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

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=== 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.

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.

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 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.

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We may further add that ‘S’ (supra) and ‘s’ (small) may raise diminished, and lower augmented intervals by 81/80 as they do to minor and major respectively and that when ‘S’ (super) raised an augmented interval, or ‘s’ (sub) lowers it, the change is by 64/63.

~~Although 'S' and 's', if applied~~, ~~may not always raise~~ or ~~lower an interval~~ by a single step ~~of the edo~~, ~~in every case considered so far~~, ~~when~~ 'S' and 's' ~~appear~~ in the ~~primary~~ interval ~~names for~~ edos, they ~~do exactly this~~. This is true ~~also~~ for ~~the following edos:~~

In edos where 81/80 or 64/63 are represented by a single step, or when the apotome is represented by a single step, Ups and Downs with 'S', 's', and 'N' can name the intervals equivalently to this system. This can be seen in add edos considered thus far, as well as in those listed directly below. What this system adds is that it may also describe the intervals of MOS scales (as well as JI scales), such that these interval can be identically named in edos that approximate the scales they belong to. This is true for all MOS scales mentioned so far, as well as those listed below.

=== Further application in edos ===

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46edo: P1 S1 sm2 m2 Sm2 A1 d3 sM2 M2 SM2 sm3 m3 Sm3 A2 d4 sM3 M3 SM3 s4 P4 S4 sd5 d5 4-5 A4 SA4 s5 P5 S5 sm6 m6 Sm6 A4 d7 sM6 M6 SM6 sm7 m7 Sm7 A6 d8 sM7 M7 SM7 s8 P8

In 43edo we encounter the first time we have to use double Augmented and diminished intervals. 43edo marks the first instance in which Jones' alternative ups and downs interval names do not match those from this system. In his system, for example, AA2 would simply be sM3, but in this system since sM3 implies an approximation to 5/4 and the M3 already represents 5/4, and therefore is equivalent to sM3, we cannot do this. This tells us however that no simple ratio is approximated by the interval, and perhaps it is better understood as an AA2. Larger edos contain unlabeled intervals (without resorting to extended diatonic interval names). The association of 'super' and 'sub' with 64/63 and with 'supra' and 'small' with 81/80 may effect the assignment of primary interval names, but for all of these edos, as well as all those mentioned before, when 'S' and 's' are used, they still signify a raising or lowering by a single step of the edo, and thus appear equivalent to the ~~ups~~ and ~~downs~~ version. The comma associations add that, though use of enharmonic equivalences and secondary interval names may be necessary, intervals from MOS scales may be spelled in a consistent way across tuning to different edos.

In 43edo we encounter the first time we have to use double Augmented and diminished intervals. 43edo marks the first instance in which Jones' alternative ups and downs interval names do not match those from this system. In his system, for example, AA2 would simply be sM3, but in this system since sM3 implies an approximation to 5/4 and the M3 already represents 5/4, and therefore is equivalent to sM3, we cannot do this. This tells us however that no simple ratio is approximated by the interval, and perhaps it is better understood as an AA2. Larger edos contain unlabeled intervals (without resorting to extended diatonic interval names). The association of 'super' and 'sub' with 64/63 and with 'supra' and 'small' with 81/80 may effect the assignment of primary interval names, but for all of these edos, as well as all those mentioned before, when 'S' and 's' are used, they still signify a raising or lowering by a single step of the edo, and thus appear equivalent to the Ups and Downs version. The comma associations add that, though use of enharmonic equivalences and secondary interval names may be necessary, intervals from MOS scales may be spelled in a consistent way across tuning to different edos.

=== Other rank-2 temperaments' MOS scales ===

On top of those discussed thus far, other temperaments generated by the P5 or a fraction of it are also supported to some extent, where their MOS scales may be represented, including Augmented, Porcupine, Diminished, Negri, Tetracot and Slendric.

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'''Definition 7.''' If alterations of 81/80 and of 64/63 need to be distinguished from one another in short-form, alterations of 81/80 can be written 'SR' and 'sl'.

== edos with extreme fifths ==

Thus far the best fifth of all edos described lies in the 'diatonic range', between 4 steps of 7edo and 3 steps of 5edo. The best fifths of some edos lies outside this range, in either directly. Whereas in edos where the best fifth lies between 7 steps of 12edo and 4 steps of 7edo, 81/80, the meantone comma is tempered out, wherein four fifths approximate the fifth harmonic (a sM3 raised two octave), in edos with fifths flatter than this, 135/128, a meantone chromatic semitone is instead tempered out, resulting in the four fifths instead approximate the Sm3, 6/5 (raised two ocaves). This system is called Mavila temperament. In the other direction, whereas the best fifth lies between 7 steps of 12edo and 3 steps of 5edo, 64/63, the septimal or Archytas comma is tempered out, wherein four fifths approximate a SM3, 9/7, raised two octaves, in edos who's best fifth is sharper than this, 9/7, as well as 5/4 are approximated by the perfect fourth, tempering out 16/15 and 28/27. This system is called Father temperament. The application of this system to edos of both of these fifth sizes is addressed below.

