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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 ==
Prime harmonics are in '''bold'''.  
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.  


{| class="wikitable center-1 right-2"
In the following table, odd harmonics and subharmonics 1–21 are in '''bold'''.
 
{| 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
| 1/1
| 0.00
| '''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
| 9/8
| 206.76
|
| '''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
| 15/8
| 1089.19
| '''32/17'''
| '''15/8'''
| '''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 ({{dash|1, 8/7, 21/16, 3/2|med}}) on C as {{nowrap|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.
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:Gamelismic clan]]
[[Category:Sensamagic clan]]
[[Category:Sensamagic clan]]
[[Category:Gamelismic clan]]
[[Category:Hemifamity temperaments]]
Retrieved from "https://en.xen.wiki/w/Rodan"