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{{Infobox Regtemp
{{Infobox regtemp
| Title = Rodan
| Title = Rodan
| Subgroups = 2.3.5.7, 2.3.5.7.11
| 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)
| 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
| Edo join 1 = 41 | Edo join 2 = 46
| Generator = 8/7 | Generator tuning = 234.4 | Optimization method = CWE
| Mapping = 1; 3 17 -1 -13
| Generators = 8/7 | Generators tuning = 234.4 | Optimization method = CWE
| MOS scales = [[1L 4s]], [[5L 1s]], [[5L 6s]], …, [[5L 36s]], [[41L 5s]]
| MOS scales = [[1L 4s]], [[5L 1s]], [[5L 6s]], …, [[5L 36s]], [[41L 5s]]
| 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 = 21
| Odd limit 2 = (2.3.5.7.11) 21 | Mistuning 2 = ? | Complexity 2 = 87
| Odd limit 2 = 11-limit 21 | Mistuning 2 = 5.34 | Complexity 2 = 36
}}
}}
'''Rodan''' is one of the notable [[extension]]s of the [[slendric]] [[regular temperament|temperament]], which divides the perfect fifth, [[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]].


'''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 in [[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.


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]].
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]].
Line 25: Line 24:


== 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.  
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. Such an inflection is equivalent to a quark, and due to the quark's versatile nature in rodan as a generalized comma, simple pental and septimal intervals tend to be represented by single-quark offsets from the diatonic spine. These "super" and "sub" prefixes are equivalent to "up" and "down" in [[#Notation|the notation]] system, where lifts and drops would represent a single edostep in [[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 sortable center-1 center-2 right-3"
{| class="wikitable sortable center-1 center-2 center-3 right-4"
|-
|-
! rowspan="2" | # !! rowspan="2" | Extended <br> diatonic <br> interval !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios
! rowspan="2" | # !! rowspan="2" | Extended <br> diatonic <br> interval !! rowspan="2" | Ups and downs <br> notation !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios
|-
|-
! rowspan="1" | Ratios of the 11-limit !! colspan="1" | Ratios of the 17-limit
! rowspan="1" | Ratios of the 11-limit !! colspan="1" | Ratios of the 17-limit
Line 37: Line 36:
| 0
| 0
| P1
| P1
| D
| 0.00
| 0.00
| '''1/1'''
| '''1/1'''
Line 43: Line 43:
| 1
| 1
| SM2
| SM2
| ^E
| 234.46
| 234.46
| '''8/7''', 55/48, 63/55
| '''8/7'''
| 39/34
| 39/34
|-
|-
| 2
| 2
| s4
| s4
| vG
| 468.92
| 468.92
| '''21/16''', 64/49, 55/42, 72/55
| '''21/16''', 64/49
| 17/13
| 17/13
|-
|-
| 3
| 3
| P5
| P5
| A
| 703.38
| 703.38
| '''3/2'''
| '''3/2'''
Line 61: Line 64:
| 4
| 4
| SM6
| SM6
| ^B
| 937.84
| 937.84
| 12/7, 55/32
| 12/7, 55/32
Line 67: Line 71:
| 5
| 5
| s8
| s8
| vD
| 1172.30
| 1172.30
| 55/28, 63/32, 96/49, 108/55, 160/81
| 55/28, 63/32, 96/49, 108/55, 160/81
Line 73: Line 78:
| 6
| 6
| M2
| M2
| E
| 206.76
| 206.76
| '''9/8''', 55/49
| '''9/8'''
| 44/39
| 44/39
|-
|-
| 7
| 7
| SM3
| SM3
| ^F#
| 441.22
| 441.22
| 9/7, 35/27
| 9/7, 35/27
Line 85: Line 92:
| 8
| 8
| s5
| s5
| vA
| 675.68
| 675.68
| 40/27, 49/33, 72/49
| 40/27, 49/33, 72/49
Line 91: Line 99:
| 9
| 9
| M6
| M6
| B
| 910.14
| 910.14
| 27/16, 56/33
| 27/16, 56/33
Line 97: Line 106:
| 10
| 10
| SM7
| SM7
| ^C#
| 1144.59
| 1144.59
| 27/14, 35/18, 64/33
| 27/14, 35/18, 64/33
Line 103: Line 113:
| 11
| 11
| sM2
| sM2
| vE
| 179.05
| 179.05
| 10/9, 49/44, 54/49
| 10/9
|  
|  
|-
|-
| 12
| 12
| M3
| M3
| F#
| 413.51
| 413.51
| 14/11, 80/63, 81/64
| 14/11, 81/64
| 33/26
| 33/26
|-
|-
| 13
| 13
| SA4
| SA4
| ^G#
| 647.97
| 647.97
| '''16/11''', 35/24
| '''16/11''', 35/24
Line 121: Line 134:
| 14
| 14
| sM6
| sM6
| vB
| 882.43
| 882.43
| 5/3
| 5/3
Line 127: Line 141:
| 15
| 15
| M7
| M7
| C#
| 1116.89
| 1116.89
| 21/11, 40/21
| 21/11, 40/21
Line 133: Line 148:
| 16
| 16
| SA1
| SA1
| ^D#
| 151.35
| 151.35
| 12/11, 35/32
| 12/11, 35/32
|  
| 56/51
|-
|-
| 17
| 17
| sM3
| sM3
| vF#
| 385.81
| 385.81
| '''5/4'''
| '''5/4'''
Line 145: Line 162:
| 18
| 18
| A4
| A4
| G#
| 620.27
| 620.27
| 10/7, 63/44
| 10/7
| 49/34, 56/39
| 49/34, 56/39
|-
|-
| 19
| 19
| SA5
| SA5
| ^A#
| 854.73
| 854.73
| 18/11, 80/49
| 18/11
| 28/17, 64/39
| 28/17, 64/39
|-
|-
| 20
| 20
| sM7
| sM7
| vC#
| 1089.19
| 1089.19
| '''15/8'''
| '''15/8'''
Line 163: Line 183:
| 21
| 21
| A1
| A1
| D#
| 123.65
| 123.65
| 15/14
| 15/14
Line 169: Line 190:
| 22
| 22
| SA2
| SA2
| ^E#
| 358.11
| 358.11
| 27/22, 60/49
| 27/22, 60/49
Line 175: Line 197:
| 23
| 23
| sA4
| sA4
| vG#
| 592.57
| 592.57
| 45/32
| 45/32
Line 181: Line 204:
| 24
| 24
| A5
| A5
| A#
| 827.03
| 827.03
| 45/28
| 45/28
Line 187: Line 211:
| 25
| 25
| SA6
| SA6
| ^B#
| 1061.49
| 1061.49
| 50/27, 90/49
| 50/27, 90/49
| 24/13, 63/34
| 24/13
|-
|-
| 26
| 26
| sA1
| sA1
| vD#
| 95.95
| 95.95
| 35/33
| 35/33
| 18/17
| 18/17
|-
| 27
| A2
| E#
| 330.41
| 40/33
|
|-
| 28
| SA3
| ^Fx
| 564.87
| 25/18
| 18/13
|-
| 29
| sA5
| vA#
| 799.33
| 35/22, 100/63
| 27/17
|-
| 30
| A6
| B#
| 1033.79
| 20/11
|
|-
| 31
| SA7
| ^Cx
| 68.25
| 25/24
| 27/26
|}
|}
<nowiki/>* In 11-limit CWE tuning, octave reduced
<nowiki/>* In 11-limit CWE tuning, octave reduced
Line 207: Line 268:


== 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 ==
Line 251: Line 312:


== Tunings ==
== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~8/7 = 234.4502{{c}}
| CWE: ~8/7 = 234.4259{{c}}
| POTE: ~8/7 = 234.4168{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~8/7 = 234.4628{{c}}
| CWE: ~8/7 = 234.4594{{c}}
| POTE: ~8/7 = 234.4587{{c}}
|}
=== 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|26\133]]
|
| 234.5865
|  
|  
|-
|-
| [[133edo|78\133]]
|  
|  
| 234.586
| [[5/3]]
| 234.5971
|  
|  
|-
|-
|  
|  
| 5/3
| [[21/11]]
| 234.597
| 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 ==
Line 441: Line 593:
[[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"