Mothra: Difference between revisions

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Line 8:

| Mapping = 1; 3 12 -1

| Pergen = (P8, P5/3)

| Odd limit 1 = 7 | Mistuning 1 = ? | Complexity 1 = 31

| Odd limit 1 = 7 | Mistuning 1 = 5.4 | Complexity 1 = 31

| Odd limit 2 = (2.3.5.7) 21 | Mistuning 2 = ? | Complexity 2 = 36

| Odd limit 2 = (2.3.5.7) 21 | Mistuning 2 = 10.8 | Complexity 2 = 36

}}

'''Mothra''' is a temperament in the [[7-limit]] that is a strong extension to [[slendric]], which is defined by splitting ~~the interval of~~ [[3/2]] into three [[8/7]]~~s and~~ tempering out [[1029/1024]]. The fifth of mothra is flattened to a [[meantone]] fifth, so that it reaches [[5/4]] when stacked four times and [[81/80]] is tempered out, unlike that of the other slendric extension [[rodan]], which is sharpened from just. This has the effect of bringing the generator 8/7 considerably closer to just, and also allowing [[MOS scale]]s of mothra to be more melodically usable than those of other forms of slendric, as the structurally-pervasive small step known as the [[quark]] (the residue between the octave and 5 generators, representing [[49/48]], [[64/63]], and in mothra also [[36/35]]) is larger here. [[EDOs]] that support mothra include [[26edo]], [[31edo]], and [[36edo]], and 31 is a particularly good tuning.

'''Mothra''', also known as '''cynder''', is a temperament of the [[7-limit]] that is a strong extension to [[slendric]], which is defined by splitting a perfect fifth representing [[3/2]] into three intervals of [[8/7]], tempering out [[1029/1024]]. The fifth of mothra is flattened to a [[meantone]] fifth, so that it reaches [[5/4]] when stacked four times and [[81/80]] is tempered out, unlike that of the other slendric extension [[rodan]], which is sharpened from just. This has the effect of bringing the generator 8/7 considerably closer to just, and also allowing [[MOS scale]]s of mothra to be more melodically usable than those of other forms of slendric, as the structurally-pervasive small step known as the [[quark]] (the residue between the octave and 5 generators, representing [[49/48]], [[64/63]], and in mothra also [[36/35]]) is larger here. [[EDOs]] that support mothra include [[26edo]], [[31edo]], and [[36edo]], and 31 is a particularly good tuning.

In the [[11-limit]], two extensions are of note: undecimal mothra (26 & 31), which tempers out [[99/98]], [[385/384]] and [[441/440]] to find the 11th harmonic at 8 generators down, and mosura (31 & 36), which tempers out [[176/175]] to find the 11th harmonic at 23 generators up. These two mappings merge at 31edo, which is therefore a uniquely suitable tuning for 11-limit mothra.

In higher limits, one may note that the two-generator interval closely approximates [[17/13]], and that the six-generator interval - the meantone whole tone of [[9/8]][[~]][[10/9]], approximates [[19/17]], so that the 13:17:19 chord is well-~~approximated~~; it is worth noting also that this chord is entirely included within the subtemperament obtained from taking every other generator of mothra, which is [[A-team]]. This can be combined with the canonical mapping of 13 for each undecimal extension, which tempers out [[144/143]], to provide a natural route to the [[19-limit]].

In higher limits, one may note that the two-generator interval closely approximates [[17/13]], and that the six-generator interval - the meantone whole tone of [[9/8]][[~]][[10/9]], approximates [[19/17]] - so that the 13:17:19 chord is well-represented; it is worth noting also that this chord is entirely included within the subtemperament obtained from taking every other generator of mothra, which is [[A-team]] (the crawma, [[83521/83486]], is the relevant comma tempered out here). This can be combined with the canonical mapping of 13 for each undecimal extension, which tempers out [[144/143]], to provide a natural route to the [[19-limit]].

For technical data, see [[Gamelismic clan #Mothra]].

