User:Lhearne/Extra-Diatonic Intervals: Difference between revisions
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Done the extension of the lit review. This page will be 'frozen' here for now, and probably deleted later, while the lit review will be put into a new page 'On the naming of musical interval', after which my new system will be introduced. |
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Ellis, in a footnote to his translation of Helmholtz,'s treatise also provides names for a single 11-limit interval. The interval 22:27, of 355c, introduced by Zalzal, says Ellis was termed a ''neutral Third'' by Herr J. P. N. Land originally in ''Over de Toonladders der Arabische Musiek'' (On the Scales of Arabic Music) in 1880. An interval a fourth higher than this is mentioned, but a ratio is not given, and it is not named. We can ourselves however find it's ratio as 11:18, and guess it's name to be a ''neutral Sixth'', given that it lies a perfect Fourth above the neutral Third. Following a similar process as in our completion of Helmholtz table above, and assuming that the octave inverse of a neutral Third should be a neutral Sixth we may introduce the following 11-limit intervals that see common use among music theorists and microtonal musicians through to today: | Ellis, in a footnote to his translation of Helmholtz,'s treatise also provides names for a single 11-limit interval. The interval 22:27, of 355c, introduced by Zalzal, says Ellis was termed a ''neutral Third'' by Herr J. P. N. Land originally in ''Over de Toonladders der Arabische Musiek'' (On the Scales of Arabic Music) in 1880. An interval a fourth higher than this is mentioned, but a ratio is not given, and it is not named. We can ourselves however find it's ratio as 11:18, and guess it's name to be a ''neutral Sixth'', given that it lies a perfect Fourth above the neutral Third. Following a similar process as in our completion of Helmholtz table above, and assuming that the octave inverse of a neutral Third should be a neutral Sixth we may introduce the following 11-limit intervals that see common use among music theorists and microtonal musicians through to today: | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Table | |+Table 2. 11-limit intervals | ||
!Intervals | !Intervals | ||
!Notation | !Notation | ||
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Considering the 11-limit otonal chord 4:5:6:7:9:11 a chain of thirds, in addition to the familiar major, minor, subminor, supermajor and neutral thirds, Dave Kennan labelled 5:7 a sub-diminished fifth and 7:11 an augmented fifth. 7:10, the inversion of 5:7, is labelled a diminished. 5:7, therefore, is also an augmented fourth. In terms of sevenths, 4:7 is subminor, 5:9 is minor and 11:6 is neutral. | Considering the 11-limit otonal chord 4:5:6:7:9:11 a chain of thirds, in addition to the familiar major, minor, subminor, supermajor and neutral thirds, Dave Kennan labelled 5:7 a sub-diminished fifth and 7:11 an augmented fifth. 7:10, the inversion of 5:7, is labelled a diminished. 5:7, therefore, is also an augmented fourth. In terms of sevenths, 4:7 is subminor, 5:9 is minor and 11:6 is neutral. | ||
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. | 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. | ||
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{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
Table 3. Fokker/Keenan Extended-diatonic interval-names indexes | |||
!Index | !Index | ||
!Prefix for unisons, fourths, fifths, octaves | !Prefix for unisons, fourths, fifths, octaves | ||
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|superaugmented | |superaugmented | ||
|} | |} | ||
The index values correspond most directly to degrees of | The index values correspond most directly to degrees of 31-tET, whose interval names by this method are given in the following table: | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Extended-diatonic interval names in 31-tET | |+Table 4. Fokker/Keenan Extended-diatonic interval names in 31-tET | ||
!31-tET degree | !31-tET degree | ||
!Ratios | !Ratios | ||
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The interval names shown in brackets could be said to be 'secondary', the others, 'primary'. | The interval names shown in brackets could be said to be 'secondary', the others, 'primary'. | ||
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 | 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 24-tET equally as directly as in 31-tET, where the prefixes correspond to degrees of the edo. Exactly the same is also true for 38-tET, twice 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, iin 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, 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 7-tET and 17-tET. | ||
