View source for User:TromboneBoi9/Generalized Dual-Fifth Notation

'''General dual-fifth notation''' (or ''GDF'' notation) is a system of notation that is designed for use with [[dual-fifth]] temperaments, intended for use with smaller dual-fifth [[EDO|EDOs]]. It can perhaps be considered a form of [[ups and downs notation]] since it uses the same nominals, symbols, and general principles, and when used for EDOs, it often materializes as a [[subset notation]].

==Overview==

The main goal of GDF notation is to describe intervals closer to what they actually "are," rather than notation systems like [[antidiatonic]] (assigning the minor fifth to the ''perfect fifth'') which switches the minor-major duality.

GDF notation does this by maintaining some fidelity to the [[Chain-of-fifths notation|chain of fifths]] as used in other approachable notation systems. To do so, it relies instead on the [[9/8|major second]], since a chain of 9/8 major seconds is the same as every other step in a chain of fifths (assuming octave equivalence). Every remaining step in between is split to accommodate the dual fifths, with the major fifths notated as raised fifths (^5) and minor fifths notated as lowered fifths (v5). This ensures that there is no fifth that resembles a ''perfect'' fifth, as dual-fifth systems do not have them.

That is to say, GDF relies on the logic that '''the major fifth and the minor fifth make a major ninth''' (major second when octave-reduced), or that <math>3^+ \cdot 3^- = 9</math>. This also means, unlike antidiatonic, neither of the major seconds generated from the off-fifths are treated as perfectly ''major'':
* Two major fifths make a doubly-raised second (^^M2).
* Two minor fifths make a doubly-lowered second (vvM2).

With split fifths, a chain of fifths would look something like this:

{|class="wikitable"
| rowspan="2" | G&flat; || ^D&flat;
| rowspan="2" | A&flat; || ^E&flat;
| rowspan="2" | B&flat; || ^F
| rowspan="2" | C || ^G
| rowspan="2" | D || ^A
| rowspan="2" | E || ^B
| rowspan="2" | F&sharp; || ^C&sharp;
|-
| vD&flat; || vE&flat; || vF || vG || vA || vB || vC&sharp;
|}

This also implies a converse system where the other set of fifths are split:
{|class="wikitable"
| ^G&flat;
| rowspan="2" | D&flat; || ^A&flat;
| rowspan="2" | E&flat; || ^B&flat;
| rowspan="2" | F || ^C
| rowspan="2" | G || ^D
| rowspan="2" | A || ^E
| rowspan="2" | B || ^F&sharp;
| rowspan="2" | C&sharp;
|-
| vG&flat; || vA&flat; || vB&flat; || vC || vD || vE || vF&sharp;
|}

===Relation to ups and downs===

When using GDF notation with EDOs (especially small EDOs), it's usually the case that the GDF notation of <math>n</math>-edo is identical to the [[subset notation]] loaned from the [[ups and downs notation]] of <math>2n</math>-edo.

That being said, it should be noted that the symbols ''^'' and ''v'' used in GDF do '''not''' have the same meaning that they do in ups and downs notation (that is, an alteration of one [[arrow|edostep]]). In GDF, these symbols indicate alterations of a ''half''-edostep; full edosteps are indicated by ''^^'' and ''vv''.

It may help to trade the ''^'' and ''v'' symbols for something else, e.g. ''/'' and ''\'' (slash and backslash), so that ''^'' and ''v'' can keep their original meaning in ups and downs while the offset chains of seconds remain visually distinct. Just remember that ''/'' + ''/'' = ''^''.

==Tables==

===13edo===
GDF notation of [[13edo]] is identical to [[26edo]] subset notation.
{|class="wikitable"
! Steps !! Cents !! Names !! Alt. Names
|- 
| 0 || 0.00 || C, vC&sharp; ^C&flat; || C, \C&sharp;, /C&flat;
|- 
| 1 || 92.31 || ^^C, ^C&sharp;, vD&flat;, vvD || ^C, /C&sharp;, \D&flat;, vD
|-
| 2 || 184.62 || D, ^D&flat;, vD&sharp; || D, /D&flat;, \D&sharp;
|-
| 3 || 276.92 || ^^D, ^D&sharp;, vE&flat;, vvE || ^D, /D&sharp;, \E&flat;, vE
|-
| 4 || 369.23 || E, ^E&flat; || E, /E&flat;
|-
| 5 || 461.54 || vF, ^^E, vvF&sharp; || \F, ^E, vF&sharp;
|-
| 6 || 553.85 || ^F, F&sharp;, vG&flat; || /F, F&sharp;, \G&flat;
|-
| 7 || 646.15 || vG, G&flat;, ^^F&sharp;, vvG&sharp; || \G, G&flat;, ^F&sharp;, vG&flat;
|-
| 8 || 738.46 || ^G, G&sharp;, ^^G&flat;, vvA&flat; || /G, G&sharp;, ^G&flat;, vA&flat;
|-
| 9 || 830.77 || vA, A&flat;, ^^G&sharp;, vvA&sharp; || \A, A&flat;, ^G&sharp;, vA&sharp;
|-
| 10 || 923.08 || ^A, A&sharp; ^^A&flat;, vvB&flat; || /A, A&sharp;, ^A&flat;, vB&flat;
|-
| 11 || 1015.38 || vB, B&flat;, ^^A&sharp; || \B, B&flat;, ^A&sharp;
|-
| 12 || 1107.69 || ^B, ^^B&flat;, vC&flat; || /B, ^B&flat;, \C&flat;
|-
| 13 || 1200.00 || C, vC&sharp;, ^C&flat; || C, \C&sharp;, /C&flat;
|}

