Macrodiatonic and microdiatonic scales: Difference between revisions

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'''Macrodiatonic scales''' are [[nonoctave]] [[MOS scale]]s that have 5 large and 2 small steps like the diatonic scale, but have a period greater than 2/1 ~~so are~~ stretched. The equivalent term for compressed diatonic scales (period < 2/1) is '''microdiatonic'''. Macrodiatonic/microdiatonic scales typically sound completely unrecognizable due to the stretching/squishing, yet are structurally identical to diatonic (with it being possible to translate from diatonic to macrodiatonic/microdiatonic scales "on the fly"). Nonoctave [[regular temperament]]s assosciated with macrodiatonic/microdiatonic scales are sometimes termed as macro[[meantone]]/micromeantones. However, it is controversial to use the name of the temperament for things that only share a scale pattern rather than harmonic structure.

'''Macrodiatonic scales'''{{idiosyncratic}} are [[nonoctave]] [[MOS scale]]s that have 5 large and 2 small steps like the diatonic scale, but have a period greater than 2/1, making them extremely [[stretched tuning|stretched]] versions of the diatonic scale. The equivalent term for compressed diatonic scales (period < 2/1) is '''microdiatonic'''{{idiosyncratic}}. Macrodiatonic/microdiatonic scales typically sound completely unrecognizable due to the stretching/squishing, yet are structurally identical to diatonic (with it being possible to translate from diatonic to macrodiatonic/microdiatonic scales "on the fly"). Nonoctave [[regular temperament]]s assosciated with macrodiatonic/microdiatonic scales are sometimes termed as macro[[meantone]]/micromeantones. However, it is controversial to use the name of the temperament for things that only share a scale pattern rather than harmonic structure. [[User:CompactStar|CompactStar]] suggests that the term macromeantone/micromeantone is only used for temperaments in which the "fifth" analogue is tuned flatter than its untempered interpretation.

In broader sense, a scale created by repeating one simple intonation ("fifth" analogue) and it has low otonal limit triad (with "third" analogue which is but not always genspan +4.) But, if 3/2 is used as generator, such as [[angel]] and [[~~obikhod~~]], ~~another name should~~ be ~~considered~~.

In a broader sense, a scale created by repeating one simple intonation ("fifth" analogue) and it has a low otonal limit triad (with "third" analogue which is but not always genspan +4.) But, if 3/2 is used as generator, such as [[angel]] and {{mos scalesig|8L 3s<3/1>|link=1}}, another name should be considered. The terms longdiatonic and shortdiatonic have been proposed to refer to the latter, most likely by [[User:Dummy index|Dummy index]].

== Notation ==

Macrodiatonic and microdiatonic scales can be notated using the standard diatonic notation, if intervals are reinterpreted to be understood as their stretched or compressed counterparts.

== Examples of macrodiatonic and microdiatonic scales ==

* [[5L 2s (3/2-equivalent)|5L 2s<3/2>]]: Generator is a neogothic major third near [[81/64]] and [[14/11]].

* [[5L 2s (4/3-equivalent)|5L 2s<4/3>]]: Generator is a neogothic minor third.

* [[Ed7/4|5L 2s<7/4>]]: ~~Dark~~ generator is approximately 5/4.

* [[10L 4s]] (5L 2s<1\2>): Generator is a neutral third. Includes the octave by virtue of its half-octave period.

* [[5L 2s (3/2-equivalent)|5L 2s<3/2>]]: Generator is a neogothic major third near [[81/64]] and [[14/11]]. In the [[half-prime subgroup]] paradigm it may be interpreted as [[63/50]] (relating to the 3/2.5/2.7/2 subgroup version of [[sirius]] temperament).

* [[Ed7/4|5L 2s<7/4>]]: Generator is approximately 7/5 (dark generator is approximately 5/4).

* [[Ed9/5|5L 2s<9/5>]]: Generator is approximately 7/5.

* [[5L 2s (12/5-equivalent)|5L 2s<12/5>]]: Generator is approximately [[5/3]], and interestingly, major triads (4:5:6) are stretched to approximately second inversion major triads (3:4:5).

* [[5L 2s (12/5-equivalent)|5L 2s<12/5>]]: Generator is approximately [[5/3]], and interestingly, major triads (4:5:6) are stretched to approximately second inversion major triads (3:4:5). This is made by cutting off the 5/3 interval range of the tetratave scale of [[tetrahanson]] [[8L 3s (4/1-equivalent)|8L 3s<4/1>]].

* [[5L 2s (3/1-equivalent)|5L 2s<3/1>]]: Generator is approximately [[17/9]], and 4 generators is vaguely equivalent to [[13/9]]. This relates this scale to the no-twos macromeantone temperament [[tempering out]] 85293/83521 in the 3.13.17 subgroup (equating 4 17/9 to 13/9, instead of 4 3/2 to 5/4), as documented on the page for [[12edt]].

