User:TromboneBoi9/Generalized Dual-Fifth Notation: Difference between revisions
Jump to navigation
Jump to search
TromboneBoi9 (talk | contribs) Created page with "'''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 a..." |
TromboneBoi9 (talk | contribs) reorganization and added alt names columns to tables |
||
| Line 1: | Line 1: | ||
'''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]]. | '''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 | 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. | ||
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>. | 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: | With split fifths, a chain of fifths would look something like this: | ||
| Line 35: | Line 38: | ||
| vG♭ || vA♭ || vB♭ || vC || vD || vE || vF♯ | | vG♭ || vA♭ || vB♭ || vC || vD || vE || vF♯ | ||
|} | |} | ||
===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== | ==Tables== | ||
===13edo=== | ===13edo=== | ||
GDF notation of [[13edo]] is identical to [[26edo]] subset notation. | GDF notation of [[13edo]] is identical to [[26edo]] subset notation. | ||
{|class="wikitable" | {|class="wikitable" | ||
! Steps !! Cents !! | ! Steps !! Cents !! Names !! Alt. Names | ||
|- | |- | ||
| 0 || 0.00 || C, vC♯ | | 0 || 0.00 || C, vC♯ ^C♭ || C, \C♯, /C♭ | ||
|- | |- | ||
| 1 || 92.31 || ^C♯, vD♭ | | 1 || 92.31 || ^^C, ^C♯, vD♭, vvD || ^C, /C♯, \D♭, vD | ||
|- | |- | ||
| 2 || 184.62 || D, ^D♭ | | 2 || 184.62 || D, ^D♭, vD♯ || D, /D♭, \D♯ | ||
|- | |- | ||
| 3 || 276.92 || ^D♯, vE♭ | | 3 || 276.92 || ^^D, ^D♯, vE♭, vvE || ^D, /D♯, \E♭, vE | ||
|- | |- | ||
| 4 || 369.23 || E, ^E♭ | | 4 || 369.23 || E, ^E♭ || E, /E♭ | ||
|- | |- | ||
| 5 || 461.54 || ^E♯, vF | | 5 || 461.54 || vF, ^^E, vvF♯ || \F, ^E, vF♯ | ||
|- | |- | ||
| 6 || 553.85 || F♯, | | 6 || 553.85 || ^F, F♯, vG♭ || /F, F♯, \G♭ | ||
|- | |- | ||
| 7 || 646.15 || G♭, vG | | 7 || 646.15 || vG, G♭, ^^F♯, vvG♯ || \G, G♭, ^F♯, vG♭ | ||
|- | |- | ||
| 8 || 738.46 || G♯, ^G | | 8 || 738.46 || ^G, G♯, ^^G♭, vvA♭ || /G, G♯, ^G♭, vA♭ | ||
|- | |- | ||
| 9 || 830.77 || A♭, vA | | 9 || 830.77 || vA, A♭, ^^G♯, vvA♯ || \A, A♭, ^G♯, vA♯ | ||
|- | |- | ||
| 10 || 923.08 || A♯, ^A | | 10 || 923.08 || ^A, A♯ ^^A♭, vvB♭ || /A, A♯, ^A♭, vB♭ | ||
|- | |- | ||
| 11 || 1015.38 || B♭, | | 11 || 1015.38 || vB, B♭, ^^A♯ || \B, B♭, ^A♯ | ||
|- | |- | ||
| 12 || 1107.69 || ^B, vC♭ | | 12 || 1107.69 || ^B, ^^B♭, vC♭ || /B, ^B♭, \C♭ | ||
|- | |- | ||
| 13 || 1200.00 || C, vC♯ | | 13 || 1200.00 || C, vC♯, ^C♭ || C, \C♯, /C♭ | ||
|} | |} | ||
| Line 78: | Line 86: | ||
GDF notation of [[18edo]] is identical to [[36edo]] subset notation. | GDF notation of [[18edo]] is identical to [[36edo]] subset notation. | ||
