Xenharmonic Wiki

Rodan: Difference between revisions

← Older edit
VisualWikitext
mNo edit summary
Fredg999 (talk | contribs)
autopatrolled, Interface administrators, Sysops
6,129 edits
m Markup
 
(35 intermediate revisions by 2 users not shown)
Line 1: Line 1:
'''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 [[5/4|5th harmonic (5/4)]]. 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]].
{{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
}}


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.
'''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]].


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.
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 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 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 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 harmonic 17 at -20 generator steps. Thus proceeds the canonical extension of rodan out to the [[17-limit]].
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 ==
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'''.  


{| class="wikitable center-1 right-2"
{| 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
| 0.00
| '''1/1'''
| '''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
| 206.76
| '''9/8'''
| '''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
| 1089.19
| '''15/8'''
| '''15/8'''
| '''32/17'''
| '''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 168: 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 (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]]'''
|  
|  
| '''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: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"