User:Unque/Dietic Minor: Difference between revisions

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This presents a problem of its own - the raised seventh is an augmented tone above the minor sixth, which creates a disjointed sound in melodies that walk up this scale. Many musicians opt to intonate the augmented step as a normal whole tone (scale pattern LsLLsLs), but this scale now falls short of the octave by one [[2187/2048|Apotome]]. In common-practice theory, the most common solution for this is to extend the preceding [[256/243|limma]] between the fifth and sixth into a [[9/8|whole tone]] (scale pattern LsLLLLs), which allows the scale to reach the octave; however, this approach largely abandons the minor tonality of the scale, with all but the third degree now being that of the Ionian scale.

[[29edo|29 equal divisions of the octave]] provides an elegant solution that retains this minor tonality, reaches the octave, and avoids disjunct augmented tones: by noticing that the Apotome is divided evenly into three steps, and there exist three limmas in the scale, we can disperse this damage across all three semitones rather than placing it all onto one of them. This creates a form of the Smitonic scale (scale pattern LsLLsLs), where the large step is 9/8 and the small step 2187/2048. This pattern can be thought of as a 3-limit essentially tempered scale that tempers out the [[Mystery comma]]. While this scale does meaningfully retain the minor tonality, the harmonies are quite distorted compared to the diatonic counterparts. The minor third and perfect fifth are both stretched a diesis wider than in diatonic, while the sixth and seventh degrees are stretched by an entire limma.

[[29edo|29 equal divisions of the octave]] provides an elegant solution that retains this minor tonality, reaches the octave, and avoids disjunct augmented tones: by noticing that the Apotome is divided evenly into three steps, and there exist three limmas in the scale, we can disperse this damage across all three semitones rather than placing it all onto one of them. This creates a form of the Smitonic scale (scale pattern LsLLsLs), where the large step is [[9/8]] and the small step [[2187/2048]]. This pattern can be thought of as a 3-limit essentially tempered scale that tempers out the [[Mystery comma]]. While this scale does meaningfully retain the minor tonality, the harmonies are quite distorted compared to the diatonic counterparts. The minor third and perfect fifth are both stretched a diesis wider than in diatonic, while the sixth and seventh degrees are stretched by an entire limma.

So perhaps instead of stretching the semitones, the dieses could be employed as step sizes ~~- these~~ smaller steps would act as commatic adjustments that ~~preserve the~~ diatonic harmony in the context of a Smitonic ~~scale~~. There are a number of arrangements that can be employed in this pattern, though they can be generalized under the form '''LmLLmLm''' with three small steps inserted at various points among the scale degrees. The name "Dietic Minor" was given to this scale, as it is a modification of the harmonic minor scale that adds dietic steps as a practical consideration.

So perhaps instead of stretching the semitones, the dieses could be employed as step sizes. These smaller steps would act as commatic adjustments that allow for diatonic-like harmony in the context of Smitonic-like melodies. There are a number of arrangements that can be employed in this pattern, though they can be generalized under the form '''LmLLmLm''' with three small steps inserted at various points among the scale degrees. The name "Dietic Minor" was given to this scale, as it is a modification of the harmonic minor scale that adds dietic steps as a practical consideration.

=== Notation ===

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As it turns out, this proposed solution is quite similar to the principles that spawned [[Groundfault]]'s [[aberrismic theory]], a concept based on the insertion of small steps called "aberrismas" into a MOS scale to solve structural issues. The small steps of the Dietic Minor scale can be conceived as aberrismas, especially in tunings outside of 29edo where these aberrismas are no longer the single step "diesis."

It should be noted, however, that Ground's approach to aberrismic scales involves certain other features that are foregone in Dietic Minor, such as formation by [[MOS substitution]] and the satisfaction of [[Monotone-MOS scale|Monotone-MOS]] conditions; the structure of Dietic Minor does not generalize well to either of these properties, and while closely related to Ground's approach, the two ~~may~~ not be considered entirely equivalent.

