Extended-diatonic interval names: Difference between revisions

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[[26edo|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

In the above primary interval names for equal tunings, it should be noted that no interval of interval-class ''n-1'' is subtended by a larger number of degrees than an interval of class ''n''. I define that an interval-name set for which this is true is said to be ''well-ordered''. The possibility for ~~Well~~-ordered interval-name sets is a desirable property for interval naming schemes to possess and is possess by all proposals discussed in this paper.

In the above primary interval names for equal tunings, it should be noted that no interval of interval-class ''n-1'' is subtended by a larger number of degrees than an interval of class ''n''. I define that an interval-name set for which this is true is said to be ''well-ordered''. The possibility for well-ordered interval-name sets is a desirable property for interval naming schemes to possess and is possess by all proposals discussed in this paper.

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.

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.

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.

==[[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.

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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 [[Superpyth|''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.

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 are still yet to find a scheme able to conserve interval arithmetic in non-meantone ETs. Though many edos can be covered, many still cannot, including the [[Superpyth|''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~~]==

==[[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 largely by [[George Secor]], with input from 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 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.

One system which in it's naming of meantone and non-meantone edos is able to conserve interval arithmetic - Sagispeak - was developed largely by [[George Secor]], with input from Dave Keenan, [[Cam Taylor]] 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 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 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.

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.

~~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, 31-tET, is shown below in sagispeak:

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[[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)

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.

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, much 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.

==Size-based systems==

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).

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== [[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. As in Keenan/Fokker Extended-diatonic interval-names, N lies exactly between M and n, splitting the apotome. 'hA' and 'hd' for 'hemi-augmented' and 'hemi-diminished' are added for similar use from perfect intervals as N from major and minor, e.g. in 31-tET 11/8 is labelled hA4 rather than S4, and 33/32 hA1 rather than S1. 'Intermediates' are also included when the limma, is split, rather than the apotome. Where neutrals are associated with the tempering out of 243/243, intermediates are associated with the tempering out of 676/675, where 13/15 is equated with half of 4/3, labeled '2-3'. Finally, 'wide' and 'narrow', with short-form 'W' and 'n' fill the role of ups and downs from Ups and Downs, and a similar role to their namesake in Fokker/Keenan Extended-interval names. This scheme conserves interval arithmetic in it~~'s application~~ to all edos and many, but not all MOS and JI scales, where names for the intervals of scales are conserved across different tunings.

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.

As in Keenan/Fokker Extended-diatonic interval-names, N lies exactly between M and n, splitting the apotome. 'hA' and 'hd' for 'hemi-augmented' and 'hemi-diminished' are added for similar use from perfect intervals as N from major and minor, e.g. in 31-tET 11/8 is labelled hA4 rather than S4, and 33/32 hA1 rather than S1. 'Intermediates' are also included when the limma, is split, rather than the apotome. Where neutrals are associated with the tempering out of 243/243, intermediates are associated with the tempering out of 676/675, where 13/15 is equated with half of 4/3, labeled '2-3'. Finally, 'wide' and 'narrow', with short-form 'W' and 'n' fill the role of ups and downs from Ups and Downs, and a similar role to their namesake in Fokker/Keenan Extended-interval names. This scheme conserves interval arithmetic wherever it may be applied, which is to all edos and many, but not all MOS and JI scales, where names for the intervals of scales are conserved across different tunings.

In addition to the 'primary names' seen below, secondary names can be included which may reveal commas tempered out, and in turn what the diatonic intervals 'sound like', as well as the correspondence between chromatic and enharmonic movement and commatic movement. For example, in 22-tet, listed below, the secondary names for the diatonic intervals show 'S' and 's' subscripts, suggesting the tempering out of 64/63 and providing the information that in 22-tET the 'structural' major third is also, or 'sound like' a super major third, for example. The classic major third is secondarily named as an augmented second, so we know how to chromatically 'find it' / how it derives from the diatonic scale. These equivalences define 7-limit Superpyth temperament.

