Extended-diatonic interval names: Difference between revisions

Besides the [[octave]], which is treated as an interval of equivalence in almost all musics globally, the [[perfect fifth]] and [[perfect fourth]] are strongest consonances. Reasonable approximations to those intervals that can be found by dividing the octave equally into 5 notes, or into 7 notes, and excellent approximations can be found by dividing the octave equally into 12 notes (corresponding to the system of [[12edo|12-tone Equal Temperament]] that sees persistent global use today). Given this, along with the limited capacity of human short-term memory, ''pentatonic'' and ''heptatonic'' (5-note and 7-note) scales are extremely common, from antiquity, to, again, almost all musics globally. The heptatonic scale in which the most number of perfect fifths and fourths is called the ''diatonic scale'', and of all pentatonic or heptatonic scales has seen more application as a basis for music notation and interval names. This article details the development of Western diatonic-based interval names. Where today a small number of competing diatonic-based interval naming schemes exist for the description of microtonal music (music that is not tuned to 12-tone Equal Temperament (12-tET), a critical review of current proposed schemes is also undertaken.  

==The origin of diatonic interval names==

==Ancient Greek interval names==

1/1 9/8 5/4 4/3 3/2 5/3 [[15/8]] 2/1

==English interval names in the Baroque==

[[File:Harmonics, or The Philosophy of Musical Sounds, Section 2 figure 3.png|thumb|517x548px|''Harmonics, or The Philosophy of Musical Sounds'', Edition 2, Smith, 1759, Section 2: On the Names and Notation of consonance and their intervals, Fig. 2 & 3, pg. 10|link=https://en.xen.wiki/w/File:Harmonics,_or_The_Philosophy_of_Musical_Sounds,_Section_2_figure_3.png]]We see these names translated into English music theorist and mathematician Robert Smith's 1749 ''Harmonics, or, The Philosophy of Musical Sounds'' (referenced figures shown on the right):<blockquote>Fig. 2. If a musical string ''CO'' and it's parts ''DO'', ''EO'', ''FO'', ''GO'', ''AO'', ''BO'', ''cO'', be in proportion to one another as the numbers 1, 8/9, 4/5, 3/4, 2/3, 3/5, 8/15, 1/2, their vibrations will exhibit the system of 8 sounds which musicians donate by the letters ''C'', ''D'', ''E'', ''F'', ''G'', ''A'', ''B'', ''c''.</blockquote><blockquote>Fig. 3. And supposing those strings to be ranged like ordinates to a right line ''Cc'', and their distances ''CD'', ''DE'', ''EF'', ''FG'', ''GA'', ''AB'', ''BC'', not to be the differences of their lengths, as in fig. 2. but to be the magnitudes proportional to the intervals of their sounds, the received Names of these intervals are shewn in the following Table; and are taken from the numbers of the strings or sounds in each interval inclusively; as a Second, Third, Fourth, Fifth, &c, with the epithet of ''major'' or ''minor'', according as the name or number belongs to a greater of smaller total interval; the difference of which results chiefly from the different magnitudes of the major and minor second, called the Tone and Hemitone.</blockquote>In contrast to Zarlino and Huygens, Smith applies the 'major' and 'minor' qualifiers also to fourths and fifths. Where they, like all other intervals of the scale but octaves and unisons, come in two different sizes in the diatonic scale, we can see this is a more consistent scheme. I believe it to be unfortunate that Smith's scheme was not favoured over the names we saw first from Zarlino, which we see still today.