It should be noted, however, that Ground's approach to aberrismic scales involves certain other features that are foregone in Dietic Minor, such as formation by [[MOS substitution]] and the satisfaction of [[Monotone-MOS scale|Monotone-MOS]] conditions; the structure of Dietic Minor does not generalize well to either of these properties, and while closely related to Ground's approach, the two might not be considered entirely equivalent.

== Tuning Ranges ==

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=== s(II-V-VII) ===

s(II-V-VII), or '''LsmLLsmLsm''', is the most obvious arrangement of the scale - an aberrisma is inserted before each semitone.

s(II-V-VII), or '''LsmLLsmLsm''', is the most obvious arrangement of the scale - an aberrisma is inserted before each semitone. When the L and m steps are the same size, this pattern becomes a [[MOS scale]], namely [[7L 3s]].

{| class="wikitable"

|+Modes of s(II-V-VII)

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=== s(II-IV-VI) ===

s(II-IV-VI), or '''LsmLsLmsLm''', is another intuitive arrangement of the scale. The aberrismas are arranged such that the scale has a maximum variety of three, which makes the scale significantly more harmonically coherent while sacrificing some of the melodic qualities of the s(II-V-VII) arrangement.

s(II-IV-VI), or '''LsmLsLmsLm''', is another intuitive arrangement of the scale. The aberrismas are arranged such that the scale has a maximum variety of three, which makes the scale significantly more harmonically coherent while sacrificing some of the melodic qualities of the s(II-V-VII) arrangement. Just like the previous example, this arrangement can collapse into 7L 3s when the L and m steps are the same size; however, this arrangement can additionally collapse into [[4L 6s]] when the m and s steps are the same size.

{| class="wikitable"

|+Modes of s(II-V-VII)

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|}

=== s(V~~-VI~~-VII) ===

=== s(I-V-VII) ===

s(V~~-VI~~-VII), or '''~~LmLLsmLsms~~''', is an arrangement that might seem unintuitive at first - after all, the aberrismas appear in a weird cluster in the upper end of the scale rather than being dispersed as one would expect. However, this placement allows for the Perfect Fourth and Fifth to be available over the same degree, which ~~in turn more precisely indicates that degree as the tonic, and~~ makes the harmonies ~~over that degree~~ significantly smoother.

s(I-V-VII), or '''sLmLLsmLsm''', is an arrangement that might seem unintuitive at first - after all, the aberrismas appear in a weird cluster in the upper end of the scale rather than being dispersed as one would expect. However, this placement allows for the Perfect Fourth and Fifth to occur over two degrees instead of one, and additionally to be available over the same degree, which makes the harmonies of this permutation significantly smoother. This arrangement notably does not collapse into a MOS scale when the m step is equated with either the L step or the s step.

{| class="wikitable"

|+Modes of s(V~~-VI~~-VII)

|+Modes of s(I-V-VII)

!Rotational Order

!Pattern

Line 188:

|-

|1

|sLmLLsmLsm

|

|-

|2

|LmLLsmLsms

|Contains Perfect Fourth and Fifth

|-

|2

|3

|mLLsmLsmsL

|

|-

|3

|4

|LLsmLsmsLm

|

|-

|4

|5

|LsmLsmsLmL

|Contains Perfect Fifth

|-

|5

|6

|smLsmsLmLL

|Contains ~~perfect~~ Fourth

|Contains Perfect Fourth

|-

|6

|7

|mLsmsLmLLs

|

|-

|7

|8

|LsmsLmLLsm

|

|-

|8

|9

|smsLmLLsmL

|

|-

|9

|10

|msLmLLsmLs

|

|-

~~|10~~

~~|sLmLLsmLsm~~

|

|}

== See Also ==

* [[Smi2s]], a closely-related scale that contains two aberrismas instead of three.

