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User:TromboneBoi9/Approaches to weird EDOs: Difference between revisions

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Using a subset of 26edo as a notation system, as you can see above, is also an option, and works best for modal or atonal music in 13edo, since it provides a much more intuitive grasp of 13edo's intervals outside of any particular scale.
Using a subset of 26edo as a notation system, as you can see above, is also an option, and works best for modal or atonal music in 13edo, since it provides a much more intuitive grasp of 13edo's intervals outside of any particular scale.
===Blackwood's "subminor"===
[[Easley Blackwood]]'s short blurb on 13edo as part of the booklet packaged with the CD copy of his ''Twelve Microtonal Etudes'' states:
<blockquote>Yet even this tuning contains a strange mode best described
as “sub-minor”.</blockquote>
He does not elaborate on the construction of this mode, but a quick analysis of his 13edo etude reveals it to be the 5|2 "Celephaïsian" mode of [[5L 3s|5L3s]] "oneirotonic". Although not in the modern sense, Blackwood aptly calls this mode "sub-minor" because its construction is ''LsLLsLLs'', which is identical to the familiar [[5L 2s|5L2s]] diatonic Aeolian (minor) mode ''LsLLsLL'' but with an extra ''s'' appended to the end, compressing all of the intervals slightly.
{|class="wikitable"
! Interval !! Cents !! Note name (in 6L1s) !! Note name (in 5L3s)
! [[26edo]]/[[User:TromboneBoi9/Generalized Dual-Fifth Notation|GDF]] Names !! Pseudo-diatonic interval name
|-
| 0\13 || 0.00 || C || C || C || Perfect unison
|-
| 2\13 || 184.62 || D || D || D || Major second
|-
| 3\13 || 276.93 || E&flat; || E&flat; || E&flat;&flat;, vE&flat; || Minor third
|-
| 5\13 || 461.54 || F&flat; || F || F&flat;, vF || Major fourth
|-
| 7\13 || 646.15 || G&flat; || G || G&flat;, vG || Minor fifth
|-
| 8\13 || 738.46 || G || H&flat; || G&sharp;, ^G || Major fifth
|-
| 10\13 || 923.08 || A || A || A&sharp;, ^A || Major sixth
|-
| 12\13 || 1107.69 || B || B || B&sharp;, ^B || Major seventh
|-
| 13\13 || 1200.00 || C || C || C || Perfect octave
|}
Blackwood continues briefly on his use of the mode:
<blockquote>The first four bars of the Etude are an arrangement of this mode into consecutive thirds — a motif that recurs later in two transposed variations. The rest of the piece is comprised of chromatic resolutions of complex altered chords.</blockquote>
Indeed, the thirds of this scale as well as the aforementioned [[6L 1s|6L1s]] remain useful, since the thirds (two-mosstep intervals) in both scales are always either 3\13 minor thirds or 4\13 major thirds, which aren't all that inaccurate from JI.


===A note on fifths===
===A note on fifths===
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|*
|*
|}
|}
===Blackwood's "subminor"===
[[Easley Blackwood]]'s short blurb on 13edo as part of the booklet packaged with the CD copy of his ''Twelve Microtonal Etudes'' states:
<blockquote>Yet even this tuning contains a strange mode best described
as “sub-minor”.<ref>https://www.cedillerecords.org/wp-content/uploads/2020/01/018-blackwood-microtonal-booklet.pdf</ref></blockquote>
He does not elaborate on the construction of this mode, but a quick analysis of his 13edo etude reveals it to be the 5|2 "Celephaïsian" mode of [[5L 3s|5L3s]] "oneirotonic". Although not in the modern sense, Blackwood aptly calls this mode "sub-minor" because its construction is ''LsLLsLLs'', which is identical to the familiar [[5L 2s|5L2s]] diatonic Aeolian (minor) mode ''LsLLsLL'' but with an extra ''s'' appended to the end, compressing all of the intervals slightly.
{|class="wikitable"
! Interval !! Cents !! Note name (in 6L1s) !! Note name (in 5L3s)
! [[26edo]]/[[User:TromboneBoi9/Generalized Dual-Fifth Notation|GDF]] Names !! Pseudo-diatonic interval name
|-
| 0\13 || 0.00 || C || C || C || Perfect unison
|-
| 2\13 || 184.62 || D || D || D || Major second
|-
| 3\13 || 276.93 || E&flat; || E&flat; || E&flat;&flat;, vE&flat; || Minor third
|-
| 5\13 || 461.54 || F&flat; || F || F&flat;, vF || Major fourth
|-
| 7\13 || 646.15 || G&flat; || G || G&flat;, vG || Minor fifth
|-
| 8\13 || 738.46 || G || H&flat; || G&sharp;, ^G || Major fifth
|-
| 10\13 || 923.08 || A || A || A&sharp;, ^A || Major sixth
|-
| 12\13 || 1107.69 || B || B || B&sharp;, ^B || Major seventh
|-
| 13\13 || 1200.00 || C || C || C || Perfect octave
|}
Blackwood continues briefly on his use of the mode:
<blockquote>The first four bars of the Etude are an arrangement of this mode into consecutive thirds — a motif that recurs later in two transposed variations. The rest of the piece is comprised of chromatic resolutions of complex altered chords.</blockquote>
Indeed, the thirds of this scale as well as the aforementioned [[6L 1s|6L1s]] remain useful, since the thirds (two-mosstep intervals) in both scales are always either 3\13 minor thirds or 4\13 major thirds, which aren't all that inaccurate from JI.


===A note on the 7/4===
===A note on the 7/4===
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This system is similar to antidiatonic in the same way that 6L1s from earlier is similar to diatonic: B-C is the only ''large'' step here rather than it being the only small step. This 1L6s system might work well if you find yourself preferring the 14\23 major fifth rather than the 13\23 minor fifth, since it features as the fifth in this scale.
This system is similar to antidiatonic in the same way that 6L1s from earlier is similar to diatonic: B-C is the only ''large'' step here rather than it being the only small step. This 1L6s system might work well if you find yourself preferring the 14\23 major fifth rather than the 13\23 minor fifth, since it features as the fifth in this scale.
<!-- Next write a section about all the split primes WITH AUDIO DEMONSTRATIONS -->
 
===Relative to 24edo===
 
It might help to perceive 23edo as a system that "phases out" from [[24edo]] in the same way that [[11edo]] "phases out" from [[12edo]].
 
The 23edo's "quarter tone" is barely more than two cents larger than 24edo's, which means that:
 
* "Semitones" (2\23) are four cents larger.
* Whole tones (4\23) are eight cents larger, making them about as good of a [[9/8]] as [[12edo|12]] or [[24edo]]'s but sharp instead of flat.
* Minor thirds (6\23) are thirteen cents larger, making them very good [[6/5]]'s.
* "Neutral thirds" (7\23) are fifteen cents larger, pushing them closer to [[5/4]]'s.
* "Major thirds" (8\23) are now seventeen cents larger, making them distinctively Pythagorean.
* "Perfect fourths" (10\23) are 21 cents larger, pushing the boundary between ''perfect fourth'' and ''[[superfourth]]''.
 
Although by no means consistent across keys, this also introduces the idea of a notation system like [[24edo]]'s but with a missing step between F and G where the 1\2 tritone would be, which introduces the enharmonic equivalences:
 
* '''F''' = '''G{{sesquiflat2}}'''
* '''F{{demisharp2}}''' = '''G&flat;'''
* '''F&sharp;''' = '''G{{demiflat2}}'''
* '''F{{sesquisharp2}}''' = '''G'''
 
This system naturally favors the major fourth, minor fifth, and the ''proper'' major second, although this only applies for a couple of keys: The G&harr;D fifth, for instance, is major, and the F&harr;G second is obviously neutral.
 
===On Blackwood's approach===
 
The 23edo etude was the only etude of [[Easley Blackwood|Blackwood]]'s twelve etudes that uses entirely non-Western scales, specifically because, as he states:
 
<blockquote>&hellip;23-note tuning contains no diatonic configurations and no chromatic structures in common with any of the other tunings explored in this study. However, it does contain an intriguing arrangement of the two distinct pentatonic modes of Java and Bali, known as ''pelog'' and ''slendro'' — modes that cannot be realistically approximated in 12-note tuning.</blockquote>
 
With some (too) quick analysis, we see that the modes he used are as follows:
 
* Pelog: '''7 3 3 7 3'''
* Slendro: '''5 4 5 4 5'''
 
{|class="wikitable"
|+ Blackwood's ''pelog''
! Interval !! Cents
! Antidiatonic name<ref>When I use the term "antidiatonic" rather than "2L5s", I'm using '''harmonic''' notation, where flats sharpen and sharps flatten.</ref>
! [[46edo]]/[[User:TromboneBoi9/Generalized Dual-Fifth Notation|GDF]] name(s)
! Pseudo-diatonic interval name
|-
| 0\23 || 0.00 || C || C || Perfect unison
|-
| 7\23 || 365.22 || E&flat; || ^D&sharp;, vvE || Neutral third
|-
| 10\23 || 521.74 || F || ^F || Major fourth
|-
| 13\23 || 678.26 || G || vG || Minor fifth
|-
| 20\23 || 1043.48 || B&flat; || ^^B&flat; || Neutral seventh
|-
| 23\23 || 1200.00 || C || C || Perfect octave
|}
 
{|class="wikitable"
|+ Blackwood's ''slendro''
! Interval !! Cents
! Antidiatonic name(s) !! 46edo/GDF name(s)
! Pseudo-diatonic interval name
|-
| 0\23 || 0.00 || C || C || Perfect unison
|-
| 5\23 || 260.87 || D&flat;&flat;, E&sharp; || ^^D, vE&flat; || Supermajor second, subminor third
|-
| 9\23 || 469.57 || F&sharp; || vF || Minor fourth
|-
| 14\23 || 730.43 || G&flat; || ^G || Major fifth
|-
| 18\23 || 939.13 || A&flat;&flat;, B&sharp; || ^A, vvB&flat; || Supermajor sixth, subminor seventh
|-
| 23\23 || 1200.00 || C || C || Perfect octave
|}
 
What's interesting about these two modes is that both kinds of fifth (major and minor) are featured in some form in each scale, and by extension, both types of fourths as well, which ostensibly demonstrates that ''both'' kinds of fourth/fifth can be used tastefully, even if in an "exotic" context.
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