===18edo===
GDF notation of [[18edo]] is identical to [[36edo]] subset notation.
{|class="wikitable"
! Steps !! Cents !! Names !! Alt. Names
|- 
| 0 || 0.00 || C || C
|-
| 1 || 66.67 || vC&sharp;, vD&flat;, ^^C || \C&sharp;, \D&flat;, ^C
|-
| 2 || 133.33 || ^C&sharp;, ^D&flat;, vvD || /C&sharp;, /D&flat;, vD
|-
| 3 || 200.00 || D || D
|-
| 4 || 266.67 || vD&sharp;, vE&flat;, ^^D || \D&sharp;, \E&flat;, ^D
|-
| 5 || 333.33 || ^D&sharp;, ^E&flat;, vvE || /D&sharp;, /E&flat;, vE
|-
| 6 || 400.00 || E || E
|-
| 7 || 466.67 || vF, ^^E || \F, ^E
|-
| 8 || 533.33 || ^F, vvF&sharp;, vvG&flat; || /F, vF&sharp;, vG&flat;
|-
| 9 || 600.00 || F&sharp;, G&flat; || F&sharp;, G&flat;
|-
| 10 || 666.67 || vG, ^^F&sharp;, ^^G&flat; || \G, ^F&sharp;, ^G&flat;
|-
| 11 || 733.33 || ^G, vvG&sharp;, vvA&flat; || /G, vG&sharp;, vA&flat;
|-
| 12 || 800.00 || G&sharp;, A&flat; || G&sharp;, A&flat;
|-
| 13 || 866.67 || vA, ^^G&sharp;, ^^A&flat; || \A, ^G&sharp;, ^A&flat;
|-
| 14 || 933.33 || ^A, vvA&sharp;, vvB&flat; || /A, vA&sharp;, vB&flat;
|-
| 15 || 1000.00 || A&sharp;, B&flat; || A&sharp;, B&flat;
|-
| 16 || 1066.67 || vB, ^^A&sharp;, ^^B&flat; || \B, ^A&sharp;, ^B&flat;
|-
| 17 || 1133.33 || ^B, vvC || /B, vC
|-
| 18 || 1200.00 || C || C
|}

===23edo===
GDF notation of [[23edo]] is identical to [[46edo]] subset notation.
{|class="wikitable"
! Steps !! Cents !! Names !! Alt. Names
|- 
| 0 || 0.00 || C || C
|-
| 1 || 52.17 || ^^C, vD&flat; || ^C, \D&flat;
|-
| 2 || 104.35 || vC&sharp;, ^D&flat; || \C&sharp;, /D&flat;
|-
| 3 || 156.52 || ^C&sharp;, vvD || /C&sharp;, vD
|-
| 4 || 208.70 || D || D
|-
| 5 || 260.87 || ^^D, vE&flat; || ^D, \E&flat;
|-
| 6 || 313.04 || vD&sharp;, ^E&flat; || \D&sharp;, /E&flat;
|-
| 7 || 365.22 || ^D&sharp;, vvE || /D&sharp;, vE
|-
| 8 || 417.39 || E || E
|-
| 9 || 469.57 || ^^E, vF || ^E, \F
|-
| 10 || 521.74 || ^F, vvG&flat; || /F, vG&flat;
|-
| 11 || 573.91 || vvF&sharp;, G&flat; || vF&sharp;, G&flat;
|-
| 12 || 626.09 || F&sharp;, ^^G&flat; || F&sharp;, ^G&flat;
|-
| 13 || 678.26 || ^^F&sharp;, vG || ^F&sharp;, \G
|-
| 14 || 730.43 || ^G, vvA&flat; || /G, vA&flat;
|-
| 15 || 782.61 || vvG&sharp;, A&flat; || vG&sharp;, A&flat;
|-
| 16 || 834.78 || G&sharp;, ^^A&flat; || G&sharp;, ^A&flat;
|-
| 17 || 886.96 || ^^G&sharp;, vA || ^G&sharp;, \A
|-
| 18 || 939.13 || ^A, vvB&flat; || /A, vB&flat;
|-
| 19 || 991.30 || B&flat; || B&flat;
|-
| 20 || 1043.48 || ^^B&flat;, vC&flat; || ^B&flat;, \C&flat;
|-
| 21 || 1095.65 || vB, ^C&flat; || \B, /C&flat;
|-
| 22 || 1147.83 || ^B, vvC || \B, vC
|-
| 23 || 1200.00 || C || C
|}