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'''Rodan''' is a [[regular temperament|temperament]] of the [[gamelismic clan]] and one of the notable [[extension]]s to [[slendric]]. Like slendric, it is generated by an [[8/7]], three of which gives [[3/2]]. In addition it finds [[5/1|harmonic 5]] at +17 generator steps, and the second defining comma for it is [[245/243]].  
{{Infobox Regtemp
| Title = Rodan
| Subgroups = 2.3.5.7, 2.3.5.7.11
| Comma basis = [[245/243]], [[1029/1024]] (7-limit); <br> [[245/243]], [[385/384]], [[441/440]] (11-limit)
| Edo join 1 = 41 | Edo join 2 = 46
| Generator = 8/7 | Generator tuning = 234.4 | Optimization method = CWE
| MOS scales = [[1L 4s]], [[5L 1s]], [[5L 6s]], …, [[5L 36s]], [[41L 5s]]
| Mapping = 1; 3 17 -1 -13
| Pergen = (P8, P5/3)
| Odd limit 1 = 9 | Mistuning 1 = 5.05 | Complexity 1 = 41
| Odd limit 2 = (2.3.5.7.11) 21 | Mistuning 2 = 5.34 | Complexity 2 = 87
}}


Reasonable tunings would be between [[41edo]] and [[46edo]]. [[87edo]] makes for a very recommendable option.  
'''Rodan''' is one of the notable [[extension]]s of the [[slendric]] [[regular temperament|temperament]], which divides [[3/2]] into three equal intervals representing [[8/7]] ([[tempering out]] the gamelisma, [[1029/1024]]), reaching the full [[7-limit]] such that 17 of these [[generators]] [[stacking|stack]] to reach the interval class of the [[5/1|5th harmonic]]. It tempers out [[245/243]], making it a [[sensamagic clan|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 temperaments|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]] (with [[87edo]] being an essentially optimal tuning). As a result, the [[256/243|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 scale]]s of rodan [[cluster MOS|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 (see [[#As a detemperament of 5et]]). 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 can be elucidated by [[S-expression]]s, rodan is very much an "opposed counterpart" to mothra: the basic equivalence of slendric tempers S7 (49/48) = S8 (64/63), and mothra proceeds to equate it to S6 ([[36/35]]) as well; meanwhile, rodan extends the equivalence in the opposite direction to add S9 (81/80) to it, making it one of the five [[rank-2 temperament]]s definable by equating three adjacent square superparticulars.
 
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 [[11/1|harmonic 11]] can easily be found at -13 generator steps. It is also worth mentioning that this equates the diatonic major third to [[14/11]], tempering out [[896/891]]. A [[2.3.7.11 subgroup]] version of rodan, known as ''radon'', uses exclusively this mapping and forgoes interpreting the 5th harmonic.
 
Toward the [[13-limit]], 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 [[13/1|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 [[17/1|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.  
See [[Gamelismic clan #Rodan]] for more information.  


== Interval chain ==
== Interval chain ==
When speaking of interval categories, as an extension of slendric it is possible to use a system [[Slendric#Interval categories|designed for slendric]], where notes are inflected from the diatonic [[chain of fifths]] by means of the prefixes "super" and "sub", such that three of these inflections stack to a diatonic minor second. These "super" and "sub" prefixes are equivalent to "up" and "down" in [[#Notation|the notation]] system, but can disambiguate from "up" and "down" symbols from [[ups and downs notation]] in the context of larger rodan edos that map this comma to multiple edosteps.
In the following table, odd harmonics and subharmonics 1–21 are in '''bold'''.  
In the following table, odd harmonics and subharmonics 1–21 are in '''bold'''.  