{|class="wikitable" | {|class="wikitable" | ||
! Steps !! Cents !! | ! Steps !! Cents !! Names !! Alt. Names | ||
|- | |- | ||
| 0 || 0.00 || C | | 0 || 0.00 || C || C | ||
|- | |- | ||
| 1 || 66.67 || vC♯, vD♭ | | 1 || 66.67 || vC♯, vD♭, ^^C || \C♯, \D♭, ^C | ||
|- | |- | ||
| 2 || 133.33 || ^C♯, ^D♭ | | 2 || 133.33 || ^C♯, ^D♭, vvD || /C♯, /D♭, vD | ||
|- | |- | ||
| 3 || 200.00 || D | | 3 || 200.00 || D || D | ||
|- | |- | ||
| 4 || 266.67 || vD♯, vE♭ | | 4 || 266.67 || vD♯, vE♭, ^^D || \D♯, \E♭, ^D | ||
|- | |- | ||
| 5 || 333.33 || ^D♯, ^E♭ | | 5 || 333.33 || ^D♯, ^E♭, vvE || /D♯, /E♭, vE | ||
|- | |- | ||
| 6 || 400.00 || E | | 6 || 400.00 || E || E | ||
|- | |- | ||
| 7 || 466.67 || vF | | 7 || 466.67 || vF, ^^E || \F, ^E | ||
|- | |- | ||
| 8 || 533.33 || ^F | | 8 || 533.33 || ^F, vvF♯, vvG♭ || /F, vF♯, vG♭ | ||
|- | |- | ||
| 9 || 600.00 || F♯, G♭ | | 9 || 600.00 || F♯, G♭ || F♯, G♭ | ||
|- | |- | ||
| 10 || 666.67 || vG | | 10 || 666.67 || vG, ^^F♯, ^^G♭ || \G, ^F♯, ^G♭ | ||
|- | |- | ||
| 11 || 733.33 || ^G | | 11 || 733.33 || ^G, vvG♯, vvA♭ || /G, vG♯, vA♭ | ||
|- | |- | ||
| 12 || 800.00 || G♯, A♭ | | 12 || 800.00 || G♯, A♭ || G♯, A♭ | ||
|- | |- | ||
| 13 || 866.67 || vA | | 13 || 866.67 || vA, ^^G♯, ^^A♭ || \A, ^G♯, ^A♭ | ||
|- | |- | ||
| 14 || 933.33 || ^A | | 14 || 933.33 || ^A, vvA♯, vvB♭ || /A, vA♯, vB♭ | ||
|- | |- | ||
| 15 || 1000.00 || A♯, B♭ | | 15 || 1000.00 || A♯, B♭ || A♯, B♭ | ||
|- | |- | ||
| 16 || 1066.67 || vB | | 16 || 1066.67 || vB, ^^A♯, ^^B♭ || \B, ^A♯, ^B♭ | ||
|- | |- | ||
| 17 || 1133.33 || ^B | | 17 || 1133.33 || ^B, vvC || /B, vC | ||
|- | |- | ||
| 18 || 1200.00 || C | | 18 || 1200.00 || C || C | ||
|} | |} | ||
| Line 122: | Line 130: | ||
GDF notation of [[23edo]] is identical to [[46edo]] subset notation. | GDF notation of [[23edo]] is identical to [[46edo]] subset notation. | ||
{|class="wikitable" | {|class="wikitable" | ||
! Steps !! Cents !! | ! Steps !! Cents !! Names !! Alt. Names | ||
|- | |- | ||
| 0 || 0.00 || C | | 0 || 0.00 || C || C | ||
|- | |- | ||
| 1 || 52.17 || ^^C, vD♭ | | 1 || 52.17 || ^^C, vD♭ || ^C, \D♭ | ||
|- | |- | ||
| 2 || 104.35 || vC♯, ^D♭ | | 2 || 104.35 || vC♯, ^D♭ || \C♯, /D♭ | ||
|- | |- | ||
| 3 || 156.52 || ^C♯, vvD | | 3 || 156.52 || ^C♯, vvD || /C♯, vD | ||
|- | |- | ||
| 4 || 208.70 || D | | 4 || 208.70 || D || D | ||
|- | |- | ||
| 5 || 260.87 || ^^D, vE♭ | | 5 || 260.87 || ^^D, vE♭ || ^D, \E♭ | ||
|- | |- | ||
| 6 || 313.04 || vD♯, ^E♭ | | 6 || 313.04 || vD♯, ^E♭ || \D♯, /E♭ | ||
|- | |- | ||
| 7 || 365.22 || ^D♯, vvE | | 7 || 365.22 || ^D♯, vvE || /D♯, vE | ||
|- | |- | ||
| 8 || 417.39 || E | | 8 || 417.39 || E || E | ||
|- | |- | ||