* [[5L 2s (3/1-equivalent)|5L 2s<3/1>]]: Generator is approximately [[17/9]], and 4 generators is vaguely equivalent to [[13/9]]. This relates this scale to the no-twos macromeantone temperament (or rather more like macroarchy, with the 17/9 being stretched) [[tempering out]] 85293/83521 in the 3.13.17 subgroup (equating 4 17/9 to 13/9, instead of 4 3/2 to 5/4), as documented on the page for [[12edt]]. Harder varieties of the scale can instead be interpreted where the generator is a sharp [[21/11]], and 3 generators is equivalent to [[7/3]], tempering out [[1331/1323]] and producing [[Mintaka]] temperament.

* [[5L 2s (4/1-equivalent)|5L 2s<4/1>]]: This is precisely a 200% stretched diatonic scale. Generator is approximately [[9/4]] ([[3/2]]2). This can very clearly be interpreted as a temperament tempering out [[6561/6400]] ([[81/80]]2) in 4.9.25 for the meantone equivalent or 4096/3969 ([[64/63]]2) in 4.9.49 for the archy equivalent.

* [[5L 2s (13/4-equivalent)|5L 2s<13/4>]]: Generator is approximately [[2/1]]. This may be suitable for stretching 4:5:6 into 2:3:4, but the "[[3/2]]" in this chord is not very accurate and [[mavila]]-like in size.

* [[5L 2s (4/1-equivalent)|5L 2s<4/1>]]: This is precisely a 200% stretched diatonic scale. Generator is approximately [[9/4]] ([[3/2]]2). This can very clearly be interpreted as a temperament tempering out [[6561/6400]] ([[81/80]]2) in 4.9.25 for the meantone equivalent or 4096/3969 ([[64/63]]2) in 4.9.49 for the archy equivalent. This scale is extremely widely spaced, so it may preferable to use the macrochromatic scales [[7L 5s (4/1-equivalent)|7L 5s<4/1>]] and [[5L 7s (4/1-equivalent)|5L 7s<4/1>]] instead.

* [[5L 2s (5/1-equivalent)|5L 2s<5/1>]]: Generator is approximately [[13/5]] or [[18/7]]. Relates to [[Hyperpyth]] temperament.

== In broader sense ==

=== Macrodiatonic/microdiatonic ===

* [[EDF|9L 2s<3/2>]]: Generator is approximately 5/4 and 6 generators is 9/8.

* [[Ed5/3|7L 2s<5/3>]]: Generator is approximately 4/3 and 4 generators is 8/7.

* [[Ed5/3|7L 2s<5/3>]]: Generator is approximately 4/3 and 4 generators is 8/7. 6:7:8 triad is homologous to minor triad on the known circle of fifths.

=== Longdiatonic/shortdiatonic ===

* [[~~Ed8~~/5|6L 1s<8/5>]]: Dark generator is approximately 3/2 and 4 generators is 6/5.

* 2L 1s<4/3>

* [[3L 4s (9/5-equivalent)|3L 4s<9/5>]]: Dark generator is just 3/2 and 2 generators is just 5/4. Merely this is a lattice by 5/4 and 6/5.

* [[3L 1s (3/2-equivalent)|3L 1s<3/2>]]

* [[6L 1s (8/5-equivalent)|6L 1s<8/5>]]: Dark generator is approximately 3/2 and 4 generators is 6/5.

* 4L 1s<5/3>: = 7L 2s<5/2> - 3L 1s<3/2>.

* [[3L 4s (7/4-equivalent)|3L 4s<7/4>]]: Dark generator is just 3/2 and 2 generators is just 9/7. Merely this is a lattice by 9/7 and 7/6.

* [[3L 4s (9/5-equivalent)|3L 4s<9/5>]]: Dark generator is just 3/2 and 2 generators is just 5/4. Merely this is a lattice by 5/4 and 6/5. The minor triad are more affinity for equave (10:12:15:18) than the major triad (20:25:30:36).

* [[7L 2s (5/2-equivalent)|7L 2s<5/2>]]: Dark generator is approximately 3/2 and 5 generators is 6/5.

* [[8L 3s (3/1-equivalent)|8L 3s<3/1>]]: Generator is approximately 3/2 and 6 generators is 5/4.

