Rodan
Rodan is a temperament of the gamelismic clan and one of the notable extensions to slendric. Like slendric, it is generated by an 8/7, three of which gives 3/2. In addition it finds harmonic 5 at +17 generator steps, and the second defining comma for it is 245/243.
A reasonable tuning would be between 41edo and 46edo. 87edo makes for a very recommendable option.
See Gamelismic clan #Rodan for more information.
Interval chain
Prime harmonics are in bold.
| # | Cents* | Approximate Ratios | |
|---|---|---|---|
| 13-limit | 17-limit Extension | ||
| 0 | 0.000 | 1/1 | |
| 1 | 234.482 | 8/7 | |
| 2 | 468.964 | 21/16 | 17/13 |
| 3 | 703.446 | 3/2 | |
| 4 | 937.929 | 12/7 | |
| 5 | 1172.411 | 63/32, 160/81 | |
| 6 | 206.893 | 9/8 | |
| 7 | 441.375 | 9/7 | 22/17 |
| 8 | 675.857 | 40/27 | |
| 9 | 910.339 | 22/13, 27/16, 33/28 | |
| 10 | 1144.821 | 27/14, 35/18, 64/33 | 33/17 |
| 11 | 179.304 | 10/9 | |
| 12 | 413.786 | 14/11, 33/26, 80/63 | |
| 13 | 648.268 | 16/11 | |
| 14 | 882.750 | 5/3 | |
| 15 | 1117.232 | 21/11, 40/21 | |
| 16 | 151.714 | 12/11 | |
| 17 | 386.196 | 5/4 | |
| 18 | 620.679 | 10/7 | |
| 19 | 855.161 | 18/11, 64/39 | 28/17 |
| 20 | 1089.643 | 15/8 | 32/17 |
| 21 | 124.125 | 14/13, 15/14 | |
| 22 | 358.607 | 16/13, 27/22 | 21/17 |
| 23 | 593.088 | 45/32 | 24/17 |
| 24 | 827.570 | 21/13 | |
| 25 | 1062.052 | 24/13 | |
* in 13-limit POTE tuning
Notation
A notation for rodan can be devised by identifying what is common in 41edo and 46edo. Below is one using circle-of-fifths notation with an additional module of arrow accidentals to represent the syntonic~septimal comma. This notation looks exactly like that for cassandra, but it is peculiar to rodan that three comma steps make up a diatonic semitone, and thus the slendric tetrad (1-8/7-21/16-3/2) on C is C-^D-vF-G, for example.
| Ratio | Nominal | Example |
|---|---|---|
| 3/2 | Perfect fifth | C-G |
| 5/4 | Down major third | C-vE |
| 7/4 | Down minor seventh | C-vBb |
| 11/8 | Double-up fourth | C-^^F |
| 13/8 | Double-up minor sixth | C-^^Ab |
Chords
- Main article: Chords of rodan
Scales
Tuning spectrum
Gencom: [2 8/7; 154/153 196/195 245/243 273/272 364/363]
Gencom mapping: [⟨1 1 -1 3 6 8 8], ⟨0 3 17 -1 -13 -22 -20]]
| Edo Generator |
Eigenmonzo (Unchanged-interval) |
Fifth (¢) |
Comments |
|---|---|---|---|
| 8/7 | 693.522 | ||
| 17/13 | 696.642 | ||
| 7/6 | 699.847 | ||
| 21\36 | 700.000 | Lower bound of 7- and 9-odd-limit diamond monotone | |
| 9/7 | 700.750 | ||
| 4/3 | 701.955 | ||
| 24\41 | 702.439 | Lower bound of 11- to 17-odd-limit diamond monotone | |
| 15/14 | 702.778 | ||
| 7/5 | 702.915 | 7- and 9-odd-limit minimax | |
| 11/9 | 703.041 | 11-odd-limit minimax | |
| 75\128 | 703.125 | ||
| 18/13 | 703.220 | 13- and 15-odd-limit minimax | |
| 16/15 | 703.240 | ||
| 12/11 | 703.244 | ||
| 15/11 | 703.359 | ||
| 13/12 | 703.371 | ||
| 15/13 | 703.410 | ||
| 51\87 | 703.448 | ||
| 5/4 | 703.467 | 5-odd-limit minimax | |
| 11/10 | 703.500 | ||
| 13/10 | 703.522 | ||
| 11/8 | 703.542 | ||
| 16/13 | 703.564 | ||
| 13/11 | 703.597 | ||
| 18/17 | 703.726 | 17-odd-limit minimax | |
| 17/15 | 703.748 | ||
| 78\133 | 703.759 | ||
| 6/5 | 703.791 | ||
| 20/17 | 703.894 | ||
| 24/17 | 703.956 | ||
| 17/16 | 704.257 | ||
| 10/9 | 704.292 | ||
| 27\46 | 704.348 | Upper bound of 11- to 17-odd-limit diamond monotone | |
| 14/11 | 704.377 | ||
| 3\5 | 720.000 | Upper bound of 7- and 9-odd-limit diamond monotone |
Music
- Pianodactyl (archived 2010) – SoundCloud | detail | play – rodan[26] in 87edo tuning