to play, most commonly used for analytical clarity, especially where performers are expected to improvise. It utilizes the numbers 1 through 7 to indicate the degrees of a scale. One notable constraint of figured bass is that the symbols for degrees must be singular digits or letters, disallowing constructions like "1/2" and "15"; this makes it rather painful to generalize to microtonal scales while keeping numeric indices for intuition. One system works around this by utilizing the letters of the Greek alphabet, which allow for 24 distinct degrees to be notated. Another option is to utilize alternative systems to figured bass, such as placing chord symbols above the score.
Scale degrees vs. intervals
A scale degree refers to a note defined by an offset from the tonal center, tonic, or otherwise a fixed reference point that does not change throughout a song or section. Scale degrees are conventionally notated by specifying alterations from the degrees of the characteristic scale. Intervals are the offsets themselves, which may be applied either to the tonal center (specifying a scale degree) or to any other pitch. Intervals are notated with qualities such as major or minor, so that "m3" ("minor third") is an interval while "b3" ("flat three") is a degree. Note that outside of major and minor/sharp and flat, additional accidentals tend to share their names with qualities.
However, note that this is somewhat disputed:
- The established convention, with which Unque agrees, is to notate degrees with alterations.
- However, Inthar, HKM, and Vector prefer the use of qualities to notate degrees, as if they are intervals.
- Wad Wizard uses alterations as if they are qualities.
Additionally, conceptualization schemes vary massively as to the difference between scale degrees and intervals:
- Unque characterizes scale degrees as "points in space" and intervals as "types of motion within that space", characterizing pitch space relative to the tonal center as a form of "secondary absolute" space.
- Wad Wizard characterizes degrees as positions in a structure (reserving 'points in space' for notes i.e. absolute pitches) and intervals as separations between notes or degrees.
- Vector sees degrees and intervals as fundamentally the same kind of object, with intervals having unambiguous readings as degrees and vice versa.
- Inthar treats the distinction between the two as analogous to that between points in 2D coordinate space and 2D vectors.
Microtonal notations
Diatonic notation
This page covers commonly accepted extensions to standard diatonic notation used in microtonality. The 7 nominals C, D, E, F, G, A, B form the C major scale, and # and b raise and lower notes by a chroma.
KISS notation
KISS notation generalizes the principles of diatonic notation to any arbitrary MOS, involving alphabetic note naming based on the MOS in question, and accidentals based on the difference between that MOS' step sizes. KISS notation refers to a family of closely-related notation schemes.
Standard KISS notation
KISS notation, in its standard form, uses numbers to name notes, but the standard # and b symbols as accidentals, and places the treble and bass clefs around the middle note as they are in diatonic notation.
Quasi-diatonic MOS notation
Quasi-diatonic MOS notation uses letters as nominals, starting from A at 440Hz as the final note of the darkest mode of the scale (much as A is the final note of the Locrian scale before it repeats at B) uses standard accidentals, and has a custom system of clefs corresponding to each nominal.
Diamond-MOS notation
Diamond-MOS notation aims to disambiguate itself from diatonic notation by using different nominals (from J onwards), different accidentals (& = #, @ = b), and a custom clef system that makes it visually apparent where J is in different octaves on the staff and what MOS is being used.
Just notation systems
Conventionally, just intonation notation systems rely on a series of accidentals that move up or down a "formal comma" for each prime from a basic Pythagorean interval.
Common formal commas include 81/80 for 5 (the difference between the classical and diatonic major thirds), 64/63 for 7 (the difference between the diatonic and septimal major thirds), 33/32 (the difference between the perfect fourth and 11th harmonic) for 11, and 513/512 (the difference between the diatonic minor third and 19th harmonic) for 19.
todo: neuFJS or floraFJS as the standard for notating JI on this wiki?
| FJS | HEJI | Neutral FJS | Ben Johnston | |
|---|---|---|---|---|
| Basic diatonic | Pythagorean | Zarlino | ||
| 7 | 64/63 | 36/35 | ||
| 11 | 33/32 | sqrt(243/242) | 33/32 | |
| 13 | 1053/1024 | 27/26 | sqrt(507/512) | 65/64 |
| 17 | 4131/4096 | 2187/2176 | 4131/4096 | 51/50 |
| 19 | 513/512 | 96/95 | ||
| Accidentals used | Numbers that stack multiplicatively, with otonal intervals all inflected in the same direction | A selection of symbols | Numbers that stack multiplicatively, with otonal intervals all inflected in the same direction | Numbers and inverted numbers, with otonal intervals all inflected in the same direction, except for the syntonic comma (+/-) and prime 11 (^/v), |
| Other notes | All formal commas are derived systematically. Each prime harmonic is inflected from the simplest Pythagorean interval within a semiaugmented unison (56.8 cents) (or, as originally described, 65/63 (54.1 cents)). | More aesthetically pleasing than FJS in certain contexts and more widely supported, however its symbol choices are very unintuitive and arbitrary. | Similar to FJS, but includes "dicot" intervals for more intuitive notation of neutral intervals. Each prime harmonic is inflected from the simplest Pythagorean or neutral interval within 33.4 cents. | Uses Zarlino diatonic as nominals, 36/35 as the accidental for 7, and 96/95 as the accidental for 19. |
Color notation
Color notation is a notation system that uses a series of syllables to notate just intervals, fully replacing the standard interval qualities. An interval's degree is determined from its mapping to 24edo.
| Syllable | Intervals | Notes |
|---|---|---|
| la | Raised by a chromatic semitone | Not used, # is used instead |
| sa | Lowered by a chromatic semitone | Not used, b is used instead |
| wa | 3-limit, no alterations | 3-limit, no alterations |
| ru | 7 under, generally supermajor | +64/63 |
| yo | 5 over, generally ptolemaic major | -81/80 |
| gu | 5 under, generally ptolemaic minor | +81/80 |
| zo | 7 over, generally subminor | -64/63 |
| lu | 11 under | -33/32, +729/704 |
| lo | 11 over | +33/32, -729/704 |
Ambiguous just intonation intervals
Systems of conceptualizing just intonation in relation to the diatonic scale disagree on the identities of certain intervals.
- 17/16 is a minor second in the Functional Just System and an augmented unison in the Helmholtz-Ellis system; all intervals of 17 are affected by this disparity. It is between the two in terms of size - the minor second is a simpler diatonic interval but it is closer in size to the augmented unison.
- 14/11 is often conceptualized as a major third (specifically, a neogothic major third), but most notation systems that use formal commas recognize it as an inflected perfect fourth. This is because it is the difference between 11/8 and 7/4 - 11/8 is about as sharp as a perfect fourth can reasonably be, and 7/4 is about as flat as a minor seventh can reasonably be. Therefore, their difference, another perfect fourth, is narrowed considerably. Systems like sagittal notation attempt to accurately capture 14/11's melodic quality, at the cost of sacrificing some stacking logic.