| 9 || 469.57 || ^^E, vF | | 9 || 469.57 || ^^E, vF || ^E, \F | ||
|- | |- | ||
| 10 || 521.74 || ^F, vvG♭ | | 10 || 521.74 || ^F, vvG♭ || /F, vG♭ | ||
|- | |- | ||
| 11 || 573.91 || vvF♯, G♭ | | 11 || 573.91 || vvF♯, G♭ || vF♯, G♭ | ||
|- | |- | ||
| 12 || 626.09 || F♯, ^^G♭ | | 12 || 626.09 || F♯, ^^G♭ || F♯, ^G♭ | ||
|- | |- | ||
| 13 || 678.26 || ^^F♯, vG | | 13 || 678.26 || ^^F♯, vG || ^F♯, \G | ||
|- | |- | ||
| 14 || 730.43 || ^G, vvA♭ | | 14 || 730.43 || ^G, vvA♭ || /G, vA♭ | ||
|- | |- | ||
| 15 || 782.61 || vvG♯, A♭ | | 15 || 782.61 || vvG♯, A♭ || vG♯, A♭ | ||
|- | |- | ||
| 16 || 834.78 || G♯, ^^A♭ | | 16 || 834.78 || G♯, ^^A♭ || G♯, ^A♭ | ||
|- | |- | ||
| 17 || 886.96 || ^^G♯, vA | | 17 || 886.96 || ^^G♯, vA || ^G♯, \A | ||
|- | |- | ||
| 18 || 939.13 || ^A, vvB♭ | | 18 || 939.13 || ^A, vvB♭ || /A, vB♭ | ||
|- | |- | ||
| 19 || 991.30 || B♭ | | 19 || 991.30 || B♭ || B♭ | ||
|- | |- | ||
| 20 || 1043.48 || ^^B♭, vC♭ | | 20 || 1043.48 || ^^B♭, vC♭ || ^B♭, \C♭ | ||
|- | |- | ||
| 21 || 1095.65 || vB, ^C♭ | | 21 || 1095.65 || vB, ^C♭ || \B, /C♭ | ||
|- | |- | ||
| 22 || 1147.83 || ^B | | 22 || 1147.83 || ^B, vvC || \B, vC | ||
|- | |- | ||
| 23 || 1200.00 || C | | 23 || 1200.00 || C || C | ||
|} | |} | ||
Latest revision as of 08:21, 16 November 2025
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.
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 as used in other approachable notation systems. To do so, 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]. 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:
| 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♯ |
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]\displaystyle{ n }[/math]-edo is identical to the subset notation loaned from the ups and downs notation of [math]\displaystyle{ 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 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.
| Steps | Cents | Names | Alt. Names |
|---|---|---|---|
| 0 | 0.00 | C, vC♯ ^C♭ | C, \C♯, /C♭ |
| 1 | 92.31 | ^^C, ^C♯, vD♭, vvD | ^C, /C♯, \D♭, vD |
| 2 | 184.62 | D, ^D♭, vD♯ | D, /D♭, \D♯ |
| 3 | 276.92 | ^^D, ^D♯, vE♭, vvE | ^D, /D♯, \E♭, vE |
| 4 | 369.23 | E, ^E♭ | E, /E♭ |
| 5 | 461.54 | vF, ^^E, vvF♯ | \F, ^E, vF♯ |
| 6 | 553.85 | ^F, F♯, vG♭ | /F, F♯, \G♭ |
| 7 | 646.15 | vG, G♭, ^^F♯, vvG♯ | \G, G♭, ^F♯, vG♭ |
| 8 | 738.46 | ^G, G♯, ^^G♭, vvA♭ | /G, G♯, ^G♭, vA♭ |
| 9 | 830.77 | vA, A♭, ^^G♯, vvA♯ | \A, A♭, ^G♯, vA♯ |
| 10 | 923.08 | ^A, A♯ ^^A♭, vvB♭ | /A, A♯, ^A♭, vB♭ |
| 11 | 1015.38 | vB, B♭, ^^A♯ | \B, B♭, ^A♯ |
| 12 | 1107.69 | ^B, ^^B♭, vC♭ | /B, ^B♭, \C♭ |
| 13 | 1200.00 | C, vC♯, ^C♭ | C, \C♯, /C♭ |
18edo
GDF notation of 18edo is identical to 36edo subset notation.