The primary interval names resulting in this system's application to these ETs is now show for easy comparison, where 'M', 'm', 'P', 'N', 'A', 'd', 'S' and 's' are shorthand for major, minor, perfect, neutral, augmented, diminished, super and sub, respectively: | The primary interval names resulting in this system's application to these ETs is now show for easy comparison, where 'M', 'm', 'P', 'N', 'A', 'd', 'S' and 's' are shorthand for major, minor, perfect, neutral, augmented, diminished, super and sub, respectively: | ||
7-tET: P1 N2 N3 P4 P5 N6 N7 P8 | |||
17-tET: P1 m2 N2 M2 m3 N3 M3 P4 S4 s5 P5 m6 N6 M6 m7 N7 M7 P8 | |||
24-tET: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 S4 A4/d5 s5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 SM7/s8 P8 | |||
31-tET: P1 S1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 S4 A4 d5 s5 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 s8 P8 | |||
38-tET: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 S4 A4 SA4/sd5 d5 s5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 P8 | |||
to Meantone tunings that are not Mohajira tunings, the regular diatonic interval names can be applied, but with the addition of double augmented and double diminished from Fokker/Keenan's system. | to Meantone tunings that are not Mohajira tunings, the regular diatonic interval names can be applied, but with the addition of double augmented and double diminished from Fokker/Keenan's system. | ||
19-tET: P1 A1 m2 M2 A2/d3 m3 M3 d4 P4 A4 d5 P5 A5 m6 M6 A6/d7 m7 d8 P8 (every second step of 38edo) | |||
26-tET: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 AA4/AA5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8 | |||
Keenan adds further that if it is desired to distinguish between ratios that are in 31-tET approximated by the same number of steps, an addition prefix be added to describe the prime limit of the approximated interval. For 3-limit intervals, the obvious choice is 'Pythagorean', for 5-limit Keenan chooses 'classic', for 7, 'septimal, 11, 'undecimal' and 13, 'tridecimal'. When the highest prime is the same, Keenan suggests adding 'small' and 'large' as final prefixes for this purpose. | Keenan adds further that if it is desired to distinguish between ratios that are in 31-tET approximated by the same number of steps, an addition prefix be added to describe the prime limit of the approximated interval. For 3-limit intervals, the obvious choice is 'Pythagorean', for 5-limit Keenan chooses 'classic', for 7, 'septimal, 11, 'undecimal' and 13, 'tridecimal'. When the highest prime is the same, Keenan suggests adding 'small' and 'large' as final prefixes for this purpose. | ||
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In non-Meantone tunings, the two definitions of major third - 4:5 and 64:81, the 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. 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. | In non-Meantone tunings, the two definitions of major third - 4:5 and 64:81, the 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. 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. | ||
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 41-tET and 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 | 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 41-tET and 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. | ||
41-tET: P1 S1 nsm2 sm2 m2 N2 nM2 M2 SM2 sm3 nm3 m3 N3 M3 nSM3 SM3 s4 P4 nS4 S4 A4 d5 s5 Ws5 P5 S5 sm6 Wsm6 m6 N6 M6 Wm6 SM6 sm7 m7 Wm7 N7 M7 SM7 WSM7 s8 P8 | |||
In 41-tET, fourth fifths make a wide major third, rather than a major third, and interval arithmetic is no longer conserved. The same is true for 72-tET, so we have still yet to a system able to conserve interval arithmetic in non-meantone ETs. Though many edos can be covered, many still cannot, including the Superpythagorean edos, where the fifth is sharper than just, and four fifths give an approximation to 7:9, the super major third, tempering out the septimal comma, 63:64. | |||
=== [[Sagittal notation|Sagittal]] - [http://forum.sagittal.org/viewforum.php?f=9 sagispeak] === | |||
One system which in it's naming of meantone and non-meantone edos is able to conserve interval arithmetic, sagispeak, was developed by [[Dave Keenan]] and others as an interval naming system that maps 1-1 with the Sagittal microtonal music notation system. Sagittal notation was developed as a generalised diatonic-based notation system applicable equally to [[just intonation]], [[Equal Temperaments|equal tunings]] and rank-''n'' [[temperaments]]. 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 the 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 for 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. | 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. | ||