* [[7L 3s (4/1-equivalent)|7L 3s<4/1>]]

== See also ==

* [[Stretched tuning]]

* [[Translated scale]]

[[Category:7-tone scales]]

[[Category:Nonoctave]]

@@ Line 1: / Line 1: @@
-'''Macrodiatonic scales''' are [[nonoctave]]  [[MOS scale]]s that have 5 large and 2 small steps like the diatonic scale, but have a period greater than 2/1 so are stretched. The equivalent term for compressed diatonic scales (period < 2/1) is '''microdiatonic'''. Macrodiatonic/microdiatonic scales typically sound completely unrecognizable due to the stretching/squishing, yet are structurally identical to diatonic (with it being possible to translate from diatonic to macrodiatonic/microdiatonic scales "on the fly"). Nonoctave [[regular temperament]]s assosciated with macrodiatonic/microdiatonic scales are sometimes termed as macro[[meantone]]/micromeantones. However, it is controversial to use the name of the temperament for things that only share a scale pattern rather than harmonic structure.
+'''Macrodiatonic scales'''{{idiosyncratic}} are [[nonoctave]]  [[MOS scale]]s that have 5 large and 2 small steps like the diatonic scale, but have a period greater than 2/1, making them extremely [[stretched tuning|stretched]] versions of the diatonic scale. The equivalent term for compressed diatonic scales (period < 2/1) is '''microdiatonic'''{{idiosyncratic}}. Macrodiatonic/microdiatonic scales typically sound completely unrecognizable due to the stretching/squishing, yet are structurally identical to diatonic (with it being possible to translate from diatonic to macrodiatonic/microdiatonic scales "on the fly"). Nonoctave [[regular temperament]]s assosciated with macrodiatonic/microdiatonic scales are sometimes termed as macro[[meantone]]/micromeantones. However, it is controversial to use the name of the temperament for things that only share a scale pattern rather than harmonic structure. [[User:CompactStar|CompactStar]] suggests that the term macromeantone/micromeantone is only used for temperaments in which the "fifth" analogue is tuned flatter than its untempered interpretation.
-In broader sense, a scale created by repeating one simple intonation ("fifth" analogue) and it has low otonal limit triad (with "third" analogue which is but not always genspan +4.) But, if 3/2 is used as generator, such as [[angel]] and [[obikhod]], another name should be considered.
+In a broader sense, a scale created by repeating one simple intonation ("fifth" analogue) and it has a low otonal limit triad (with "third" analogue which is but not always genspan +4.) But, if 3/2 is used as generator, such as [[angel]] and {{mos scalesig|8L 3s<3/1>|link=1}}, another name should be considered. The terms longdiatonic and shortdiatonic have been proposed to refer to the latter, most likely  by [[User:Dummy index|Dummy index]].
+== Notation ==
+Macrodiatonic and microdiatonic scales can be notated using the standard diatonic notation, if intervals are reinterpreted to be understood as their stretched or compressed counterparts.
 == Examples of macrodiatonic and microdiatonic scales ==
-* [[5L 2s (3/2-equivalent)|5L 2s<3/2>]]: Generator is a neogothic major third near [[81/64]] and [[14/11]].
+* [[5L 2s (4/3-equivalent)|5L 2s<4/3>]]: Generator is a neogothic minor third.
-* [[Ed7/4|5L 2s<7/4>]]: Dark generator is approximately 5/4.
+* [[10L 4s]] (5L 2s<1\2>): Generator is a neutral third. Includes the octave by virtue of its half-octave period.
+* [[5L 2s (3/2-equivalent)|5L 2s<3/2>]]: Generator is a neogothic major third near [[81/64]] and [[14/11]]. In the [[half-prime subgroup]] paradigm it may be interpreted as [[63/50]] (relating to the 3/2.5/2.7/2 subgroup version of [[sirius]] temperament).
+* [[Ed7/4|5L 2s<7/4>]]: Generator is approximately 7/5 (dark generator is approximately 5/4).
 * [[Ed9/5|5L 2s<9/5>]]: Generator is approximately 7/5.
-* [[5L 2s (12/5-equivalent)|5L 2s<12/5>]]: Generator is approximately [[5/3]], and interestingly, major triads (4:5:6) are stretched to approximately second inversion major triads (3:4:5).