Smith adds that<blockquote>Any one of the ratios in the third column of the foregoing Table, except 80 to 81, or any one of them compounded once of oftener with the ratio 2 to 1 or 1 to 2, is called a Perfect ratio when reduced to it's least terms. And when the times of the single vibrations of any two sounds have a perfect ratio, the consonance and it's interval too after called Perfect; and is called Imperfect or Tempered when that perfect ratio and interval is a little increased or decreased.</blockquote><blockquote>...</blockquote><blockquote>Any small increment of decrement of a perfect interval is called respectively the Sharp or Flat Temperament of the imperfect consonance, and is measured most conveniently by the proportion it bears to the comma</blockquote>Therefore, in this system 3/2 is the ''Perfect major Fifth'' and 5/4 the ''Perfect major Third''. 81/64 might be labelled a ''comma sharp major Third'', [[32/27]] a ''comma flat minor Third'', and the 1/4-comma Meantone fifth a ''1/4-comma flat major Fifth''. The interval naming scheme Smith describes may be immediately applied to 5-limit microtonal systems. There is an inconsistency, however, where it seems that 9/8 should be called a ''Perfect major Second,'' but that, while [[9/5]] be named a ''comma sharp minor Seventh'', it's inverse, 10/9, is a ''Perfect minor Tone.''

==Helmholtz and Ellis==

[[File:Helmholtz consonances table.png|thumb|617x469px|link=https://en.xen.wiki/w/File:Helmholtz_consonances_table.png|Table detailing the influence of the different consonances on each other, from ''On the Sensations of Tone as a Psychological Basis for the Theory of Music'', Helmholtz, 1863, Translated by Ellis, 1875, Chap. X, pg. 187.]]Through the investigations of Galileo (1638), Newton, Euler (1729), and Bernoulli (1771), theorist Hermann von Helmholtz was aware that ratios governing the lengths of strings existed also for the vibrations of the tones they produced. His investigation of the harmonic series associated with these ratios of vibration led him to the consideration of ratios above the [[5-limit]]. In his seminal ''On the Sensations of Tone as a Physiological Basis for the Theory of Music'', published in German in 1863 and translated into English in 1875 by Alexander Ellis, he listed just intervals as show in the table to the right. It is interesting to note that 8:9 is labelled a 'Second' rather than a 'major Second'. The minor Seventh is shown as 5:9 rather than as [[16/9|9:16]] seemingly because of the 9 partial limit imposed on the table. It is also worth noting that [[7/5|5:7]] is labelled a subminor Fifth. 'Super', indicated in notation with a '+', raises an interval by [[36/35|35:36]], the septimal quarter tone, and 'sub', indicated with a '-' lowers by the same interval with the exception of the Supersecond [[8/7|7:8]], which lies [[64/63|63:64]], the septimal comma above the Second. The subminor fifth is not included in this as no minor Fifth is shown. If we assume that 'sub' lowers an interval 35:36, then the minor Fifth would be [[25/18|18:25]], 80:81 above Smith's [[64/45|45:64]] minor Fifth, however in table 2 below, Ellis labels 18:25 a ''superfluous Fourth'', and it's inverse, [[32/25|25:32]], an ''acute diminished Fifth'', whilst 64:45 is labelled a ''diminished Fifth'' and its inverse [[45/32|32:45]] a ''false Fourth or Tritone.'' If we label 9:10 as a 'major Second', and 7:8 as a 'supermajor Second' then they differ by 35:36, the major Second is the inverse of the minor Seventh, and the supermajor Second is the octave inverse of the subminor Seventh. We can assume that Smith's 'major Fifth' and 'minor Fourth' names for 3/2 and 4/3 were not wholly taken up. We can add to this table the remaining octave inversions as well as the super Fourth and sub Fifth.

==Fokker/[[Dave Keenan|Keenan]] [http://www.dkeenan.com/Music/IntervalNaming.htm Extended-diatonic interval-names]==

From this, Keenan defines a consistent interval naming system, meaning one which obeys diatonic interval arithmetic (In each column, the parenthesised prefix is the one that is implied when there is no prefix). When adding intervals the indexes are added together to give the index of the resulting interval. Keenan also adds corrections for each interval class to the indexes in order to account for inconsistencies that occur within diatonic interval arithmetic when concerning intervals greater than an octave, so that his system, unlike regular diatonic interval names, may be completely consistent.

|}The index values correspond most directly to degrees of 31-tET, whose interval names by this method are given in the following table:

|}The interval names shown in brackets could be said to be 'secondary', the others, 'primary'.