== Intervals ==

As a strong extension of slendric, mothra's intervals can be expressed using the same system of extended diatonic interval naming [[Slendric#Interval categories|used for slendric]]. It is particularly convenient to use diatonic conventions for mothra, because its chain of fifths is meantone, and therefore 5/4 is simply read as a major third.

As a strong extension of slendric, mothra's intervals can be expressed using the same system of extended diatonic interval naming [[Slendric #Interval categories|used for slendric]]. It is particularly convenient to use diatonic conventions for mothra, because its chain of fifths is meantone, and therefore 5/4 is simply read as a major third.

In the following table, odd harmonics and subharmonics 1–21 are labeled in '''bold'''.

{| class="wikitable sortable center-~~all~~ right-3"

{| class="wikitable sortable center-1 center-2 right-3"

|-

! rowspan="3" | &#~~35;~~ !! rowspan="3" | Extended <br /> diatonic <br /> interval !! rowspan="3" | Cents* !! colspan="3" | Approximate ratios

! rowspan="3" | # !! rowspan="3" | Extended <br> diatonic <br> interval !! rowspan="3" | Cents* !! colspan="3" | Approximate ratios

|-

! rowspan="2" | 7-limit intervals !! colspan="2" | Intervals of ~~undecimal~~ extensions

! rowspan="2" | 7-limit intervals !! colspan="2" | Intervals of 11-limit extensions

|-

! Undecimal mothra !! Mosura

Line 222:

| 33/32, 55/54

|}

<nowiki/>* In 7-limit [[CWE tuning]]

<nowiki/>* In 7-limit [[CWE tuning]], octave reduced

== Tuning spectrum ==

Vals refer to the appropriate undecimal extension in the EDO's range.

