Extended-diatonic interval names: Difference between revisions

[[File:Mesopotamian interval names table.jpg|thumb|500x500px|Mesopotamian interval names, from http://www.historyofmusictheory.com/?page_id=130, accessed October 7, 2018.|link=https://en.xen.wiki/w/File:Mesopotamian_interval_names_table.jpg]]Music theory describing the use of heptatonic-diatonic scales, including interval names, has been traced back as far as 2000BC, deciphered from a Sumerian cuneiform tablet from Nippur by Kilmer (1986). From Kummel (1970) we know that 'the names given to the seven tunings/scales were derived from the specific intervals on which the tuning procedure started' (Kilmer, 1986). This formed the basis of their musical notation ([http://www.jstor.org/stable/985853. Kilmer, 2016]). The table to the right following table displays the Ancient Mesopotamian interval names accompanied by their modern names.

Such mathematical and musical ideas are attributed to Pythagoras, who undoubtedly made them popular., although many scholars suggest he may have learned these ideas from his Babylonian and Egyptian mentors. Nonetheless, Pythagoras' idea that that by dividing the length of a string into ratios of halves, thirds, quarters and fifths created the musical intervals of an octave, a perfect fifth, an octave again, and a major third form the basis of Ancient Greek music theory (http://www.historyofmusictheory.com/?page_id=20). His tuning of the diatonic scale by only octaves and perfect fifths ([[Pythagorean tuning]]) is influential through to today.

''ditone'' referred to the interval made by stacking two [[9/8]] whole tones, resulting in [[81/64]], the Pythagorean major third. ([[Joe Monzo|Monzo]], [http://www.tonalsoft.com/ http://www.tonalsoft.com])

Included in this scale, however, were ''wolf intervals:'' imperfect consonances that occurred as tunings of the same interval as perfect consonances. For example, between 1/1 and 3/2, 4/3 and 1/1, 5/3 and 5/4; and 5/4 and 15/8 occurs the perfect fifth, 3/2, whereas between 9/8 and 5/3 occurs the wolf fifth, [[40/27]], flat of 3/2 by [[81/80]]. This was also the interval by which four 3/2 fifths missed [[5/1]] (the interval two octaves above 5/4). It was named the ''syntonic comma'' after Ptolemy's ''syntonus'' or ''intense diatonic tetrachord'' which consists of the intervals 9/8, [[10/9]] and [[16/15]], where 9/8 and 10/9 differ by this interval. By making the syntonic comma a unison the wolf fifth could be made a perfect fifth. It was discovered that this could be achieved by flattening (tempering) the perfect fifth by some fraction of this comma such that four of these fifths less two octaves gave an approximation of 5/4. Where two fifths less an octave give 9/8, the next two add another 10/9 to result in the 5/4. 9/8 and 10/9 were referred to as the ''major tone'' (''tunono maggiore'') and ''minor tone'' (''tunono minore''), respectively, and where this tuning led to them being equated, it was referred to as Meantone temperament, which is said to 'temper out' the syntonic comma. Zarlino advocated the flattening of the fifth by 2/7 of a comma, leading to [[2-7 Syntonic Comma Meantone|2/7-comma Meantone]], but also described [[1-3 Syntonic Comma Meantone|1/3-comma]] and 1/4-comma Meantone as usable (Zarlino, 1558).

The diagram on the right, from Zarlino's 1558 treatise ''Le istitutioni harmoniche'' associates many intervals with their tuning as perfect consonances. The perfect tuning for the ditone was considered then to be 5/4, rather than 81/64. The interval for which 6/5 is considered a perfect tuning was referred to as a ''semiditone'' (labelled also in ''Le istitutioni harmoniche'' by as ''Trihemituono)''. This may seem odd to us now, but in Latin 'semi' referred not to 'half', but to 'smaller', so 'semiditone' translated to something like 'smaller ditone'. Additionally 'semitone' referred to the interval smaller than the 'tone'. Like the tone, this interval possessed two alternative perfect tunings: 16/15, the difference between 15/8 and 2/1, or 5/4 and 4/3, and [[25/24]], the difference between 6/5 and 5/4. 16/15 was referred to as the ''major semitone'' (''semituono maggiore'') and 25/24 as the ''minor semitone (semituono maggiore'').

|}Each interval name has two sizes that differ by the comma 243/242. The notation included in the table is from HEWM notation, developed as an extension to the Helmholtz-Ellis use of '+' and '-' by Joe Monzo (http://www.tonalsoft.com/enc/h/hewm.aspx<nowiki/>).'^' indicates raising 'v' a lowered of [[33/32]]. In HEWM notation '+' and '-' are refined to mean raising and lowering of 81/80 respectively and '>' and '<' are added instead to indicate raising and lowering of 64/63. Letter names correspond instead of the Ptolemaic sequence, as in Smith's and Helmholtz' descriptions, but to a Pythagorean tuning of the diatonic scale, where '#' and '♭' and respectively raise and lower the apotome, 2187/2048. HEWM notation is not accompanied by an interval naming system.

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 pythagorean intervals. 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). The mid-4th ~4 is halfway between P4 and A4, and ~5 is likewise half-way diminished. 'Up' and 'down' prefixes may be used before mid also, e.g. 'v~ 3' in 72-tET. P1, P4, P5 and P8 when upped or downed (or midded, in the case of P4 and P5) are simply labelled '1', '4', '5' and '8'. This system benefits from its simplicity as well as its conservation of interval arithmetic. The latter makes possible the naming of chords, e.g. downminor 7th. Rank-2 temperaments may also be described, with the possible addition of an additional pair of qualifiers, lifts and drops - '/' 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 given different names.

== SHEFKHED interval names ==

{{Main| SHEFKHED interval names }}

This review was both motivated by, and has been integral to, the development of the author's own interval naming scheme. SHEFKHED Interval names, or ''Smith/Helmholtz/Ellis/Fokker/Keenan/Hearne Extended-diatonic interval names'', are essentially an extension of Fokker/Keenan Extended-diatonic interval-names, with a nod to Smith, Helmholtz and Ellis, redesigned with Pythagorean intonation at the core, from Sagittal/Sagispeak, where prefixes correspond to alteration by specific commas, into non-meantone edos, keeping interval arithmetic conserved. 'S' and 's', for 'super' and 'sub' suggest alteration by 64/63, and for seconds, thirds, sixths and sevenths, sub and super intervals remain the same as they have been since Helmholtz/Ellis. Similarly, 'C' and 'c' suggest alteration by 81/80. For perfect intervals 'C' and 'c' are short for 'comma-wide' and 'comma-narrow' respectively, derivative of part of Smith's interval naming scheme, and for all other intervals they are short for 'classic', after Keenan's use of the word. In this way the 5/4 major third in non-meantone system receives a label that is still suggestive of it being a familiar major third.