==[[Miracle]] [http://dkeenan.com/Music/Miracle/MiracleIntervalNaming.txt interval naming]==

Keenan later describes how the scheme can be extended to also cover 72-tET and 41-tET. In 31-tET the fifth may divided into six minor seconds. This temperament is called ''Miracle'', and is also supported by [[41edo|41-tET]] and [[72edo|72-tET]]. The first six generators of Miracle give the following intervals: P1 m2 SM2 m3 S4 P5, as can be seen in the table above. 31-tET may be covered by 15 generators downwards and 15 generators upwards from 1:1. In 72-tET, either side of the intervals that that result from these 31 notes, called Miracle[31] 15|15, lie unnamed intervals that may be found first at either 31 or 41 generators further upwards or downwards and in 41-tET, either at 10 or 31 generators. If, one degree of 41 or 72-tET above an interval or Miracle[31] 15|15 lies an unnamed interval that can be first found by an additional 31 generators upwards, it is given the same name as the interval directly below it, with the addition of the prefix 'n', for 'narrow'. Similarly, 'W' for 'wide' prefixes an unnamed interval one degree, 31 generators below.

==[[Sagittal notation|Sagittal]] - [http://forum.sagittal.org/viewforum.php?f=9 Sagispeak]==

Sagispeak, one system which in its naming of meantone and non-meantone edos is able to conserve interval arithmetic was developed initially by [[George Secor]] as a way to pronounce accidentals used in Saggital notation, a generalised diatonic-based notation system applicable equally to just intonation, equal tunings and rank-''n'' [[temperaments]]. With input from Dave Keenan and others, [[Cam Taylor]] extended it for use as an interval naming system. In Saggital, dozens of different accidentals can be used on a regular diatonic [[Staff notation|staff]] to notate up to extremely fine divisions, however in most cases only a handful are needed. In Sagispeak, each accidental is presented by a prefix, made up of a single letter, in most cases, followed by either 'ai' if the accidental raises a note, or 'ao' if it lowers a note. As in HEWM notation, Pythagorean intonation is assumed as a basis. Then prefixes depart from Pythaogrean intonation, altering by commas and introducing other primes. In place of the prefixes 'sub' and 'super', generally signifying an alteration of 36/35 from 5-limit intervals or 64/63 from 3-limit, Sagittal features an accidental of 64/63, which may be used to take a Pythagorean major interval to a supermajor, minor to subminor, or perfect to super or sub. The prefix 'tao' indicates a decrease of 64/63 and the prefix 'tai' an increase. Whereas in previous interval naming schemes 'major' and 'minor' were synonymous with the 5-limit tunings, in Sagispeak they map instead to Pythagorean. A prefix is needed then to take a Pythagorean intoned interval to a 5-limit tuning. Where 5/4 is 81/80 below the Pythagorean third, the prefixes 'pai' and 'pao' (where 'p' is for 'pental', as in, involving prime 5), which raise or lower a note by 81/80 respectively. Similarly, 'vai' and 'vao', which raise or lower a note by 33/32 respectively, leading to ratios of 11.

==Dave Keenan's most recent system==

[[File:Dave Keenan edo interval names prefix diagram.png|thumb|434x581px|Prefix diagram from ''One way to name the interval of any EDO from 5 to 72'', Keenan, 2016, pg. 4.|link=https://en.xen.wiki/w/File:Dave_Keenan_edo_interval_names_prefix_diagram.png]]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 symmetry of 3rds in a fifth. The interval arithmetic associated with these symmetries which may be summarised by the rule 'x + y = Pz and x + Pz = y where x and y are both perfect, both major/minor, or both dim/aug', is also conserved. 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)

==Size-based systems==

==Ups and Downs==

== [[SHEFKHED interval names]] ==