16-tET, 22-tET, 41-tET and 50-tET are given below as examples:

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 [[26edo|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
-In the above primary interval names for equal tunings, it should be noted that no interval of interval-class ''n-1'' is subtended by a larger number of degrees than an interval of class ''n''. I define that an interval-name set for which this is true is said to be ''well-ordered''. The possibility for Well-ordered interval-name sets is a desirable property for interval naming schemes to possess and is possess by all proposals discussed in this paper.
+In the above primary interval names for equal tunings, it should be noted that no interval of interval-class ''n-1'' is subtended by a larger number of degrees than an interval of class ''n''. I define that an interval-name set for which this is true is said to be ''well-ordered''. The possibility for well-ordered interval-name sets is a desirable property for interval naming schemes to possess and is possess by all proposals discussed in this paper.
 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.
-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.
+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.
 ==[[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.
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 -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 [[Superpyth|''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.
+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 are still yet to find a scheme able to conserve interval arithmetic in non-meantone ETs. Though many edos can be covered, many still cannot, including the [[Superpyth|''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]==
+==[[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 largely by [[George Secor]], with input from 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 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.
+One system which in it's naming of meantone and non-meantone edos is able to conserve interval arithmetic - Sagispeak - was developed largely by [[George Secor]], with input from Dave Keenan, [[Cam Taylor]] 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 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 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.
+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.
-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, 31-tET, is shown below in sagispeak:
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 [[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)
-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.
+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, much 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.
 ==Size-based systems==
 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).
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 == [[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. As in Keenan/Fokker Extended-diatonic interval-names, N lies exactly between M and n, splitting the apotome. 'hA' and 'hd' for 'hemi-augmented' and 'hemi-diminished' are added for similar use from perfect intervals as N from major and minor, e.g. in 31-tET 11/8 is labelled hA4 rather than S4, and 33/32 hA1 rather than S1. 'Intermediates' are also included when the limma, is split, rather than the apotome. Where neutrals are associated with the tempering out of 243/243, intermediates are associated with the tempering out of 676/675, where 13/15 is equated with half of 4/3, labeled '2-3'. Finally, 'wide' and 'narrow', with short-form 'W' and 'n' fill the role of ups and downs from Ups and Downs, and a similar role to their namesake in Fokker/Keenan Extended-interval names. This scheme conserves interval arithmetic in it's application to all edos and many, but not all MOS and JI scales, where names for the intervals of scales are conserved across different tunings.
+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.
+As in Keenan/Fokker Extended-diatonic interval-names, N lies exactly between M and n, splitting the apotome. 'hA' and 'hd' for 'hemi-augmented' and 'hemi-diminished' are added for similar use from perfect intervals as N from major and minor, e.g. in 31-tET 11/8 is labelled hA4 rather than S4, and 33/32 hA1 rather than S1. 'Intermediates' are also included when the limma, is split, rather than the apotome. Where neutrals are associated with the tempering out of 243/243, intermediates are associated with the tempering out of 676/675, where 13/15 is equated with half of 4/3, labeled '2-3'. Finally, 'wide' and 'narrow', with short-form 'W' and 'n' fill the role of ups and downs from Ups and Downs, and a similar role to their namesake in Fokker/Keenan Extended-interval names. This scheme conserves interval arithmetic wherever it may be applied, which is to all edos and many, but not all MOS and JI scales, where names for the intervals of scales are conserved across different tunings.
+In addition to the 'primary names' seen below, secondary names can be included which may reveal commas tempered out, and in turn what the diatonic intervals 'sound like', as well as the correspondence between chromatic and enharmonic movement and commatic movement. For example, in 22-tet, listed below, the secondary names for the diatonic intervals show 'S' and 's' subscripts, suggesting the tempering out of 64/63 and providing the information that in 22-tET the 'structural' major third is also, or 'sound like' a super major third, for example. The classic major third is secondarily named as an augmented second, so we know how to chromatically 'find it' / how it derives from the diatonic scale. These equivalences define 7-limit Superpyth temperament.
 -tET, 22-tET, 41-tET and 50-tET are given below as examples: