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.
Reasonable tunings would be between 41edo and 46edo. 87edo makes for a very recommendable option.
See Gamelismic clan #Rodan for more information.
Interval chain
In the following table, odd harmonics and subharmonics 1–21 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 is listed in the notation guide for rank-2 pergens under pergen #8, (P8, P5/3). The generator is an upmajor 2nd. The enharmonic unison is a trudminor 2nd. Thus three ups equals a diatonic semitone, and three generators equals a perfect 5th. In rodan in particular, ^1 equals ~81/80 and ~64/63, and ^^1 equals ~33/32 and ~1053/1024.
| Ratio | Nominal | Example |
|---|---|---|
| 3/2 | Perfect fifth | C−G |
| 5/4 | Downmajor third | C−vE |
| 7/4 | Downminor seventh | C−vB♭ |
| 11/8 | Dup fourth | C−^^F |
| 13/8 | Dupminor sixth | C−^^A♭ |
Rodan's notation has much in common with that for 41edo and 46edo, since both edos map a minor 2nd to three edosteps. It also resembles the notation for cassandra. All four notations notate the slendric tetrad (1–8/7–21/16–3/2) on C as C–^D–vF–G, and all four notations notate 5/4, 7/4, 11/8, and 13/8 as in the table above. But the notations diverge for other intervals, such as 11/10.
Chords
Scales
Tunings
Tuning spectrum
| Edo generator |
Eigenmonzo (unchanged-interval) |
Generator (¢) | Comments |
|---|---|---|---|
| 7/4 | 231.174 | ||
| 17/13 | 232.214 | ||
| 7/6 | 232.282 | ||
| 21\36 | 233.333 | 36cfg val, lower bound of 7- and 9-odd-limit diamond monotone | |
| 9/7 | 233.583 | ||
| 3/2 | 233.985 | ||
| 24\41 | 234.146 | Lower bound of 11- to 17-odd-limit diamond monotone | |
| 15/14 | 234.259 | ||
| 7/5 | 234.305 | 7- and 9-odd-limit minimax | |
| 11/9 | 234.347 | 11-odd-limit minimax | |
| 75\128 | 234.375 | 128g val | |
| 13/9 | 234.407 | 13- and 15-odd-limit minimax | |
| 15/8 | 234.413 | ||
| 11/6 | 234.415 | ||
| 15/11 | 234.453 | ||
| 13/12 | 234.457 | ||
| 15/13 | 234.470 | ||
| 51\87 | 234.483 | ||
| 5/4 | 234.489 | 5-odd-limit minimax | |
| 11/10 | 234.500 | ||
| 13/10 | 234.507 | ||
| 11/8 | 234.514 | ||
| 13/8 | 234.521 | ||
| 13/11 | 234.532 | ||
| 17/9 | 234.575 | 17-odd-limit minimax | |
| 17/15 | 234.583 | ||
| 78\133 | 234.586 | ||
| 5/3 | 234.597 | ||
| 17/10 | 234.631 | ||
| 17/12 | 234.652 | ||
| 17/16 | 234.752 | ||
| 9/5 | 234.764 | ||
| 27\46 | 234.783 | Upper bound of 11- to 17-odd-limit diamond monotone | |
| 11/7 | 234.792 | ||
| 3\5 | 240.000 | 5f val, upper bound of 7- and 9-odd-limit diamond monotone |
Music
- Pianodactyl (archived 2010) – SoundCloud | detail | play – rodan[26] in 87edo tuning