=== Mavila ===

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in 28edo 81/80 is represented by -1 steps. We en-devour to maintain as best we can in our primary interval names the original premise behind the notation - that the prefix 's' takes an interval down a single step of the edo and 'S' a single step up. In our primary interval names for 28edo we have S4 below P4 and d5 above P5. We can avoid this confusion however by using the secondary interval names for P4 and P5 in 28edo - N4 and N5. In our list of edos below this change is made.

== How to ==

Though we have derived our interval names through many examples thus far, the process has not been explained as yet such that the reader may immediately apply them. This will be addressed here, with a step-by-step derivation guide:.

For ''n-''edo:

# Find the best approximations of 3/2, 5/4 and 7/4, which are to be labelled P5, sM3 and sm7 (This is equivalent to finding the [[7-limit]] [[patent val]]).

#* P5 = round(ln(3/2)/ln(''n'')) steps

#* sM3 = round(ln(5/4)/ln(''n'')) steps

#* sm7 = round(ln(7/4)/ln(''n'')) steps

# Using the best fifth, label the diatonic intervals. i.e.

#* M2 = (2*P5) mod ''n''

#* M6 = (3*P5) mod ''n''

#* M3 = (4*P5) mod ''n''

#* M7 = (5*P5) mod ''n''

#* (P4, m7, m3, m6, m2) = ''n'' - (P4, M2, M6, M3, M7)

#

== Lists of edos and MOS scales ==

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== 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. One does not need to understand the comma associations to make use of the interval names. 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.

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. One does not need to understand the comma associations to make use of the interval names. While the intervals of some MOS scales may hold consistent names in edos in an Ups and Downs based scheme, there are many common scales that do not 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.