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| 232.193

|

| 1/4-comma meantone fifth, 5-odd-limit minimax

| 1/4-comma meantone fifth, (7-limit) 5- through 21-odd-limit minimax

|-

|

@@ Line 8: / Line 8: @@
 | Mapping = 1; 3 12 -1
 | Pergen = (P8, P5/3)
-| Odd limit 1 = 7 | Mistuning 1 = ? | Complexity 1 = 31
+| Odd limit 1 = 7 | Mistuning 1 = 5.4 | Complexity 1 = 31
-| Odd limit 2 = (2.3.5.7) 21 | Mistuning 2 = ? | Complexity 2 = 36
+| Odd limit 2 = (2.3.5.7) 21 | Mistuning 2 = 10.8 | Complexity 2 = 36
 }}
-'''Mothra''' is a temperament in the [[7-limit]] that is a strong extension to [[slendric]], which is defined by splitting the interval of [[3/2]] into three [[8/7]]s and tempering out [[1029/1024]]. The fifth of mothra is flattened to a [[meantone]] fifth, so that it reaches [[5/4]] when stacked four times and [[81/80]] is tempered out, unlike that of the other slendric extension [[rodan]], which is sharpened from just. This has the effect of bringing the generator 8/7 considerably closer to just, and also allowing [[MOS scale]]s of mothra to be more melodically usable than those of other forms of slendric, as the structurally-pervasive small step known as the [[quark]] (the residue between the octave and 5 generators, representing [[49/48]], [[64/63]], and in mothra also [[36/35]]) is larger here. [[EDOs]] that support mothra include [[26edo]], [[31edo]], and [[36edo]], and 31 is a particularly good tuning.
+'''Mothra''', also known as '''cynder''', is a temperament of the [[7-limit]] that is a strong extension to [[slendric]], which is defined by splitting a perfect fifth representing [[3/2]] into three intervals of [[8/7]], tempering out [[1029/1024]]. The fifth of mothra is flattened to a [[meantone]] fifth, so that it reaches [[5/4]] when stacked four times and [[81/80]] is tempered out, unlike that of the other slendric extension [[rodan]], which is sharpened from just. This has the effect of bringing the generator 8/7 considerably closer to just, and also allowing [[MOS scale]]s of mothra to be more melodically usable than those of other forms of slendric, as the structurally-pervasive small step known as the [[quark]] (the residue between the octave and 5 generators, representing [[49/48]], [[64/63]], and in mothra also [[36/35]]) is larger here. [[EDOs]] that support mothra include [[26edo]], [[31edo]], and [[36edo]], and 31 is a particularly good tuning.
 In the [[11-limit]], two extensions are of note: undecimal mothra (26 & 31), which tempers out [[99/98]], [[385/384]] and [[441/440]] to find the 11th harmonic at 8 generators down, and mosura (31 & 36), which tempers out [[176/175]] to find the 11th harmonic at 23 generators up. These two mappings merge at 31edo, which is therefore a uniquely suitable tuning for 11-limit mothra.
-In higher limits, one may note that the two-generator interval closely approximates [[17/13]], and that the six-generator interval - the meantone whole tone of [[9/8]][[~]][[10/9]], approximates [[19/17]], so that the 13:17:19 chord is well-approximated; it is worth noting also that this chord is entirely included within the subtemperament obtained from taking every other generator of mothra, which is [[A-team]]. This can be combined with the canonical mapping of 13 for each undecimal extension, which tempers out [[144/143]], to provide a natural route to the [[19-limit]].
+In higher limits, one may note that the two-generator interval closely approximates [[17/13]], and that the six-generator interval - the meantone whole tone of [[9/8]][[~]][[10/9]], approximates [[19/17]] - so that the 13:17:19 chord is well-represented; it is worth noting also that this chord is entirely included within the subtemperament obtained from taking every other generator of mothra, which is [[A-team]] (the crawma, [[83521/83486]], is the relevant comma tempered out here). This can be combined with the canonical mapping of 13 for each undecimal extension, which tempers out [[144/143]], to provide a natural route to the [[19-limit]].
 For technical data, see [[Gamelismic clan #Mothra]].
 == Intervals ==
-As a strong extension of slendric, mothra's intervals can be expressed using the same system of extended diatonic interval naming [[Slendric#Interval categories|used for slendric]]. It is particularly convenient to use diatonic conventions for mothra, because its chain of fifths is meantone, and therefore 5/4 is simply read as a major third.
+As a strong extension of slendric, mothra's intervals can be expressed using the same system of extended diatonic interval naming [[Slendric #Interval categories|used for slendric]]. It is particularly convenient to use diatonic conventions for mothra, because its chain of fifths is meantone, and therefore 5/4 is simply read as a major third.
 In the following table, odd harmonics and subharmonics 1–21 are labeled in '''bold'''.
-{| class="wikitable sortable center-all right-3"
+{| class="wikitable sortable center-1 center-2 right-3"
 |-
-! rowspan="3" | &#35; !! rowspan="3" | Extended <br /> diatonic <br /> interval !! rowspan="3" | Cents* !! colspan="3" | Approximate ratios
+! rowspan="3" | # !! rowspan="3" | Extended <br> diatonic <br> interval !! rowspan="3" | Cents* !! colspan="3" | Approximate ratios
 |-
-! rowspan="2" | 7-limit intervals !! colspan="2" | Intervals of undecimal extensions
+! rowspan="2" | 7-limit intervals !! colspan="2" | Intervals of 11-limit extensions
 |-
 ! Undecimal mothra !! Mosura
@@ Line 222: / Line 222: @@
 | 33/32, 55/54
 |}
-<nowiki/>* In 7-limit [[CWE tuning]]
+<nowiki/>* In 7-limit [[CWE tuning]], octave reduced
 == Tuning spectrum ==
+{{see also|Slendric #Tuning spectrum}}
 Vals refer to the appropriate undecimal extension in the EDO's range.
@@ Line 329: / Line 331: @@
 | 232.193
 |
-| 1/4-comma meantone fifth, 5-odd-limit minimax
+| 1/4-comma meantone fifth, (7-limit) 5- through 21-odd-limit minimax
 |-
 |