SHEFKHED interval names: Difference between revisions

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|}Each interval name has two sizes that differ by the comma 242~~:243~~. 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.

|}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.

===Common interval names today===

These interval names are used by theorists and microtonal musicians today, though 7:5 and 10:7 are given many different names, today also considered to be an augmented fourth and diminished fifth, lesser septimal tritone and greater septimal tritone, or simply as tritones. The fourth and fifth are today called perfect fourth and perfect fifth, and Smith's major Fourth and minor Fifth referred to as augmented fourth and diminished fifth respectively. As can be seen in Tchaikovsky's ''A Guide to the Practical Study of Harmony,'' by the beginning of end of the 19th century the familiar conventions for the naming of intervals were set, wherein

These interval names are used by theorists and microtonal musicians today, though 7/5 and 10/7 are given many different names, today also considered to be an augmented fourth and diminished fifth, lesser septimal tritone and greater septimal tritone, or simply as tritones. The fourth and fifth are today called perfect fourth and perfect fifth, and Smith's major Fourth and minor Fifth referred to as augmented fourth and diminished fifth respectively. As can be seen in Tchaikovsky's ''A Guide to the Practical Study of Harmony,'' by the beginning of end of the 19th century the familiar conventions for the naming of intervals were set, wherein

*Seconds, thirds, sixths and sevenths appear in the diatonic in two sizes, the larger labelled 'major' and the smaller, 'minor'.

*Major, when raised by a semitone, becomes 'augmented', and minor, lowered by a semitone, 'diminished'.

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31-tET: P1 tai-1/vai-1 tao-m2 m2 vai-m2/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===

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 and Miracle interval naming when applied to any of the ETs they were able to cover.

[[File:Dave Keenan edo interval names prefix diagram.png|thumb|580.99x580.99px|Prefix diagram from ''One way to name the interval of any EDO from 5 to 72'', Keenan, 2016, pg. 4.]]

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 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.

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.

@@ Line 151: / Line 151: @@
 |44:81
 |1057
-|}Each interval name has two sizes that differ by the comma 242:243. 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.
+|}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.
 ===Common interval names today===
-These interval names are used by theorists and microtonal musicians today, though 7:5 and 10:7 are given many different names, today also considered to be an augmented fourth and diminished fifth, lesser septimal tritone and greater septimal tritone, or simply as tritones. The fourth and fifth are today called perfect fourth and perfect fifth, and Smith's major Fourth and minor Fifth referred to as augmented fourth and diminished fifth respectively. As can be seen in Tchaikovsky's ''A Guide to the Practical Study of Harmony,'' by the beginning of end of the 19th century the familiar conventions for the naming of intervals were set, wherein
+These interval names are used by theorists and microtonal musicians today, though 7/5 and 10/7 are given many different names, today also considered to be an augmented fourth and diminished fifth, lesser septimal tritone and greater septimal tritone, or simply as tritones. The fourth and fifth are today called perfect fourth and perfect fifth, and Smith's major Fourth and minor Fifth referred to as augmented fourth and diminished fifth respectively. As can be seen in Tchaikovsky's ''A Guide to the Practical Study of Harmony,'' by the beginning of end of the 19th century the familiar conventions for the naming of intervals were set, wherein
 *Seconds, thirds, sixths and sevenths appear in the diatonic in two sizes, the larger labelled 'major' and the smaller, 'minor'.
 *Major, when raised by a semitone, becomes 'augmented', and minor, lowered by a semitone, 'diminished'.
@@ Line 417: / Line 417: @@
 -tET: P1 tai-1/vai-1 tao-m2 m2 vai-m2/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===
-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 and Miracle interval naming when applied to any of the ETs they were able to cover.
+[[File:Dave Keenan edo interval names prefix diagram.png|thumb|580.99x580.99px|Prefix diagram from ''One way to name the interval of any EDO from 5 to 72'', Keenan, 2016, pg. 4.]]
+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 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.
 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.