Rodan: Difference between revisions

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! Edo<br>generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! Fifth<br>(¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|  
|  
| 7/4
| 7/4
| 693.522
| 231.174
|  
|  
|-
|-
|  
|  
| 17/13
| 17/13
| 696.642
| 232.214
|  
|  
|-
|-
|  
|  
| 7/6
| 7/6
| 699.847
| 232.282
|  
|  
|-
|-
| [[36edo|21\36]]
| [[36edo|21\36]]
|  
|  
| 700.000
| 233.333
| Lower bound of 7- and 9-odd-limit diamond monotone
| Lower bound of 7- and 9-odd-limit diamond monotone
|-
|-
|  
|  
| 9/7
| 9/7
| 700.750
| 233.583
|  
|  
|-
|-
|  
|  
| 3/2
| 3/2
| 701.955
| 233.985
|  
|  
|-
|-
| [[41edo|24\41]]
| [[41edo|24\41]]
|  
|  
| 702.439
| 234.146
| Lower bound of 11- to 17-odd-limit diamond monotone
| Lower bound of 11- to 17-odd-limit diamond monotone
|-
|-
|  
|  
| 15/14
| 15/14
| 702.778
| 234.259
|  
|  
|-
|-
|  
|  
| 7/5
| 7/5
| 702.915
| 234.305
| 7- and 9-odd-limit minimax
| 7- and 9-odd-limit minimax
|-
|-
|  
|  
| 11/9
| 11/9
| 703.041
| 234.347
| 11-odd-limit minimax
| 11-odd-limit minimax
|-
|-
| [[128edo|75\128]]
| [[128edo|75\128]]
|  
|  
| 703.125
| 234.375
|
|
|-
|-
|  
|  
| 13/9
| 13/9
| 703.220
| 234.407
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
|-
|-
|  
|  
| 15/8
| 15/8
| 703.240
| 234.413
|  
|  
|-
|-
|  
|  
| 11/6
| 11/6
| 703.244
| 234.415
|  
|  
|-
|-
|  
|  
| 15/11
| 15/11
| 703.359
| 234.453
|  
|  
|-
|-
|  
|  
| 13/12
| 13/12
| 703.371
| 234.457
|  
|  
|-
|-
|  
|  
| 15/13
| 15/13
| 703.410
| 234.470
|  
|  
|-
|-
| [[87edo|51\87]]
| [[87edo|51\87]]
|  
|  
| 703.448
| 234.483
|  
|  
|-
|-
|  
|  
| 5/4
| 5/4
| 703.467
| 234.489
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
|  
|  
| 11/10
| 11/10
| 703.500
| 234.500
|  
|  
|-
|-
|  
|  
| 13/10
| 13/10
| 703.522
| 234.507
|  
|  
|-
|-
|  
|  
| 11/8
| 11/8
| 703.542
| 234.514
|  
|  
|-
|-
|  
|  
| 13/8
| 13/8
| 703.564
| 234.521
|  
|  
|-
|-
|  
|  
| 13/11
| 13/11
| 703.597
| 234.532
|  
|  
|-
|-
|  
|  
| 17/9
| 17/9
| 703.726
| 234.575
| 17-odd-limit minimax
| 17-odd-limit minimax
|-
|-
|  
|  
| 17/15
| 17/15
| 703.748
| 234.583
|  
|  
|-
|-
| [[133edo|78\133]]
| [[133edo|78\133]]
|  
|  
| 703.759
| 234.586
|  
|  
|-
|-
|  
|  
| 5/3
| 5/3
| 703.791
| 234.597
|  
|  
|-
|-
|  
|  
| 17/10
| 17/10
| 703.894
| 234.631
|  
|  
|-
|-
|  
|  
| 17/12
| 17/12
| 703.956
| 234.652
|  
|  
|-
|-
|  
|  
| 17/16
| 17/16
| 704.257
| 234.752
|  
|  
|-
|-
|  
|  
| 9/5
| 9/5
| 704.292
| 234.764
|  
|  
|-
|-
| [[46edo|27\46]]
| [[46edo|27\46]]
|  
|  
| 704.348
| 234.783
| Upper bound of 11- to 17-odd-limit diamond monotone
| Upper bound of 11- to 17-odd-limit diamond monotone
|-
|-
|  
|  
| 11/7
| 11/7
| 704.377
| 234.792
|  
|  
|-
|-
| [[5edo|3\5]]
| [[5edo|3\5]]
|  
|  
| 720.000
| 240.000
| Upper bound of 7- and 9-odd-limit diamond monotone
| Upper bound of 7- and 9-odd-limit diamond monotone
|}
|}

Revision as of 11:46, 2 January 2025

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

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 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, med, 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

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 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
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 Upper bound of 7- and 9-odd-limit diamond monotone

Music

Gene Ward Smith
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