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 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.
The main goal of GDF notation is to describe intervals closer to what they actually "are," rather than notation systems like antidiatonic which make minor or neutral seconds look like major seconds.
The idea of GDF notation is to maintain some semblance of a chain of fifths as used in other approachable notation systems. To do this, it relies instead on the 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]\displaystyle{ 3^+ \cdot 3^- = 9 }[/math].
With split fifths, a chain of fifths would look something like this:
| G♭ | ^D♭ | A♭ | ^E♭ | B♭ | ^F | C | ^G | D | ^A | E | ^B | F♯ | ^C♯ |
| vD♭ | vE♭ | vF | vG | vA | vB | vC♯ |
This also implies a converse system where the other set of fifths are split:
| ^G♭ | D♭ | ^A♭ | E♭ | ^B♭ | F | ^C | G | ^D | A | ^E | B | ^F♯ | C♯ |
| vG♭ | vA♭ | vB♭ | vC | vD | vE | vF♯ |
Tables
For EDOs, it's usually the case that the GDF notation of [math]\displaystyle{ n }[/math]-edo is identical to the subset notation taken from [math]\displaystyle{ 2n }[/math]-edo.
Please note that when GDF for EDOs, the symbols ^ and v do not represent single edosteps as they do in ups and downs notation. Instead, they refer to half-edosteps; full edosteps are represented by ^^ and vv.
13edo
GDF notation of 13edo is identical to 26edo subset notation.
| Steps | Cents | Name(s) |
|---|---|---|
| 0 | 0.00 | C, vC♯ |
| 1 | 92.31 | ^C♯, vD♭ |
| 2 | 184.62 | D, ^D♭ |
| 3 | 276.92 | ^D♯, vE♭ |
| 4 | 369.23 | E, ^E♭ |
| 5 | 461.54 | ^E♯, vF |
| 6 | 553.85 | F♯, ^F |
| 7 | 646.15 | G♭, vG |
| 8 | 738.46 | G♯, ^G |
| 9 | 830.77 | A♭, vA |
| 10 | 923.08 | A♯, ^A |
| 11 | 1015.38 | B♭, vB |
| 12 | 1107.69 | ^B, vC♭ |
| 13 | 1200.00 | C, vC♯ |
18edo
GDF notation of 18edo is identical to 36edo subset notation.
| Steps | Cents | Name(s) |
|---|---|---|
| 0 | 0.00 | C |
| 1 | 66.67 | vC♯, vD♭ |
| 2 | 133.33 | ^C♯, ^D♭ |
| 3 | 200.00 | D |
| 4 | 266.67 | vD♯, vE♭ |
| 5 | 333.33 | ^D♯, ^E♭ |
| 6 | 400.00 | E |
| 7 | 466.67 | vF |
| 8 | 533.33 | ^F |
| 9 | 600.00 | F♯, G♭ |
| 10 | 666.67 | vG |
| 11 | 733.33 | ^G |
| 12 | 800.00 | G♯, A♭ |
| 13 | 866.67 | vA |
| 14 | 933.33 | ^A |
| 15 | 1000.00 | A♯, B♭ |
| 16 | 1066.67 | vB |
| 17 | 1133.33 | ^B |
| 18 | 1200.00 | C |
23edo
GDF notation of 23edo is identical to 46edo subset notation.
| Steps | Cents | Name(s) |
|---|---|---|
| 0 | 0.00 | C |
| 1 | 52.17 | ^^C, vD♭ |
| 2 | 104.35 | vC♯, ^D♭ |
| 3 | 156.52 | ^C♯, vvD |
| 4 | 208.70 | D |
| 5 | 260.87 | ^^D, vE♭ |
| 6 | 313.04 | vD♯, ^E♭ |
| 7 | 365.22 | ^D♯, vvE |
| 8 | 417.39 | E |
| 9 | 469.57 | ^^E, vF |
| 10 | 521.74 | ^F, vvG♭ |
| 11 | 573.91 | vvF♯, G♭ |
| 12 | 626.09 | F♯, ^^G♭ |
| 13 | 678.26 | ^^F♯, vG |
| 14 | 730.43 | ^G, vvA♭ |
| 15 | 782.61 | vvG♯, A♭ |
| 16 | 834.78 | G♯, ^^A♭ |
| 17 | 886.96 | ^^G♯, vA |
| 18 | 939.13 | ^A, vvB♭ |
| 19 | 991.30 | B♭ |
| 20 | 1043.48 | ^^B♭, vC♭ |
| 21 | 1095.65 | vB, ^C♭ |
| 22 | 1147.83 | ^B |
| 23 | 1200.00 | C |