[[Category:29edo]]

[[Category:Aberrismic_theory]]

@@ Line 8: / Line 8: @@
 This presents a problem of its own - the raised seventh is an augmented tone above the minor sixth, which creates a disjointed sound in melodies that walk up this scale.  Many musicians opt to intonate the augmented step as a normal whole tone (scale pattern LsLLsLs), but this scale now falls short of the octave by one [[2187/2048|Apotome]].  In common-practice theory, the most common solution for this is to extend the preceding [[256/243|limma]] between the fifth and sixth into a [[9/8|whole tone]] (scale pattern LsLLLLs), which allows the scale to reach the octave; however, this approach largely abandons the minor tonality of the scale, with all but the third degree now being that of the Ionian scale.
-[[29edo|29 equal divisions of the octave]] provides an elegant solution that retains this minor tonality, reaches the octave, and avoids disjunct augmented tones: by noticing that the Apotome is divided evenly into three steps, and there exist three limmas in the scale, we can disperse this damage across all three semitones rather than placing it all onto one of them.  This creates a form of the Smitonic scale (scale pattern LsLLsLs), where the large step is 9/8 and the small step 2187/2048.  This pattern can be thought of as a 3-limit essentially tempered scale that tempers out the [[Mystery comma]].  While this scale does meaningfully retain the minor tonality, the harmonies are quite distorted compared to the diatonic counterparts.  The minor third and perfect fifth are both stretched a diesis wider than in diatonic, while the sixth and seventh degrees are stretched by an entire limma.
+[[29edo|29 equal divisions of the octave]] provides an elegant solution that retains this minor tonality, reaches the octave, and avoids disjunct augmented tones: by noticing that the Apotome is divided evenly into three steps, and there exist three limmas in the scale, we can disperse this damage across all three semitones rather than placing it all onto one of them.  This creates a form of the Smitonic scale (scale pattern LsLLsLs), where the large step is [[9/8]] and the small step [[2187/2048]].  This pattern can be thought of as a 3-limit essentially tempered scale that tempers out the [[Mystery comma]].  While this scale does meaningfully retain the minor tonality, the harmonies are quite distorted compared to the diatonic counterparts.  The minor third and perfect fifth are both stretched a diesis wider than in diatonic, while the sixth and seventh degrees are stretched by an entire limma.
-So perhaps instead of stretching the semitones, the dieses could be employed as step sizes - these smaller steps would act as commatic adjustments that preserve the diatonic harmony in the context of a Smitonic scale.  There are a number of arrangements that can be employed in this pattern, though they can be generalized under the form '''LmLLmLm''' with three small steps inserted at various points among the scale degrees.  The name "Dietic Minor" was given to this scale, as it is a modification of the harmonic minor scale that adds dietic steps as a practical consideration.
+So perhaps instead of stretching the semitones, the dieses could be employed as step sizes.  These smaller steps would act as commatic adjustments that allow for diatonic-like harmony in the context of Smitonic-like melodies.  There are a number of arrangements that can be employed in this pattern, though they can be generalized under the form '''LmLLmLm''' with three small steps inserted at various points among the scale degrees.  The name "Dietic Minor" was given to this scale, as it is a modification of the harmonic minor scale that adds dietic steps as a practical consideration.
 === Notation ===
@@ Line 18: / Line 18: @@
 As it turns out, this proposed solution is quite similar to the principles that spawned [[Groundfault]]'s [[aberrismic theory]], a concept based on the insertion of small steps called "aberrismas" into a MOS scale to solve structural issues.  The small steps of the Dietic Minor scale can be conceived as aberrismas, especially in tunings outside of 29edo where these aberrismas are no longer the single step "diesis."