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We have seen that there are two competing definitions of a major third, the ratio '5/4' or the interval built from fourth stacked fifths, that may or may not correspond to 5/4. In meantone systems, those we are used to, they correspond, but in most edos they do not. Interval naming systems wherein the major third is defined as an approximation to 5/4 rather than as four fifths minus two octaves may benefit from a familiar name for 5/4, but they are unable to conserve diatonic interval arithmetic. | We have seen that there are two competing definitions of a major third, the ratio '5/4' or the interval built from fourth stacked fifths, that may or may not correspond to 5/4. In meantone systems, those we are used to, they correspond, but in most edos they do not. Interval naming systems wherein the major third is defined as an approximation to 5/4 rather than as four fifths minus two octaves may benefit from a familiar name for 5/4, but they are unable to conserve diatonic interval arithmetic. | ||
For comparison, | For comparison, 31-tET, is shown below in sagispeak: | ||
31-tET: P1 tai-1/vai-1 tao-m2 m2 vai-2m/vao-M2 M2 tai-M2 tao-m3 m3 vai-m3/vao-M3 M3 tai-M3 tao-4 P4 vai-4 A4 d5 vao-5 P5 tai-5 tao-m6 m6 vai-m6/vao-M6 M6 tai-M6 tao-m7 m7 vai-m7.vao-M7 M7 tai-M7 vao-8/tao-8 P8 | |||
=== Dave Keenan's most recent system === | === Dave Keenan's most recent system === | ||
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 UPD notation]]. To name an interval in an | 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 UPD notation]]. 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 following chart details this process (can't load the chart :( ). 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 when applied to any of the ETs it was able to cover. | ||
Keenan's system is an elegant way to keep the 'major 3rd' label for 5/4, where labels depend on the size of the best fifth, however it suffers from it's applicability only to | Keenan's system is an elegant way to keep the 'major 3rd' label for 5/4, where labels depend on the size of the best fifth, however it suffers from it's applicability only to ETs, and that it does not conserve interval arithmetic. Another 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. | ||
=== Size-based systems === | === Size-based systems === | ||
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=== Ups and Downs === | === Ups and Downs === | ||
One final interval naming system, associated with the [[Ups and Downs Notation|Ups and Downs Notation]] system, belonging to microtonal theorist and musicians [[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]], [[19edo]] or [[31edo]] for example) 5/4 may be a M3, and in others a vM3 (downmajor 3rd) (e.g. [[15edo]], [[22edo]], 41edo, 72edo), or even an up-major 3rd (e.g. [[21edo]]). 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). This system benefits from it's simplicity as well as it's conservation of interval arithmetic. It can be used for some MOS scales where one of the generators is a perfect fifth or a fraction of a perfect fifth, but not all of these (e.g. Diminished[8]), and not all MOS scales (if such scales are to be described, an additional pair of accidentals/qualifiers is used. Although the scales then are described, their intervals still are not given the same names in Ups and Downs' edo names). 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. | One final interval naming system, associated with the [[Ups and Downs Notation|Ups and Downs Notation]] system, belonging to microtonal theorist and musicians [[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]], [[19edo]] or [[31edo]] for example) 5/4 may be a M3, and in others a vM3 (downmajor 3rd) (e.g. [[15edo]], [[22edo]], 41edo, 72edo), or even an up-major 3rd (e.g. [[21edo]]). 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. It can be used for some MOS scales where one of the generators is a perfect fifth or a fraction of a perfect fifth, but not all of these (e.g. Diminished[8]), and not all MOS scales (if such scales are to be described, an additional pair of accidentals/qualifiers is used. Although the scales then are described, their intervals still are not given the same names in Ups and Downs' edo names). 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. | ||
[[Igliashon Jones]] is a supporter of this system, but for the relabeling of 'down' as 'sub' and 'up' as 'super' | [[Igliashon Jones]] is a supporter of this system, but for the relabeling of 'down' as 'sub' and 'up' as 'super' and 'mid' as 'neutral', so that more common names are used, wherein 'super' infers a raise of 1 step of the edo, and 'sub' a lowering of one step. In this 'Extra-diatonic' system 'super' and 'sub' may be doubly applied, as in Ups and Downs, but they may not be applied before 'neutral' where in Ups and Downs they may be applied before 'mid'. The author's own extra-diatonic system is developed as a departure with caveat that 'S' and 's' prefixes are defined not as alterations by a single step of the edo, but by comma alterations as in Sagittal, in order that interval of MOS scales may be represented consistently across different tunings. Throughout the rest of the article the development is detailed, and the system defined. | ||
== Premise: == | == Premise: == | ||