@@ Line 60: / Line 60: @@
 === 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.
+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.
 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 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.
@@ Line 198: / Line 198: @@
 We may further add that ‘S’ (supra) and ‘s’ (small) may raise diminished, and lower augmented intervals by 81/80 as they do to minor and major respectively and that when ‘S’ (super) raised an augmented interval, or ‘s’ (sub) lowers it, the change is by 64/63.
-Although 'S' and 's', if applied, may not always raise or lower an interval by a single step of the edo, in every case considered so far, when 'S' and 's' appear in the primary interval names for edos, they do exactly this. This is true also for the following edos:
+In edos where 81/80 or 64/63 are represented by a single step, or when the apotome is represented by a single step, Ups and Downs with 'S', 's', and 'N' can name the intervals equivalently to this system. This can be seen in add edos considered thus far, as well as in those listed directly below. What this system adds is that it may also describe the intervals of MOS scales (as well as JI scales), such that these interval can be identically named in edos that approximate the scales they belong to. This is true for all MOS scales mentioned so far, as well as those listed below.
 === Further application in edos ===
@@ Line 225: / Line 225: @@
 edo: P1 S1 sm2 m2 Sm2 A1 d3 sM2 M2 SM2 sm3 m3 Sm3 A2 d4 sM3 M3 SM3 s4 P4 S4 sd5 d5 4-5 A4 SA4 s5 P5 S5 sm6 m6 Sm6 A4 d7 sM6 M6 SM6 sm7 m7 Sm7 A6 d8 sM7 M7 SM7 s8 P8
-In 43edo we encounter the first time we have to use double Augmented and diminished intervals. 43edo marks the first instance in which Jones' alternative ups and downs interval names do not match those from this system. In his system, for example, AA2 would simply be sM3, but in this system since sM3 implies an approximation to 5/4 and the M3 already represents 5/4, and therefore is equivalent to sM3, we cannot do this. This tells us however that no simple ratio is approximated by the interval, and perhaps it is better understood as an AA2. Larger edos contain unlabeled intervals (without resorting to extended diatonic interval names). The association of 'super' and 'sub' with 64/63 and with 'supra' and 'small' with 81/80 may effect the assignment of primary interval names, but for all of these edos, as well as all those mentioned before, when 'S' and 's' are used, they still signify a raising or lowering by a single step of the edo, and thus appear equivalent to the ups and downs version. The comma associations add that, though use of enharmonic equivalences and secondary interval names may be necessary, intervals from MOS scales may be spelled in a consistent way across tuning to different edos.
+In 43edo we encounter the first time we have to use double Augmented and diminished intervals. 43edo marks the first instance in which Jones' alternative ups and downs interval names do not match those from this system. In his system, for example, AA2 would simply be sM3, but in this system since sM3 implies an approximation to 5/4 and the M3 already represents 5/4, and therefore is equivalent to sM3, we cannot do this. This tells us however that no simple ratio is approximated by the interval, and perhaps it is better understood as an AA2. Larger edos contain unlabeled intervals (without resorting to extended diatonic interval names). The association of 'super' and 'sub' with 64/63 and with 'supra' and 'small' with 81/80 may effect the assignment of primary interval names, but for all of these edos, as well as all those mentioned before, when 'S' and 's' are used, they still signify a raising or lowering by a single step of the edo, and thus appear equivalent to the Ups and Downs version. The comma associations add that, though use of enharmonic equivalences and secondary interval names may be necessary, intervals from MOS scales may be spelled in a consistent way across tuning to different edos.
 === Other rank-2 temperaments' MOS scales ===
 On top of those discussed thus far, other temperaments generated by the P5 or a fraction of it are also supported to some extent, where their MOS scales may be represented, including Augmented, Porcupine, Diminished, Negri, Tetracot and Slendric.
@@ Line 308: / Line 308: @@
 '''Definition 7.''' If alterations of 81/80 and of 64/63 need to be distinguished from one another in short-form, alterations of 81/80 can be written 'SR' and 'sl'.
+== edos with extreme fifths ==
+Thus far the best fifth of all edos described lies in the 'diatonic range', between 4 steps of 7edo and 3 steps of 5edo. The best fifths of some edos lies outside this range, in either directly. Whereas in edos where the best fifth lies between 7 steps of 12edo and 4 steps of 7edo, 81/80, the meantone comma is tempered out, wherein four fifths approximate the fifth harmonic (a sM3 raised two octave), in edos with fifths flatter than this, 135/128, a meantone chromatic semitone is instead tempered out, resulting in the four fifths instead approximate the Sm3, 6/5 (raised two ocaves). This system is called Mavila temperament. In the other direction, whereas the best fifth lies between 7 steps of 12edo and 3 steps of 5edo, 64/63, the septimal or Archytas comma is tempered out, wherein four fifths approximate a SM3, 9/7, raised two octaves, in edos who's best fifth is sharper than this, 9/7, as well as 5/4 are approximated by the perfect fourth, tempering out 16/15 and 28/27. This system is called Father temperament. The application of this system to edos of both of these fifth sizes is addressed below.
 === Mavila ===
@@ Line 463: / Line 466: @@
 in 28edo 81/80 is represented by -1 steps. We en-devour to maintain as best we can in our primary interval names the original premise behind the notation - that the prefix 's' takes an interval down a single step of the edo and 'S' a single step up. In our primary interval names for 28edo we have S4 below P4 and d5 above P5. We can avoid this confusion however by using the secondary interval names for P4 and P5 in 28edo - N4 and N5. In our list of edos below this change is made.
+== How to ==
+Though we have derived our interval names through many examples thus far, the process has not been explained as yet such that the reader may immediately apply them. This will be addressed here, with a step-by-step derivation guide:.
+For ''n-''edo:
+# Find the best approximations of 3/2, 5/4 and 7/4, which are to be labelled P5, sM3 and sm7 (This is equivalent to finding the [[7-limit]] [[patent val]]).
+#* P5 = round(ln(3/2)/ln(''n'')) steps
+#* sM3 = round(ln(5/4)/ln(''n'')) steps
+#* sm7 = round(ln(7/4)/ln(''n'')) steps
+# Using the best fifth, label the diatonic intervals. i.e.
+#* M2 = (2*P5) mod ''n''
+#* M6 = (3*P5) mod ''n''
+#* M3 = (4*P5) mod ''n''
+#* M7 = (5*P5) mod ''n''
+#* (P4, m7, m3, m6, m2) = ''n'' - (P4, M2, M6, M3, M7)
+#
 == Lists of edos and MOS scales ==
@@ Line 617: / Line 636: @@
 == 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. One does not need to understand the comma associations to make use of the interval names. 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.
+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. One does not need to understand the comma associations to make use of the interval names. While the intervals of some MOS scales may hold consistent names in edos in an Ups and Downs based scheme, there are many common scales that do not 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.