Rodan

Rodan is an extension of the slendric temperament, which divides 3/2 into three equal intervals representing 8/7 (tempering out the gamelisma, 1029/1024), to the full 7-limit such that 17 of these generators stack to reach the interval class of the 5th harmonic (5/4). It tempers out 245/243, making it a sensamagic temperament, so that 5/3 is divided into two intervals of 9/7; and it tempers out 5120/5103, making it also a hemifamity temperament, so that 9/8 stacks thrice into 10/7.

Unlike mothra, which flattens the fifth to a meantone fifth, the fifth of rodan is slightly sharp of just, ranging from that of 41edo to that of 46edo. As a result, the diatonic minor second is compressed, and the interval known as the quark, which represents 49/48, 64/63, and in rodan also 81/80, is even smaller than it is in tunings of slendric with a nearly just fifth. This entails that the MOS scales of rodan cluster even more strongly around 5edo, although this can be thought of as an advantage in that it simplifies the conceptualization of rodan's inventory of intervals; rather than directly using MOS scales, which are either extremely imbalanced or overly large, an approach to rodan may involve picking and choosing which intervals from each pentatonic category to keep in the scale.

As for further extensions, slendric temperaments often find 55/32 at 4 generator steps (tempering out 385/384 and 441/440), giving new interpretations to the quark as 55/54 and 56/55; 55/32 is particularly accurate in the tuning subrange appropriate for rodan, and so harmonic 11 can easily be found at -13 generator steps. The diatonic minor third (32/27) in hemifamity temperaments represents the square root of 7/5, for which 13/11 is a good interpretation (tempering out 352/351 and 847/845), which turns out to place harmonic 13 at -22 generator steps. Finally, 17/13 is a good interpretation of the slendric subfourth comprising two generators, otherwise equated to 21/16 (tempering out 273/272 and 833/832), and this places harmonic 17 at -20 generator steps. Thus proceeds the canonical extension of rodan out to the 17-limit.

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.

Rodan nomenclature
for selected intervals

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

Main article: Chords of rodan

Scales

• Radon5
• Radon11
• Radon16
• Rodan26opt
• Rodan31opt
• Rodan41opt

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

Gene Ward Smith

• Pianodactyl (archived 2010) – SoundCloud | detail | play – rodan[26] in 87edo tuning