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| Mapping = 1; 3 17 -1 -13
| Mapping = 1; 3 17 -1 -13
| Pergen = (P8, P5/3)
| Pergen = (P8, P5/3)
| Odd limit 1 = 9 | Mistuning 1 = ? | Complexity 1 = 41
| Odd limit 1 = 9 | Mistuning 1 = 5.05 | Complexity 1 = 41
| Odd limit 2 = (2.3.5.7.11) 21 | Mistuning 2 = ? | Complexity 2 = 87
| Odd limit 2 = (2.3.5.7.11) 21 | Mistuning 2 = 5.34 | Complexity 2 = 87
}}
}}


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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.
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 a "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 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.
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.
Line 44: Line 44:
| SM2
| SM2
| 234.46
| 234.46
| '''8/7''', 55/48, 63/55
| '''8/7'''
| 39/34
| 39/34
|-
|-
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| s4
| s4
| 468.92
| 468.92
| '''21/16''', 64/49, 55/42, 72/55
| '''21/16''', 64/49
| 17/13
| 17/13
|-
|-
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| M2
| M2
| 206.76
| 206.76
| '''9/8''', 55/49
| '''9/8'''
| 44/39
| 44/39
|-
|-
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| sM2
| sM2
| 179.05
| 179.05
| 10/9, 49/44, 54/49
| 10/9
|  
|  
|-
|-
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| M3
| M3
| 413.51
| 413.51
| 14/11, 80/63, 81/64
| 14/11, 81/64
| 33/26
| 33/26
|-
|-
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| 151.35
| 151.35
| 12/11, 35/32
| 12/11, 35/32
|  
| 56/51
|-
|-
| 17
| 17
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| A4
| A4
| 620.27
| 620.27
| 10/7, 63/44
| 10/7
| 49/34, 56/39
| 49/34, 56/39
|-
|-
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| SA5
| SA5
| 854.73
| 854.73
| 18/11, 80/49
| 18/11
| 28/17, 64/39
| 28/17, 64/39
|-
|-
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| 1061.49
| 1061.49
| 50/27, 90/49
| 50/27, 90/49
| 24/13, 63/34
| 24/13
|-
|-
| 26
| 26
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| 35/33
| 35/33
| 18/17
| 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 11-limit CWE tuning, octave reduced
<nowiki/>* In 11-limit CWE tuning, octave reduced
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== Chords ==
== Chords ==
{{Main| Chords of rodan }}
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 ==
== Scales ==
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* [[Rodan31opt]]
* [[Rodan31opt]]
* [[Rodan41opt]]
* [[Rodan41opt]]


== Notation ==
== Notation ==
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== 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]]
|  
|  
| 15/11
| 234.4186
| 234.453
| 215dgg val
|-
|
| [[15/11]]
| 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 ==
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[[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]]
Retrieved from "https://en.xen.wiki/w/Rodan"