Rodan: Difference between revisions

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{{Infobox ~~Regtemp~~

{{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

| 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]]

~~| Mapping = 1; 3 17 -1 -13~~

| Pergen = (P8, P5/3)

| Odd limit 1 = 9 | Mistuning 1 = 5.05 | 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 = 5.34 | 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]].

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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]].

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== 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, 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.

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'''.

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== 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 ===

{| class="wikitable center-all left-4 left-5"

|-

! ~~EDO~~<br>generator

! Edo<br>generator

! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]

! Generator (¢)

@@ Line 1: / Line 1: @@
-{{Infobox Regtemp
+{{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
+| 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]]
-| Mapping = 1; 3 17 -1 -13
 | Pergen = (P8, P5/3)
-| Odd limit 1 = 9 | Mistuning 1 = 5.05 | 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 = 5.34 | 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]].
@@ Line 18: / Line 17: @@
 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]].
@@ Line 25: / Line 24: @@
 == 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, 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.
+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'''.
@@ Line 313: / Line 312: @@
 == 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 ===
-{{see also|Slendric #Tuning spectrum}}
+{{See also| Slendric #Tuning spectrum }}
 {| class="wikitable center-all left-4 left-5"
 |-
-! EDO<br>generator
+! Edo<br>generator
 ! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
 ! Generator (¢)