{| class="wikitable center-1 right-2"
{| class="wikitable sortable center-1 center-2 right-3"
|-
|-
! rowspan="2" | #
! rowspan="2" | # !! rowspan="2" | Extended <br> diatonic <br> interval !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios
! rowspan="2" | Cents*
! colspan="2" | Approximate ratios
|-
|-
! 13-limit
! rowspan="1" | Ratios of the 11-limit !! colspan="1" | Ratios of the 17-limit
! 17-limit extension
|-
|-
| 0
| 0
| 0.000
| P1
| 0.00
| '''1/1'''
| '''1/1'''
|
|
|-
|-
| 1
| 1
| 234.482
| SM2
| 234.46
| '''8/7'''
| '''8/7'''
|
| 39/34
|-
|-
| 2
| 2
| 468.964
| s4
| '''21/16'''
| 468.92
| '''21/16''', 64/49
| 17/13
| 17/13
|-
|-
| 3
| 3
| 703.446
| P5
| 703.38
| '''3/2'''
| '''3/2'''
|
|  
|-
|-
| 4
| 4
| 937.929
| SM6
| 12/7
| 937.84
|
| 12/7, 55/32
|  
|-
|-
| 5
| 5
| 1172.411
| s8
| 63/32, 160/81
| 1172.30
|
| 55/28, 63/32, 96/49, 108/55, 160/81
| 51/26
|-
|-
| 6
| 6
| 206.893
| M2
| 206.76
| '''9/8'''
| '''9/8'''
|
| 44/39
|-
|-
| 7
| 7
| 441.375
| SM3
| 9/7
| 441.22
| 9/7, 35/27
| 22/17
| 22/17
|-
|-
| 8
| 8
| 675.857
| s5
| 40/27
| 675.68
|
| 40/27, 49/33, 72/49
|  
|-
|-
| 9
| 9
| 910.339
| M6
| 22/13, 27/16, 33/28
| 910.14
|  
| 27/16, 56/33
| 22/13
|-
|-
| 10
| 10
| 1144.821
| SM7
| 1144.59
| 27/14, 35/18, 64/33
| 27/14, 35/18, 64/33
| 33/17
| 33/17
|-
|-
| 11
| 11
| 179.304
| sM2
| 179.05
| 10/9
| 10/9
|
|  
|-
|-
| 12
| 12
| 413.786
| M3
| 14/11, 33/26, 80/63
| 413.51
|
| 14/11, 81/64
| 33/26
|-
|-
| 13
| 13
| 648.268
| SA4
| '''16/11'''
| 647.97
|
| '''16/11''', 35/24
|  
|-
|-
| 14
| 14
| 882.750
| sM6
| 882.43
| 5/3
| 5/3
|
|  
|-
|-
| 15
| 15
| 1117.232
| M7
| 1116.89
| 21/11, 40/21
| 21/11, 40/21
|
| 98/51
|-
|-
| 16
| 16
| 151.714
| SA1
| 12/11
| 151.35
|
| 12/11, 35/32
| 56/51
|-
|-
| 17
| 17
| 386.196
| sM3
| 385.81
| '''5/4'''
| '''5/4'''
|
| 49/39, 64/51
|-
|-
| 18
| 18
| 620.679
| A4
| 620.27
| 10/7
| 10/7
|
| 49/34, 56/39
|-
|-
| 19
| 19
| 855.161
| SA5
| 18/11, 64/39
| 854.73
| 28/17
| 18/11
| 28/17, 64/39
|-
|-
| 20
| 20
| 1089.643
| sM7
| 1089.19
| '''15/8'''
| '''15/8'''
| '''32/17'''
| '''32/17''', 49/26
|-
|-
| 21
| 21
| 124.125
| A1
| 14/13, 15/14
| 123.65
|  
| 15/14
| 14/13
|-
|-
| 22
| 22
| 358.607
| SA2
| '''16/13''', 27/22
| 358.11
| 21/17
| 27/22, 60/49
| '''16/13''', 21/17
|-
|-
| 23
| 23
| 593.088
| sA4
| 592.57
| 45/32
| 45/32
| 24/17
| 24/17
|-
|-
| 24
| 24
| 827.570
| A5
| 827.03
| 45/28
| 21/13
| 21/13
|
|-
|-
| 25
| 25
| 1062.052
| SA6
| 1061.49
| 50/27, 90/49
| 24/13
| 24/13
|-
| 26
| sA1
| 95.95
| 35/33
| 18/17
|-
| 27
| A2
| 330.41
| 40/33
|  
|  
|-
| 28
| SA3
| 564.87
| 25/18
| 18/13
|-
| 29
| sA5
| 799.33
| 35/22, 100/63
| 27/17
|-
| 30
| A6
| 1033.79
| 20/11
|
|-
| 31
| SA7
| 68.25
| 25/24
| 27/26
|}
|}
<nowiki/>* In 13-limit POTE tuning
<nowiki/>* In 11-limit CWE tuning, octave reduced
 
[[File: Rodan 5et Detempering.png|thumb|Rodan as a 46-tone 5et detempering]]
 
=== As a detemperament of 5et ===
Rodan is naturally a [[detemperament]] of the [[5edo|5 equal temperament]]. The diagram on the right shows a 46-tone detempered scale, with a generator range of -22 to +23. 46 is the largest number of tones for a mos where intervals in the 5 categories do not overlap. Each category is divided into eight or nine qualities separated by 5 generator steps, which represent the syntonic comma.
 