| Steps | Cents | Names | Alt. Names |
|---|---|---|---|
| 0 | 0.00 | C | C |
| 1 | 66.67 | vC♯, vD♭, ^^C | \C♯, \D♭, ^C |
| 2 | 133.33 | ^C♯, ^D♭, vvD | /C♯, /D♭, vD |
| 3 | 200.00 | D | D |
| 4 | 266.67 | vD♯, vE♭, ^^D | \D♯, \E♭, ^D |
| 5 | 333.33 | ^D♯, ^E♭, vvE | /D♯, /E♭, vE |
| 6 | 400.00 | E | E |
| 7 | 466.67 | vF, ^^E | \F, ^E |
| 8 | 533.33 | ^F, vvF♯, vvG♭ | /F, vF♯, vG♭ |
| 9 | 600.00 | F♯, G♭ | F♯, G♭ |
| 10 | 666.67 | vG, ^^F♯, ^^G♭ | \G, ^F♯, ^G♭ |
| 11 | 733.33 | ^G, vvG♯, vvA♭ | /G, vG♯, vA♭ |
| 12 | 800.00 | G♯, A♭ | G♯, A♭ |
| 13 | 866.67 | vA, ^^G♯, ^^A♭ | \A, ^G♯, ^A♭ |
| 14 | 933.33 | ^A, vvA♯, vvB♭ | /A, vA♯, vB♭ |
| 15 | 1000.00 | A♯, B♭ | A♯, B♭ |
| 16 | 1066.67 | vB, ^^A♯, ^^B♭ | \B, ^A♯, ^B♭ |
| 17 | 1133.33 | ^B, vvC | /B, vC |
| 18 | 1200.00 | C | C |
23edo
GDF notation of 23edo is identical to 46edo subset notation.
| Steps | Cents | Names | Alt. Names |
|---|---|---|---|
| 0 | 0.00 | C | C |
| 1 | 52.17 | ^^C, vD♭ | ^C, \D♭ |
| 2 | 104.35 | vC♯, ^D♭ | \C♯, /D♭ |
| 3 | 156.52 | ^C♯, vvD | /C♯, vD |
| 4 | 208.70 | D | D |
| 5 | 260.87 | ^^D, vE♭ | ^D, \E♭ |
| 6 | 313.04 | vD♯, ^E♭ | \D♯, /E♭ |
| 7 | 365.22 | ^D♯, vvE | /D♯, vE |
| 8 | 417.39 | E | E |
| 9 | 469.57 | ^^E, vF | ^E, \F |
| 10 | 521.74 | ^F, vvG♭ | /F, vG♭ |
| 11 | 573.91 | vvF♯, G♭ | vF♯, G♭ |
| 12 | 626.09 | F♯, ^^G♭ | F♯, ^G♭ |
| 13 | 678.26 | ^^F♯, vG | ^F♯, \G |
| 14 | 730.43 | ^G, vvA♭ | /G, vA♭ |
| 15 | 782.61 | vvG♯, A♭ | vG♯, A♭ |
| 16 | 834.78 | G♯, ^^A♭ | G♯, ^A♭ |
| 17 | 886.96 | ^^G♯, vA | ^G♯, \A |
| 18 | 939.13 | ^A, vvB♭ | /A, vB♭ |
| 19 | 991.30 | B♭ | B♭ |
| 20 | 1043.48 | ^^B♭, vC♭ | ^B♭, \C♭ |
| 21 | 1095.65 | vB, ^C♭ | \B, /C♭ |
| 22 | 1147.83 | ^B, vvC | \B, vC |
| 23 | 1200.00 | C | C |