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'''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]].
'''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; 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 ([[#As a detemperament of 5et|as described here]]) it simplifies the conceptualization of rodan's inventory of intervals. 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 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.
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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-all right-3"
|-
|-
! rowspan="2" | #
! rowspan="2" | &#35; !! 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" | 11-limit intervals !! colspan="1" | 17-limit intervals
! 17-limit extension
|-
|-
| 0
| 0
| 0.00
| P1
| 0.0
| '''1/1'''
| '''1/1'''
|
|
|-
|-
| 1
| 1
| 234.48
| SM2
| '''8/7'''
| 234.46
|
| '''8/7''', 55/48, 63/55
| 39/34
|-
|-
| 2
| 2
| 468.96
| s4
| '''21/16'''
| 468.92
| '''21/16''', 64/49, 55/42, 72/55
| 17/13
| 17/13
|-
|-
| 3
| 3
| 703.45
| P5
| 703.38
| '''3/2'''
| '''3/2'''
|
|  
|-
|-
| 4
| 4
| 937.93
| SM6
| 12/7
| 937.84
|
| 12/7, 55/32, 140/81
| 88/51
|-
|-
| 5
| 5
| 1172.41
| s8
| 55/28, 63/32, 96/49, <br>108/55, 160/81
| 1172.30
| 55/28, 63/32, 96/49, 108/55, 160/81
| 51/26
| 51/26
|-
|-
| 6
| 6
| 206.89
| M2
| '''9/8'''
| 206.76
|
| '''9/8''', 55/49
| 44/39
|-
|-
| 7
| 7
| 441.38
| SM3
| 9/7
| 441.22
| 9/7, 35/27
| 22/17
| 22/17
|-
|-
| 8
| 8
| 675.86
| s5
| 40/27
| 675.68
|
| 40/27, 49/33, 72/49, 81/55
|  
|-
|-
| 9
| 9
| 910.34
| M6
| 22/13, 27/16, 33/28
| 910.14
|  
| 27/16, 56/33
| 22/13
|-
|-
| 10
| 10
| 1144.82
| SM7
| 1144.59
| 27/14, 35/18, 64/33
| 27/14, 35/18, 64/33
| 33/17
| 33/17
|-
|-
| 11
| 11
| 179.30
| sM2
| 10/9
| 179.05
|
| 10/9, 49/44, 54/49
|  
|-
|-
| 12
| 12
| 413.79
| M3
| 14/11, 33/26, 80/63
| 413.51
|  
| 14/11, 80/63, 81/64
| 33/26
|-
|-
| 13
| 13
| 648.27
| SA4
| '''16/11'''
| 647.97
|
| '''16/11''', 35/24, 81/56
|  
|-
|-
| 14
| 14
| 882.75
| sM6
| 5/3
| 882.43
|
| 5/3, 81/49
|  
|-
|-
| 15
| 15
| 1117.23
| M7
| 1116.89
| 21/11, 40/21
| 21/11, 40/21
|
| 98/51
|-
|-
| 16
| 16
| 151.71
| SA1
| 12/11
| 151.35
|
| 12/11, 35/32
| 56/51
|-
|-
| 17
| 17
| 386.20
| sM3
| 385.81
| '''5/4'''
| '''5/4'''
|
| 49/39, 64/51
|-
|-
| 18
| 18
| 620.68
| A4
| 10/7
| 620.27
|
| 10/7, 63/44
| 49/34, 56/39
|-
|-
| 19
| 19
| 855.16
| SA5
| 18/11, 64/39
| 854.73
| 28/17
| 18/11, 80/49
| 28/17, 64/39
|-
|-
| 20
| 20
| 1089.64
| sM7
| 1089.19
| '''15/8'''
| '''15/8'''
| '''32/17'''
| '''32/17''', 49/26
|-
|-
| 21
| 21
| 124.13
| A1
| 14/13, 15/14
| 123.65
|  
| 15/14
| 14/13
|-
|-
| 22
| 22
| 358.61
| SA2
| '''16/13''', 27/22
| 358.11
| 21/17
| 27/22, 60/49, 100/81
| '''16/13''', 21/17
|-
|-
| 23
| 23
| 593.09
| sA4
| 592.57
| 45/32
| 45/32
| 24/17
| 24/17
|-
|-
| 24
| 24
| 827.57
| A5
| 827.03
| 45/28
| 21/13
| 21/13
|
|-
|-
| 25
| 25
| 1062.05
| SA6
| 24/13
| 1061.49
|  
| 50/27, 81/44, 90/49
| 24/13, 63/34
|-
| 26
| sA1
| 95.95
| 35/33
| 18/17
|}
|}
<nowiki/>* In 13-limit CWE tuning, octave reduced
<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 syntonic comma in 46edo. 87edo tunes it to one half the size of the syntonic comma, which may be seen as a good compromise.


[[File: Rodan 5et Detempering.png|thumb|Rodan as a 46-tone 5et detempering]]
[[File: Rodan 5et Detempering.png|thumb|Rodan as a 46-tone 5et detempering]]
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== Tunings ==
== Tunings ==
=== Tuning spectrum ===
{| class="wikitable center-all left-4 left-5"
|-
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*
! Generator (¢)
! Comments
|-
|
| [[8/7]]
| 231.174
|
| Untempered tuning
|-
| '''[[31edo|6\31]]'''
|
| '''232.258'''
| 31ceffgg val, '''lower bound of 5-odd-limit diamond monotone'''
|-
| '''[[36edo|7\36]]'''
|
| '''233.333'''
| 36cfg val, '''lower bound of 7- and 9-odd-limit diamond monotone'''
|-
|
| [[3/2]]
| 233.985
| 1/3-comma
|-
| '''[[41edo|8\41]]'''
|
| '''234.146'''
| '''Lower bound of 11- to 17-odd-limit diamond monotone'''
|-
| [[128edo|25\128]]
|
| 234.375
|
|-
|
| [[55/32]]
| 234.408
| As SM6
|-
| [[87edo|17\87]]
|
| 234.483
|
|-
|
| [[63/32]]
| 234.547
| 2/5-comma
|-
| [[133edo|26\133]]
|
| 234.586
|
|-
| '''[[46edo|9\46]]'''
|
| '''234.783'''
| '''Upper bound of 11- to 17-odd-limit diamond monotone'''
|-
| [[51edo|10\51]]
|
| 235.294
| 51ceg val
|-
|
| [[21/16]]
| 235.390
| 1/2-comma
|-
| '''[[5edo|1\5]]'''
|
| '''240.000'''
| 5f val, '''upper bound of 5- to 9-odd-limit diamond monotone'''
|}
<nowiki/>* Besides the octave
=== Tuning spectrum ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
Retrieved from "https://en.xen.wiki/w/Rodan"