Notice also the little interval between the largest of a category and the smallest of the next, which represents the differences between 16/15 and 14/13, between 11/9 and 16/13, between 7/5 and 45/32, between 13/8 and 18/11, and between 13/7 and 15/8. It spans 41 generator steps, so it vanishes in 41edo, but is tuned to the same size as the quark/syntonic comma in 46edo. 87edo tunes it to one half the size of the syntonic comma, which may be seen as a good compromise.
 
== Chords ==
11-limit rodan contains [[essentially tempered chord]]s of the commas 245/243, 385/384, 441/440, and 896/891. A list of 11-odd-limit [[dyadic chord|dyadically consonant chords]], both essentially tempered and essentially just, can be found at [[Chords of rodan]].
 
== Scales ==
* [[Radon5]]
* [[Radon11]]
* [[Radon16]]
* [[Rodan26opt]]
* [[Rodan31opt]]
* [[Rodan41opt]]


== Notation ==
== Notation ==
A notation for rodan is listed in the notation guide for rank-2 pergens under [[pergen]] #8, {{nowrap|(P8, P5/3)}}. The generator is an upmajor 2nd. The [[enharmonic unisons in ups and downs notation|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]].  
A notation for rodan is listed in the notation guide for rank-2 pergens under [[pergen]] #8, {{nowrap|(P8, P5/3)}}. The generator is an upmajor second. The [[enharmonic unisons in ups and downs notation|enharmonic unison]] is a triple-down minor second. Thus three ups equals a diatonic semitone, and three generators equals a perfect fifth. In rodan in particular, ^1 equals ~81/80 and ~64/63, and ^^1 equals ~33/32 and ~[[1053/1024]].  


{| class="wikitable center-1 center-3"
{| class="wikitable center-1 center-3"
|+ style="font-size: 105%;" | Rodan nomenclature<br />for selected intervals
|+ style="font-size: 105%;" | Rodan nomenclature<br>for selected intervals
|-
|-
! Ratio
! Ratio
Line 164: Line 262:
|-
|-
| 5/4
| 5/4
| Downmajor third
| Down major third
| C−vE
| C−vE
|-
|-
| 7/4
| 7/4
| Downminor seventh
| Down minor seventh
| C−vB♭
| C−vB♭
|-
|-
| 11/8
| 11/8
| Dup fourth
| Double-up fourth
| C−^^F
| C−^^F
|-
|-
| 13/8
| 13/8
| Dupminor sixth
| Double-up minor sixth
| C−^^A♭
| 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.
Rodan's notation has much in common with that for 41edo and 46edo, since both edos map a minor second 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| Chords of rodan }}
 
== Scales ==
* [[Radon5]]
* [[Radon11]]
* [[Radon16]]
* [[Rodan26opt]]
* [[Rodan31opt]]
* [[Rodan41opt]]


== Tunings ==
== Tunings ==
=== Tuning spectrum ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{{see also|Slendric #Tuning spectrum}}
 
{| class="wikitable center-all left-4 left-5"
|-
|-
! Edo<br>generator
! EDO<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
|  
|  
| 7/4
| [[8/7]]
| 231.174
| 231.1741
|  
| Untempered tuning
|-
|-
|  
|  
| 17/13
| [[17/13]]
| 232.214
| 232.2139
|  
|  
|-
|-
| '''[[36edo|7\36]]'''
|  
|  
| 7/6
| '''233.3333'''
| 232.282
| 36cfg val, '''lower bound of 7- and 9-odd-limit diamond monotone'''
|-
|  
|  
| [[3/2]]
| 233.9850
| 1/3-comma slendric
|-
|-
| [[36edo|21\36]]
| '''[[41edo|8\41]]'''
|  
|  
| 233.333
| '''234.1463'''
| 36cfg val, lower bound of 7- and 9-odd-limit diamond monotone
| '''Lower bound of 11- through 17-odd-limit diamond monotone'''
|-
|-
|  
|  
| 9/7
| [[22/17]]
| 233.583
| 234.1946
|  
|  
|-
|-
|  
|  
| 3/2
| [[15/14]]
| 233.985
| 234.2592
|  
|  
|-
|-
| [[41edo|24\41]]
|  
|  
| 234.146
| [[10/7]]
| Lower bound of 11- to 17-odd-limit diamond monotone
| 234.3049
| 7- and 9-odd-limit minimax
|-
|-
| [[169edo|33\169]]
|  
|  
| 15/14
| 234.3195
| 234.259
| 169fgg val
|-
|  
|  
| [[18/11]]
| 234.3470
| 11-odd-limit minimax
|-
|-
|  
|  
| 7/5
| [[40/21]]
| 234.305
| 234.3689
| 7- and 9-odd-limit minimax
|-
|  
|  
| 11/9
| 234.347
| 11-odd-limit minimax
|-
|-
| [[128edo|75\128]]
| [[128edo|25\128]]
|  
|  
| 234.375
| 234.3750
| 128g val
| 128g val
|-
|-
|  
|  
| 13/9
| [[18/13]]
| 234.407
| 234.4065
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
|-
|-
|  
|  
| 15/8
| [[55/32]]
| 234.413
| 234.4079
| As SM6
|-
|
| [[15/8]]
| 234.4134
|  
|  
|-
|-
|  
|  
| 11/6
| [[12/11]]
| 234.415
| 234.4148
|
|-
| [[215edo|42\215]]
|  
|  
| 234.4186
| 215dgg val
|-
|-
|  
|  
| 15/11
| [[15/11]]
| 234.453
| 234.4531
|  
|  
|-
|-
|  
|  
| 13/12
| [[24/13]]
| 234.457
| 234.4571
|  
|  
|-
|-
|  
|  
| 15/13
| [[15/13]]
| 234.470
| 234.4700
|  
|  
|-
|-
| [[87edo|51\87]]
| [[87edo|17\87]]
|  
|  
| 234.483
| 234.4828
|  
|  
|-
|-
|  
|  
| 5/4
| [[5/4]]
| 234.489
| 234.4890
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
|  
|  
| 11/10
| [[20/11]]
| 234.500
| 234.4999
|
|-
|
| [[20/13]]
| 234.5073
|  
|  
|-
|-
|  
|  
| 13/10
| [[16/11]]
| 234.507
| 234.5140
|  
|  
|-
|-
|  
|  
| 11/8
| [[16/13]]
| 234.514
| 234.5215
|  
|  
|-
|-
|  
|  
| 13/8
| [[22/13]]
| 234.521
| 234.5323
|  
|  
|-
|-
| [[220edo|43\220]]
|  
|  
| 13/11
| 234.5455
| 234.532
| 220dg val
|-
|  
|  
| [[63/32]]
| 234.5472
| 2/5-comma slendric
|-
|-
|  
|  
| 17/9
| [[18/17]]
| 234.575
| 234.5752
| 17-odd-limit minimax
| 17-odd-limit minimax
|-
|-
|  
|  
| 17/15
| [[30/17]]
| 234.583
| 234.5828
|  
|  
|-
|-
| [[133edo|78\133]]
| [[133edo|26\133]]
|  
|  
| 234.586
| 234.5865
|  
|  
|-
|-
|  
|  
| 5/3
| [[5/3]]
| 234.597
| 234.5971
|
|-
|
| [[21/11]]
| 234.6309
|
|-
|
| [[20/17]]
| 234.6313
|
|-
| [[179edo|35\179]]
|
| 234.6369
| 179d val
|-
|
| [[24/17]]
| 234.6522
|  
|  
|-
|-
|  
|  
| 17/10
| [[14/13]]
| 234.631
| 234.6809
|  
|  
|-
|-
|  
|  
| 17/12
| [[32/17]]
| 234.652
| 234.7522
|  
|  
|-
|-
|  
|  
| 17/16
| [[10/9]]
| 234.752
| 234.7640
|
|-
| '''[[46edo|9\46]]'''
|  
|  
| '''234.7826'''
| '''Upper bound of 11- through 17-odd-limit diamond monotone'''
|-
|-
|  
|  
| 9/5
| [[14/11]]
| 234.764
| 234.7923
|  
|  
|-
|-
| [[46edo|27\46]]
|  
|  
| 234.783
| [[28/17]]
| Upper bound of 11- to 17-odd-limit diamond monotone
| 234.9406
|  
|-
|-
| [[51edo|10\51]]
|  
|  
| 11/7
| 235.2941
| 234.792
| 51cf val
|-
|  
|  
| [[21/16]]
| 235.3905
| 1/2-comma slendric
|-
|-
| [[5edo|3\5]]
| '''[[5edo|1\5]]'''
|  
|  
| 240.000
| '''240.0000'''
| 5f val, upper bound of 7- and 9-odd-limit diamond monotone
| 5f val, '''upper bound of 5- through 9-odd-limit diamond monotone'''
|}
|}
<nowiki>*</nowiki> Besides the octave


