SHEFKHED interval names: Difference between revisions

Kilmer also writes that 'the ancient Mesopotamian musicians/“musicologists” knew what we call today the Pythagorean series of fifths, and that the series could be accomplished within a single octave by means of “inversion” '. The Mesopotamian's music and theory was passed down through the Babylonians and the Assyrians to the Ancient Greeks, as well as their mathematics, particularly concerning musical and acoustical sound [[ratios]] (Ibid, [http://math-cs.aut.ac.ir/~shamsi/HoM/Hodgkin%20-%20A%20History%20of%20Mathematics%20From%20Mesopotamia%20to%20Modernity.pdf Hodgekin, 2005]).

In 2016 Dave Keenan proposed an alternative generalised [http://dkeenan.com/Music/EdoIntervalNames.pdf microtonal interval naming system for edos]. In what might be understood as a generalisation of his extended-diatonic interval-naming system described above onto any equal tuning. Employing as prefixes the familiar 'sub', 'super', and 'neutral'. His scheme is based on the diatonic scale, however the diatonic interval names are not defined by their position in a cycle of fifths like is Sagispeak. In Keenan's system the ET's best 3/2 is first labelled P5, and the fourth P4. The interval half-way between the tonic and fifth is labelled the neutral third, or 'N3', and halfway between the fourth and the octave N6. Then the interval a perfect fifth larger than N3 is labelled N7, and the interval a fifth smaller than N6 labelled N2. The neutral intervals then lie either at a step of the ET, or between two steps. After this the remaining interval names are decided based on the distance they lie in pitch from the 7 labelled intervals, which make up the ''Neutral scale'', P1 N2 N3 P4 P5 N6 N7, which, like the diatonic, is an MOS scale, which may be labelled [[Neutral7|Neutral[7]]] 3|3 using [[Modal UDP Notation|Modal UDP notation]]. This results in the conservation of symmetry about the tetrachord, and the octave, and the interval arithmetic associated with these symmetries. To name an interval in an ET, the number of steps of 72-tET that most closely approximate the size of the interval difference from a note of the neutral scale is first found. Then the prefix corresponding to that number of steps of 72-tET is applied to the interval name. The diagram to the right details this process. An interval just smaller than a major third in Keenan's system is labelled a ''narrow major third'', and an interval just wider than a 6/5 minor third a ''wide minor third'', however he notes that 'narrow' and 'wide' are only necessary in edos greater than 31. This system is equivalent to the Fokker/Keenan Extended-diatonic interval-naming system and Miracle interval naming when applied to any of the ETs they were able to cover. In application to ETs whose best fifth lies outside of the ''regular diatonic range'' (between 4 degrees of 7-tET, and 3 degrees of 5-tET)

Keenan's system is an elegant way to keep the 'major 3rd' label for 5/4 in application to non-meantone edos, while conserving interval arithmetic that results from symmetry about the tetrachord and the octave. However most interval arithmetic remains unconserved in non-meantone ETs. A potentially undesirable result of the system is that the major second approximates 10/9, and a ''wide major second'' 9/8, where as 9/8 is almost always considered a major second, and 10/9 often a narrow or small major second. One such system that considers 10/9 a narrow major second is that of Aaron Hunt.

Microtonal theorist [[Aaron Andrew Hunt]] devised [http://musictheory.zentral.zone/huntsystem4.html the Hunt system], which includes interval name assignments for JI (just intonation) and edos based on [[41edo]]. Compared to Keenan's 72 interval names, Aaron's system includes 41. His system is based directly on 41-tET, and unlike Keenan's system, interval are given the name of the closest step of 41-tET, and no account is taken of the size of the edos fifth. In 41-tET, Major, minor, augmented and diminished intervals are those obtained through the approximately Pythagorean cycle of fifths. Intervals one step of 41-tET above these are given the prefix 'small', one step larger are given the prefix 'large', two steps smaller the prefix 'narrow' and two larger the prefix 'wide'. As a result, 5/4 is labelled a 'small major 3rd', or SM3 (not to be confused with a super major third, a label that does not exist in this system).

Neo-medieval musicians and early music historian and theorist [[Margo Schulter]] described her own [http://www.bestii.com/~mschulter/IntervalSpectrumRegions.txt interval naming scheme] built on approximations to JI intervals. Each interval names corresponds to an approximate size, and no particular ET is referenced. In her scheme middle major thirds range in size from 400-423 cents, and small major thirds from 372-400c. 5/4 is labelled a small major third, 81/64 a middle major third and 9/7 a large major third. Margo's scheme includes small, middle and large varieties of major, minor and neutral 2nds, 3rds, 6ths, 7ths; perfect fourth and fifths; and tritones, as well as a sub fifth and super fourth; a dieses and comma and an octave less dieses and comma; and ''interseptimals'', which correspond to intermediates, her name referencing the fact that they may each approximate two ratios of 7.

In Hunt's system when used in 41-tET or JI diatonic interval arithmetic is conserved, but in other tunings it may not be, and Margo's system may not conserve diatonic interval arithmetic either. Both systems may be applied to arbitrary tunings, but the same intervals (defined, perhaps by a MOS scale) may not be given the same interval names across different tunings. [[User:PiotrGrochowski/Extra-Diatonic Intervals|Other]] size-based systems also exist, but are less thoroughly described and less well known. In all these systems, interval arithmetic is not conserved across all tunings.

One final interval naming system, associated with the [[Ups and Downs Notation]] system, belonging to microtonal theorist and musician [[KiteGiedraitis|Kite Giedraitis]], like Sagittal is based on deviations from the diatonic scale. In this system however, deviations (from major, minor, perfect, augmented and diminished) are notated simply by the addition of up or down arrows: '^' or 'v', corresponding to raising or lowering of a single step of an edo. In some tunings ([[12edo|12-tET]], 19-tET or 31-tET for example) 5/4 may be a M3, and in others a vM3 (downmajor 3rd) (e.g. [[15edo|15-tET]], [[22edo|22-tET]], 41-tET, 72-tET), or even an up-major 3rd (e.g. [[21edo|21-tET]]). Ups and downs also includes neutrals, which lay exactly in-between major and minor intervals of the same degree, labelled '~' (mid). 'Up' and 'down' prefixes may be used before mid also, i.e. 'v~ 3). P1, P4, P5 and P8 are simply labelled '1', '4', '5' and '8'. This system benefits from it's simplicity as well as it's conservation of interval arithmetic. Rank-2 temperaments may also be described, with the addition of of an additional pair of qualifiers - '/' and '\'. A rank-2 scale, such as a MOS scale may appear different than this rank-2 notation when approximated in an equal (rank-1) tuning. Another criticism of Kite's system that does not apply to the others is the fact that when an edo is doubled or multiplied by some simple fraction, and the best fifth is constant across the two edos, the same intervals may be be given different names.

31-tET in Ups and Downs: