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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. 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]].
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'''.  


Line 29: Line 33:
! 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" | Cents* !! colspan="2" | Approximate ratios
|-
|-
! rowspan="1" | 11-limit intervals !! colspan="1" | 17-limit intervals
! rowspan="1" | Ratios of the 11-limit !! colspan="1" | Ratios of the 17-limit
|-
|-
| 0
| 0
Line 40: Line 44:
| SM2
| SM2
| 234.46
| 234.46
| '''8/7''', 55/48, 63/55
| '''8/7'''
| 39/34
| 39/34
|-
|-
Line 46: Line 50:
| s4
| s4
| 468.92
| 468.92
| '''21/16''', 64/49, 55/42, 72/55
| '''21/16''', 64/49
| 17/13
| 17/13
|-
|-
Line 58: Line 62:
| SM6
| SM6
| 937.84
| 937.84
| 12/7, 55/32, 140/81
| 12/7, 55/32
| 88/51
|  
|-
|-
| 5
| 5
Line 70: Line 74:
| M2
| M2
| 206.76
| 206.76
| '''9/8''', 55/49
| '''9/8'''
| 44/39
| 44/39
|-
|-
Line 82: Line 86:
| s5
| s5
| 675.68
| 675.68
| 40/27, 49/33, 72/49, 81/55
| 40/27, 49/33, 72/49
|  
|  
|-
|-
Line 100: Line 104:
| sM2
| sM2
| 179.05
| 179.05
| 10/9, 49/44, 54/49
| 10/9
|  
|  
|-
|-
Line 106: Line 110:
| M3
| M3
| 413.51
| 413.51
| 14/11, 80/63, 81/64
| 14/11, 81/64
| 33/26
| 33/26
|-
|-
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| SA4
| SA4
| 647.97
| 647.97
| '''16/11''', 35/24, 81/56
| '''16/11''', 35/24
|  
|  
|-
|-
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| sM6
| sM6
| 882.43
| 882.43
| 5/3, 81/49
| 5/3
|  
|  
|-
|-
Line 142: Line 146:
| 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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| SA2
| SA2
| 358.11
| 358.11
| 27/22, 60/49, 100/81
| 27/22, 60/49
| '''16/13''', 21/17
| '''16/13''', 21/17
|-
|-
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| SA6
| SA6
| 1061.49
| 1061.49
| 50/27, 81/44, 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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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.
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 ==
Line 234: Line 279:


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.
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
|  
|  
|-
|-
|  
|  
| 13/10
| [[20/13]]
| 234.507
| 234.5073
|  
|  
|-
|-
|  
|  
| 11/8
| [[16/11]]
| 234.514
| 234.5140
|
|-
|
| [[16/13]]
| 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
|
|-
|
| [[5/3]]
| 234.5971
|
|-
|
| [[21/11]]
| 234.6309
|  
|  
|-
|-
| [[133edo|78\133]]
|  
|  
| 234.586
| [[20/17]]
| 234.6313
|
|-
| [[179edo|35\179]]
|  
|  
| 234.6369
| 179d val
|-
|-
|  
|  
| 5/3
| [[24/17]]
| 234.597
| 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]]'''
|  
|  
| 9/5
| '''234.7826'''
| 234.764
| '''Upper bound of 11- through 17-odd-limit diamond monotone'''
|-
|
| [[14/11]]
| 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 437: Line 529:
[[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"