== Music ==
== Music ==
; [[Gene Ward Smith]]
; [[Gene Ward Smith]]
* ''Pianodactyl'' (archived 2010) – [https://soundcloud.com/genewardsmith/pianodactyl SoundCloud] | [http://www.archive.org/details/Pianodactyl detail] | [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] – rodan[26] in 87edo tuning
* ''Pianodactyl'' (archived 2010) – [https://soundcloud.com/genewardsmith/pianodactyl SoundCloud] | [http://www.archive.org/details/Pianodactyl detail] | [http://www.archive.org/download/Pianodactyl/pianodactyl.mp3 play] – in Rodan[26], 87edo tuning


[[Category:Temperaments]]
[[Category:Rodan| ]] <!-- main article -->
[[Category:Rodan| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Gamelismic clan]]
[[Category:Sensamagic clan]]
[[Category:Sensamagic clan]]
[[Category:Gamelismic clan]]
[[Category:Hemifamity temperaments]]

Latest revision as of 18:57, 28 July 2025

Rodan
Subgroups 2.3.5.7, 2.3.5.7.11
Comma basis 245/243, 1029/1024 (7-limit);
245/243, 385/384, 441/440 (11-limit)
Reduced mapping ⟨1; 3 17 -1 -13]
Edo join 41 & 46
Generator (CWE) ~8/7 = 234.4 ¢
MOS scales 1L 4s, 5L 1s, 5L 6s, …, 5L 36s, 41L 5s
Ploidacot tricot
Pergen (P8, P5/3)
Minimax error (9-odd limit) 5.05 ¢;
((2.3.5.7.11) 21-odd limit) 5.34 ¢
Target scale size (9-odd limit) 41 notes;
((2.3.5.7.11) 21-odd limit) 87 notes

Rodan is one of the notable extensions of the slendric temperament, which divides 3/2 into three equal intervals representing 8/7 (tempering out the gamelisma, 1029/1024), reaching the full 7-limit such that 17 of these generators stack to reach the interval class of the 5th harmonic. 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 (with 87edo being an essentially optimal tuning). 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 (see #As a detemperament of 5et). 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 can be elucidated by S-expressions, rodan is very much an "opposed counterpart" to mothra: the basic equivalence of slendric tempers S7 (49/48) = S8 (64/63), and mothra proceeds to equate it to S6 (36/35) as well; meanwhile, rodan extends the equivalence in the opposite direction to add S9 (81/80) to it, making it one of the five rank-2 temperaments definable by equating three adjacent square superparticulars.

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. It is also worth mentioning that this equates the diatonic major third to 14/11, tempering out 896/891. A 2.3.7.11 subgroup version of rodan, known as radon, uses exclusively this mapping and forgoes interpreting the 5th harmonic.

Toward the 13-limit, 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

When speaking of interval categories, as an extension of slendric it is possible to use a system designed for slendric, where notes are inflected from the diatonic chain of fifths by means of the prefixes "super" and "sub", such that three of these inflections stack to a diatonic minor second. These "super" and "sub" prefixes are equivalent to "up" and "down" in the notation system, but can disambiguate from "up" and "down" symbols from ups and downs notation in the context of larger rodan edos that map this comma to multiple edosteps.

In the following table, odd harmonics and subharmonics 1–21 are in bold.