-It should be noted, however, that Ground's approach to aberrismic scales involves certain other features that are foregone in Dietic Minor, such as formation by [[MOS substitution]] and the satisfaction of [[Monotone-MOS scale|Monotone-MOS]] conditions; the structure of Dietic Minor does not generalize well to either of these properties, and while closely related to Ground's approach, the two may not be considered entirely equivalent.
+It should be noted, however, that Ground's approach to aberrismic scales involves certain other features that are foregone in Dietic Minor, such as formation by [[MOS substitution]] and the satisfaction of [[Monotone-MOS scale|Monotone-MOS]] conditions; the structure of Dietic Minor does not generalize well to either of these properties, and while closely related to Ground's approach, the two might not be considered entirely equivalent.
 == Tuning Ranges ==
@@ Line 82: / Line 82: @@
 === s(II-V-VII) ===
-s(II-V-VII), or '''LsmLLsmLsm''', is the most obvious arrangement of the scale - an aberrisma is inserted before each semitone.
+s(II-V-VII), or '''LsmLLsmLsm''', is the most obvious arrangement of the scale - an aberrisma is inserted before each semitone.  When the L and m steps are the same size, this pattern becomes a [[MOS scale]], namely [[7L 3s]].
 {| class="wikitable"
 |+Modes of s(II-V-VII)
@@ Line 131: / Line 131: @@
 === s(II-IV-VI) ===
-s(II-IV-VI), or '''LsmLsLmsLm''', is another intuitive arrangement of the scale.  The aberrismas are arranged such that the scale has a maximum variety of three, which makes the scale significantly more harmonically coherent while sacrificing some of the melodic qualities of the s(II-V-VII) arrangement.
+s(II-IV-VI), or '''LsmLsLmsLm''', is another intuitive arrangement of the scale.  The aberrismas are arranged such that the scale has a maximum variety of three, which makes the scale significantly more harmonically coherent while sacrificing some of the melodic qualities of the s(II-V-VII) arrangement.  Just like the previous example, this arrangement can collapse into 7L 3s when the L and m steps are the same size; however, this arrangement can additionally collapse into [[4L 6s]] when the m and s steps are the same size.
 {| class="wikitable"
 |+Modes of s(II-V-VII)
@@ Line 179: / Line 179: @@
 |}
-=== s(V-VI-VII) ===
+=== s(I-V-VII) ===
-s(V-VI-VII), or '''LmLLsmLsms''', is an arrangement that might seem unintuitive at first - after all, the aberrismas appear in a weird cluster in the upper end of the scale rather than being dispersed as one would expect.  However, this placement allows for the Perfect Fourth and Fifth to be available over the same degree, which in turn more precisely indicates that degree as the tonic, and makes the harmonies over that degree significantly smoother.
+s(I-V-VII), or '''sLmLLsmLsm''', is an arrangement that might seem unintuitive at first - after all, the aberrismas appear in a weird cluster in the upper end of the scale rather than being dispersed as one would expect.  However, this placement allows for the Perfect Fourth and Fifth to occur over two degrees instead of one, and additionally to be available over the same degree, which makes the harmonies of this permutation significantly smoother.  This arrangement notably does not collapse into a MOS scale when the m step is equated with either the L step or the s step.
 {| class="wikitable"
-|+Modes of s(V-VI-VII)
+|+Modes of s(I-V-VII)
 !Rotational Order
 !Pattern
@@ Line 188: / Line 188: @@
 |-
 |1
+|sLmLLsmLsm
+|
+|-
+|2
 |LmLLsmLsms
 |Contains Perfect Fourth and Fifth
 |-
-|2
+|3
 |mLLsmLsmsL
 |
 |-
-|3
+|4
 |LLsmLsmsLm
 |
 |-
-|4
+|5
 |LsmLsmsLmL
 |Contains Perfect Fifth
 |-
-|5
+|6
 |smLsmsLmLL
-|Contains perfect Fourth
+|Contains Perfect Fourth
 |-
-|6
+|7
 |mLsmsLmLLs
 |
 |-
-|7
+|8
 |LsmsLmLLsm
 |
 |-
-|8
+|9
 |smsLmLLsmL
 |
 |-
-|9
+|10
 |msLmLLsmLs
-|
-|-
-|10
-|sLmLLsmLsm
 |
 |}
+== See Also ==
+* [[Smi2s]], a closely-related scale that contains two aberrismas instead of three.
+[[Category:29edo]]
+[[Category:Aberrismic_theory]]