Rodan

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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.

Rodan nomenclature
for selected intervals
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

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

Gene Ward Smith