# Extended
diatonic
interval		 Cents* Approximate ratios
Ratios of the 11-limit Ratios of the 17-limit
0 P1 0.00 1/1
1 SM2 234.46 8/7 39/34
2 s4 468.92 21/16, 64/49 17/13
3 P5 703.38 3/2
4 SM6 937.84 12/7, 55/32
5 s8 1172.30 55/28, 63/32, 96/49, 108/55, 160/81 51/26
6 M2 206.76 9/8 44/39
7 SM3 441.22 9/7, 35/27 22/17
8 s5 675.68 40/27, 49/33, 72/49
9 M6 910.14 27/16, 56/33 22/13
10 SM7 1144.59 27/14, 35/18, 64/33 33/17
11 sM2 179.05 10/9
12 M3 413.51 14/11, 81/64 33/26
13 SA4 647.97 16/11, 35/24
14 sM6 882.43 5/3
15 M7 1116.89 21/11, 40/21 98/51
16 SA1 151.35 12/11, 35/32 56/51
17 sM3 385.81 5/4 49/39, 64/51
18 A4 620.27 10/7 49/34, 56/39
19 SA5 854.73 18/11 28/17, 64/39
20 sM7 1089.19 15/8 32/17, 49/26
21 A1 123.65 15/14 14/13
22 SA2 358.11 27/22, 60/49 16/13, 21/17
23 sA4 592.57 45/32 24/17
24 A5 827.03 45/28 21/13
25 SA6 1061.49 50/27, 90/49 24/13
26 sA1 95.95 35/33 18/17
27 A2 330.41 40/33
28 SA3 564.87 25/18 18/13
29 sA5 799.33 35/22, 100/63 27/17
30 A6 1033.79 20/11
31 SA7 68.25 25/24 27/26

* In 11-limit CWE tuning, octave reduced

Rodan as a 46-tone 5et detempering

As a detemperament of 5et

Rodan is naturally a detemperament of the 5 equal temperament. The diagram on the right shows a 46-tone detempered scale, with a generator range of -22 to +23. 46 is the largest number of tones for a mos where intervals in the 5 categories do not overlap. Each category is divided into eight or nine qualities separated by 5 generator steps, which represent the syntonic comma.

Notice also the little interval between the largest of a category and the smallest of the next, which represents the differences between 16/15 and 14/13, between 11/9 and 16/13, between 7/5 and 45/32, between 13/8 and 18/11, and between 13/7 and 15/8. It spans 41 generator steps, so it vanishes in 41edo, but is tuned to the same size as the quark/syntonic comma in 46edo. 87edo tunes it to one half the size of the syntonic comma, which may be seen as a good compromise.

Chords

11-limit rodan contains essentially tempered chords of the commas 245/243, 385/384, 441/440, and 896/891. A list of 11-odd-limit dyadically consonant chords, both essentially tempered and essentially just, can be found at Chords of rodan.

Scales

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 second. The enharmonic unison is a triple-down minor second. Thus three ups equals a diatonic semitone, and three generators equals a perfect fifth. 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 Down major third C−vE
7/4 Down minor seventh C−vB♭
11/8 Double-up fourth C−^^F
13/8 Double-up minor sixth C−^^A♭

Rodan's notation has much in common with that for 41edo and 46edo, since both edos map a minor second 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.

Tunings

Tuning spectrum

See also: Slendric § Tuning spectrum
EDO
generator 		Unchanged interval
(eigenmonzo) 		Generator (¢) Comments
8/7 231.1741 Untempered tuning
17/13 232.2139
7\36 233.3333 36cfg val, lower bound of 7- and 9-odd-limit diamond monotone
3/2 233.9850 1/3-comma slendric
8\41 234.1463 Lower bound of 11- through 17-odd-limit diamond monotone
22/17 234.1946
15/14 234.2592
10/7 234.3049 7- and 9-odd-limit minimax
33\169 234.3195 169fgg val
18/11 234.3470 11-odd-limit minimax
40/21 234.3689
25\128 234.3750 128g val
18/13 234.4065 13- and 15-odd-limit minimax
55/32 234.4079 As SM6
15/8 234.4134
12/11 234.4148
42\215 234.4186 215dgg val
15/11 234.4531
24/13 234.4571
15/13 234.4700
17\87 234.4828
5/4 234.4890 5-odd-limit minimax
20/11 234.4999
20/13 234.5073
16/11 234.5140
16/13 234.5215
22/13 234.5323
43\220 234.5455 220dg val
63/32 234.5472 2/5-comma slendric
18/17 234.5752 17-odd-limit minimax
30/17 234.5828
26\133 234.5865
5/3 234.5971
21/11 234.6309
20/17 234.6313
35\179 234.6369 179d val
24/17 234.6522
14/13 234.6809
32/17 234.7522
10/9 234.7640
9\46 234.7826 Upper bound of 11- through 17-odd-limit diamond monotone
14/11 234.7923
28/17 234.9406
10\51 235.2941 51cf val
21/16 235.3905 1/2-comma slendric
1\5 240.0000 5f val, upper bound of 5- through 9-odd-limit diamond monotone

* Besides the octave

Music

Gene Ward Smith
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