Extended-diatonic interval names: Difference between revisions

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. None the less, 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.

The Ancient greek term ''diatonon'', meaning 'through tones', refers to the genus with two whole tones and a semitone, or any genus in which no interval is greater than one half of the fourth (Chalmers, 1993). The Pythaogrean diatonic scale is the scale that may be built from one two Pythagorean tetrachords, and the left over interval of 9/8.

[[File:Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1.png|thumb|566x573px|''Le institutioni harmoniche,'' Zarlino, 1558, Parte Prima, Cap. 15: Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1, pg. 25.|link=https://en.xen.wiki/w/File:Della_propriet%C3%A0_del_numero_Senario_&_della_sue_parti;_&_come_in_esse_si_ritroua_ogni_consonanze_musicale,_figura_1.png]]Intervals were referred to by the Ancient Greek names through the the 18th century, as Latin names. By the Renaissance it had been discovered that a Pythagorean diminished fourth sounded sweet, and approximated the string length ratio [[5/4]]. This just tuning for the major third was sought after, along with the complementary [[6/5]] tuning for the minor third, and octave complements to both - [[8/5]] for the minor sixth and [[5/3]] for the major sixth. Influential Italian music theorist and composer Gioseffo Zarlino put forth that choirs tuned the diatonic scale to the tuning built from this tetrachord, the ''[[Zarlino|intense diatonic scale]]'', also known as the ''syntonic or syntonus diatonic scale'' or the ''Ptolemaic sequence'':

[[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.''

[[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 Bernouilli (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.

|}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 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.

After releasing his system Keenan was informed that is was identical to the extended-diatonic interval-naming scheme of Adriaan Fokker but for the acknowledgment of more 11-limit ratios.This system depends on the tempering out of 81/80, where the diatonic major third, from four stacked fifths, approximates the just major third, 5/4. It also depends on the existence of neutral intervals, i.e., that the perfect fifth or equivalently, the chromatic semitone, subtends an even number of degrees of the ET. To simply to our familiar naming scheme for 12-tET, we observe that it applies to [[24edo|24-tET]] equally as directly as in 31-tET, where the prefixes correspond to degrees of the ET. Exactly the same is also true for [[38edo|38-tET]], twice [[19edo|19-tET]], a meantone which very closely approximates 1/3-comma Meantone. Meantone temperament wherein the fifth is divided into two equally sized neutral thirds is referred to as neutral temperament. Whereas meantone temperament is generated by the fifth, in neutral temperament the generator is half this interval, the neutral third. Where it was seen above that there are two neutral thirds, 9:11 and 22:27 that differ by 243/242, neutral temperament is at its most simple the temperament defined by this equivalence: the tempering out of 243/242, as meantone is defined by the tempering out of 81/80. The temperament that tempers out both 81/80 and 243/242 is called [[Mohajira|''Mohajira'']], upon which Keenan's scheme can be said to be based. As well as 24-tET, 31-tET and 38-tET, Mohajira is supported by [[7edo|7-tET]] and [[17edo|17-tET]].

The modern interval names were built on the assumption of meantone tempering, where the major third built from four fifths is the approximation of 5/4. In non-Meantone tunings, these two definitions of major third, just (or classic) and Pythagorean major thirds no longer correspond. If intervals are to receive unique names then to one or both of these major thirds must be added a prefix. A prefix to a major third might suggest it is not considered the 'true' major third. Keenan has been involved with the development of both types of systems. Only when the major is defined by it's mapping as fourth fifths, i.e. 81/64, can conserve interval arithmetic, but that may lead to a scheme that goes against what people believe the intervals to sound like.

Sagispeak, one system which in it's 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 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 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.

Because it is built off of the diatonic scale, Sagispeak conserves diatonic interval arithmetic, i.e. familiar relations in the diatonic scale, i.e. M2 + m3 = P4. As in Fokker/Keenan Extended-diatonic Interval-names, diatonic interval arithmetic is also extended, where, for example, tai-major 2 + tao-minor 3 = P4 (8/7 + 7/6 = 4/3), where opposite alterations cancel each other out, and diatonic interval arithmetic is conserved, a very useful property for a microtonal interval naming system to possess. Another helpful property of Sagispeak is its generalised applicability to edos, just intonation and other tunings, where the same intervals maintain their spelling across different tunings. Despite these benefits however, many see Sagittal and Sagispeak as overly complex (even though the entire extended system need hardly ever be applied), and requiring too many new terms to be learnt. It is also worth noting that since 5/4 is in his system referred to as a pao-M3, and the major third, in systems with sharper fifth particularly, may be fairly sharp of this familiar tuning for a major third, intervals names may no longer correspond to what they 'sound like'. In superpythagorean systems, for example, the major third approximates 9/7, which is familiar from meantone-based naming as a super major third. This is true of any scheme in which the major third is defined by it's generation from fifths. On the other hand, any scheme in which the major third is defined instead as an approximation to 5/4 does not preserve interval arithmetic in non-meantone systems, but may conserve existing associations between interval names and sound / size.

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.