User:Lhearne/Extra-Diatonic Intervals: Difference between revisions

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Today a small under of competing interval naming schemes exist for the description of microtonal music. More common than any particular defined standard are certain tendencies for microtonal interval naming, or names for specific intervals. While risking the creation of simply another competing standard, an effort is made to develop a scheme that is able to take the best aspects of the existing standards and apply them in a formal interval naming system built on common undefined practice. Such a system is developed, where in addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', only the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees. Using this system all intervals in each equal division of the octave between up to 29, and several larger commonly used equal temperaments can be named such that 'S' and 's' correspond to a displacement of an interval up or down a single interval of the edo, respectively. Many commonly used MOS scales may also be described using this scheme such that these scales' interval names are consistent expression in any tuning that supports them. The resultant scheme can also be easily mapped to any of the current naming standards, and may even facilitate translation between. The resulting scheme should improve pedagogy and communication in microtonal music.

Today a small under of competing interval naming schemes exist for the description of microtonal music. More common than any particular defined standard are certain tendencies for microtonal interval naming, or names for specific intervals. While risking the creation of simply another competing standard, an effort is made to develop a scheme that is able to take the best aspects of the existing standards and apply them in a formal interval naming system built on common undefined practice. Such a system is developed, where in addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', only the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees. Using this system all intervals in each equal division of the octave between up to 29, and several larger commonly used equal temperaments can be named such that 'S' and 's' correspond to a displacement of an interval up or down a single interval of the edo, respectively. Many commonly used [[MOS scale|MOS scales]] may also be described using this scheme such that these scales' interval names are consistent expression in any tuning that supports them. The resultant scheme can also be easily mapped to any of the current naming standards, and may even facilitate translation between. The resulting scheme should improve pedagogy and communication in microtonal music.

== Background ==

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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 edos, 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.

=== ~~Measurement~~ based systems ===

=== 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 and edos based on 41edo. Compared to Keenan's 72 interval names, Aaron's system includes 41. His system is based directly on 41edo, and unlike Keenan's system, interval are given the name of the closest step of 41edo, and no account is taken of the size of the edos fifth. In 41edo, Major, minor, augmented and diminished intervals are those obtained through the approximately Pythagorean cycle of fifths. Intervals one step of 41edo 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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In Hunt's system when used in 41edo or JI diatonic interval arithmetic is conserved, but in other tunings it may not be, and Margo's system may not conserve diatonic interval arithmetic either. Both systems may be applied to arbitrary tunings, but the same intervals (defined, perhaps by a MOS scale) may not be given the same interval names across different tunings.

=== 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. 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' (or supra) 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 Saggital, 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: ==

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Tetracot[13] 6|6: P1 N2 sM2 Sm3 N3 P4 S4 s5 P5 N6 sM6 Sm7 N7 P8

== Conclusion ==

Using only 'S' and 's' as qualifiers to M, m, P, A and d, along with N and intermediate degrees, a system is developed wherein for most edos below 50 intervals can be systematically named such that for primary interval names, 'S' and 's' raise and lower, respectively, by 1 step of an edo , whilst the intervals of many MOS scales may be consistently named across different tunings, taking the best of both Igliashon Jones' extra diatonic interval names and Sagispeak. While the intervals of some MOS scales may hold consistent names in an ups and downs based scheme, there are many common scales that cannot be in such a system that do in this one, such as scales of Diminished and Augmented temperament. What's more, the use of neutrals and intermediates leads to quicker recognition of MOS scales that may be supported in edos.

== Further divergent scheme ==

@@ Line 1: / Line 1: @@
-Today a small under of competing interval naming schemes exist for the description of microtonal music. More common than any particular defined standard are certain tendencies for microtonal interval naming, or names for specific intervals. While risking the creation of simply another competing standard, an effort is made to develop a scheme that is able to take the best aspects of the existing standards and apply them in a formal interval naming system built on common undefined practice. Such a system is developed, where in addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', only the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees. Using this system all intervals in each equal division of the octave between up to 29, and several larger commonly used equal temperaments can be named such that 'S' and 's' correspond to a displacement of an interval up or down a single interval of the edo, respectively. Many commonly used MOS scales may also be described using this scheme such that these scales' interval names are consistent expression in any tuning that supports them. The resultant scheme can also be easily mapped to any of the current naming standards, and may even facilitate translation between. The resulting scheme should improve pedagogy and communication in microtonal music.
+Today a small under of competing interval naming schemes exist for the description of microtonal music. More common than any particular defined standard are certain tendencies for microtonal interval naming, or names for specific intervals. While risking the creation of simply another competing standard, an effort is made to develop a scheme that is able to take the best aspects of the existing standards and apply them in a formal interval naming system built on common undefined practice. Such a system is developed, where in addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', only the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees. Using this system all intervals in each equal division of the octave between up to 29, and several larger commonly used equal temperaments can be named such that 'S' and 's' correspond to a displacement of an interval up or down a single interval of the edo, respectively. Many commonly used [[MOS scale|MOS scales]] may also be described using this scheme such that these scales' interval names are consistent expression in any tuning that supports them. The resultant scheme can also be easily mapped to any of the current naming standards, and may even facilitate translation between. The resulting scheme should improve pedagogy and communication in microtonal music.
 == Background ==
@@ Line 50: / Line 50: @@
 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 edos, 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.
-=== Measurement based systems ===
+=== 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 and edos based on 41edo. Compared to Keenan's 72 interval names, Aaron's system includes 41. His system is based directly on 41edo, and unlike Keenan's system, interval are given the name of the closest step of 41edo, and no account is taken of the size of the edos fifth. In 41edo, Major, minor, augmented and diminished intervals are those obtained through the approximately Pythagorean cycle of fifths. Intervals one step of 41edo 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).
@@ Line 56: / Line 56: @@
 In Hunt's system when used in 41edo or JI diatonic interval arithmetic is conserved, but in other tunings it may not be, and Margo's system may not conserve diatonic interval arithmetic either. Both systems may be applied to arbitrary tunings, but the same intervals (defined, perhaps by a MOS scale) may not be given the same interval names across different tunings.
+=== 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. 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' (or supra) 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 Saggital, 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: ==
@@ Line 601: / Line 606: @@
 Tetracot[13] 6|6: P1 N2 sM2 Sm3 N3 P4 S4 s5 P5 N6 sM6 Sm7 N7 P8
+== Conclusion ==
+Using only 'S' and 's' as qualifiers to M, m, P, A and d, along with N and intermediate degrees, a system is developed wherein for most edos below 50 intervals can be systematically named such that for primary interval names, 'S' and 's' raise and lower, respectively, by 1 step of an edo , whilst the intervals of many MOS scales may be consistently named across different tunings, taking the best of both Igliashon Jones' extra diatonic interval names and Sagispeak. While the intervals of some MOS scales may hold consistent names in an ups and downs based scheme, there are many common scales that cannot be in such a system that do in this one, such as scales of Diminished and Augmented temperament. What's more, the use of neutrals and intermediates leads to quicker recognition of MOS scales that may be supported in edos.
 == Further divergent scheme ==