+* [[5L 2s (12/5-equivalent)|5L 2s<12/5>]]: Generator is approximately [[5/3]], and interestingly, major triads (4:5:6) are stretched to approximately second inversion major triads (3:4:5). This is made by cutting off the 5/3 interval range of the tetratave scale of [[tetrahanson]] [[8L 3s (4/1-equivalent)|8L 3s<4/1>]].
-* [[5L 2s (3/1-equivalent)|5L 2s<3/1>]]: Generator is approximately [[17/9]], and 4 generators is vaguely equivalent to [[13/9]]. This relates this scale to the no-twos macromeantone temperament [[tempering out]] 85293/83521 in the 3.13.17 subgroup (equating 4 17/9 to 13/9, instead of 4 3/2 to 5/4), as documented on the page for [[12edt]].
+* [[5L 2s (3/1-equivalent)|5L 2s<3/1>]]: Generator is approximately [[17/9]], and 4 generators is vaguely equivalent to [[13/9]]. This relates this scale to the no-twos macromeantone temperament (or rather more like macroarchy, with the 17/9 being stretched) [[tempering out]] 85293/83521 in the 3.13.17 subgroup (equating 4 17/9 to 13/9, instead of 4 3/2 to 5/4), as documented on the page for [[12edt]]. Harder varieties of the scale can instead be interpreted where the generator is a sharp [[21/11]], and 3 generators is equivalent to [[7/3]], tempering out [[1331/1323]] and producing [[Mintaka]] temperament.
-* [[5L 2s (4/1-equivalent)|5L 2s<4/1>]]: This is precisely a 200% stretched diatonic scale. Generator is approximately [[9/4]] ([[3/2]]<sup>2</sup>). This can very clearly be interpreted as a temperament tempering out [[6561/6400]] ([[81/80]]<sup>2</sup>) in 4.9.25 for the meantone equivalent or 4096/3969 ([[64/63]]<sup>2</sup>) in 4.9.49 for the archy equivalent.
+* [[5L 2s (13/4-equivalent)|5L 2s<13/4>]]: Generator is approximately [[2/1]]. This may be suitable for stretching 4:5:6 into 2:3:4, but the "[[3/2]]" in this chord is not very accurate and [[mavila]]-like in size.
+* [[5L 2s (4/1-equivalent)|5L 2s<4/1>]]: This is precisely a 200% stretched diatonic scale. Generator is approximately [[9/4]] ([[3/2]]<sup>2</sup>). This can very clearly be interpreted as a temperament tempering out [[6561/6400]] ([[81/80]]<sup>2</sup>) in 4.9.25 for the meantone equivalent or 4096/3969 ([[64/63]]<sup>2</sup>) in 4.9.49 for the archy equivalent. This scale is extremely widely spaced, so it may preferable to use the macrochromatic scales [[7L 5s (4/1-equivalent)|7L 5s<4/1>]] and [[5L 7s (4/1-equivalent)|5L 7s<4/1>]] instead.
+* [[5L 2s (5/1-equivalent)|5L 2s<5/1>]]: Generator is approximately [[13/5]] or [[18/7]]. Relates to [[Hyperpyth]] temperament.
 == In broader sense ==
 === Macrodiatonic/microdiatonic ===
 * [[EDF|9L 2s<3/2>]]: Generator is approximately 5/4 and 6 generators is 9/8.
-* [[Ed5/3|7L 2s<5/3>]]: Generator is approximately 4/3 and 4 generators is 8/7.
+* [[Ed5/3|7L 2s<5/3>]]: Generator is approximately 4/3 and 4 generators is 8/7. 6:7:8 triad is homologous to minor triad on the known circle of fifths.
 === Longdiatonic/shortdiatonic ===
-* [[Ed8/5|6L 1s<8/5>]]: Dark generator is approximately 3/2 and 4 generators is 6/5.
+* 2L 1s<4/3>
-* [[3L 4s (9/5-equivalent)|3L 4s<9/5>]]: Dark generator is just 3/2 and 2 generators is just 5/4. Merely this is a lattice by 5/4 and 6/5.
+* [[3L 1s (3/2-equivalent)|3L 1s<3/2>]]
+* [[6L 1s (8/5-equivalent)|6L 1s<8/5>]]: Dark generator is approximately 3/2 and 4 generators is 6/5.
+* 4L 1s<5/3>: = 7L 2s<5/2> - 3L 1s<3/2>.
+* [[3L 4s (7/4-equivalent)|3L 4s<7/4>]]: Dark generator is just 3/2 and 2 generators is just 9/7. Merely this is a lattice by 9/7 and 7/6.
+* [[3L 4s (9/5-equivalent)|3L 4s<9/5>]]: Dark generator is just 3/2 and 2 generators is just 5/4. Merely this is a lattice by 5/4 and 6/5. The minor triad are more affinity for equave (10:12:15:18) than the major triad (20:25:30:36).
+* [[7L 2s (5/2-equivalent)|7L 2s<5/2>]]: Dark generator is approximately 3/2 and 5 generators is 6/5.
+* [[8L 3s (3/1-equivalent)|8L 3s<3/1>]]: Generator is approximately 3/2 and 6 generators is 5/4.
+* [[7L 3s (4/1-equivalent)|7L 3s<4/1>]]
+== See also ==
+* [[Stretched tuning]]
+* [[Translated scale]]
 [[Category:7-tone scales]]
+[[Category:Nonoctave]]