Quasisuper: Difference between revisions

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{{Infobox ~~Regtemp~~

{{Infobox regtemp

| Title = Quasisuper; quasisupra

| Subgroups = 2.3.5.7, 2.3.5.7.11

Line 5:

| Edo join 1 = 17c | Edo join 2 = 22

| Mapping = 1; 1 -13 -2 -6

| ~~Generator~~ = 3/2

| Generators = 3/2

| ~~Generator~~ tuning = 708.3

| Generators tuning = 708.3

| Optimization method = CWE

| MOS scales = [[5L 2s]], [[5L 7s]], [[5L 12s]]

| MOS scales = [[5L 2s]], [[5L 7s]], [[5L 12s]], [[17L 5s]]

| Pergen = (P8, P5)

| Color name = Sasaguti

| Odd limit 1 = 9 | Mistuning 1 = 13.7 | Complexity 1 = 17

| Odd limit 2 = (11-limit) 15 | Mistuning 2 = 14.9 | Complexity 2 = 17

| Odd limit 2 = 11-limit 15 | Mistuning 2 = 14.9 | Complexity 2 = 17

}}

'''Quasisuper''' is an alternative [[extension]] of the [[archy]] [[chain of fifths]] to [[superpyth]]. Like superpyth, it is a [[regular temperament|temperament]] generated by a perfect fifth, where stacking two of them reaches the interval of [[8/7]][[~]][[9/8]], [[tempering out]] [[64/63]]. The difference is that this extension maps [[prime interval|prime]] [[5/1|5]] to -13 [[generator]]s, as a double-diminished fifth (C–G𝄫). This extension works in the range [[17edo|17c-edo]] to [[22edo|22-edo]]. In contrast, full 7-limit [[superpyth]] does not work in this range, as tunings with a flatter fifth than 22edo swap the sizes of [[7/5]] and [[10/7]]. This extension may be preferred over superpyth due to having a softer [[5L 2s|diatonic]] scale, with a small step of around 60 [[cent]]s compared to about 50 cents in regular 7-limit superpyth.

~~The best~~ extension to ~~the~~ [[~~11-limit~~]], ~~'''quasisupra'''~~, ~~uses~~ the [[~~supra~~]] ~~mapping of~~ prime [[11/1|11]] to ~~-6 generators~~, as a diminished fifth (~~C–G♭~~). This ~~tempers out~~ [[~~99/98~~]] as in ~~supra~~, as ~~well as~~ [[~~121~~/~~120~~]] and [[~~540~~/~~539~~]].

'''Quasisuper''' is an alternative [[extension]] of the [[archy]] [[chain of fifths]] to [[superpyth]]. Like superpyth, it is a [[regular temperament|temperament]] generated by a perfect fifth, where stacking two of them reaches the interval of [[8/7]][[~]][[9/8]], [[tempering out]] [[64/63]]. The difference is that this extension maps [[prime interval|prime]] [[5/1|5]] to −13 [[generator]]s, as a double-diminished fifth (C–G𝄫). This extension works in the range [[17edo|17c-edo]] to [[22edo|22-edo]]. In contrast, full 7-limit [[superpyth]] does not work in this range, as tunings with a flatter fifth than 22edo swap the sizes of [[7/5]] and [[10/7]]. This extension may be preferred over superpyth due to having a softer [[5L 2s|diatonic]] scale, with a small step of around 60 [[cent]]s compared to about 50 cents in regular 7-limit superpyth.

For technical data see [[Archytas clan #Quasisuper]].

The best extension to the [[11-limit]], '''quasisupra''', maps prime [[11/1|11]] to −6 generators as a diminished fifth (C–G♭), tempering out [[99/98]] as well as [[121/120]] and [[540/539]]. Removing prime 5 from quasisupra results in a 2.3.7.11-subgroup restriction, called '''supra''', which is notable for its simplicity. Finally, taking every other step of supra gives a 2.9.7.11-subgroup restriction, called [[machine]].

For technical data see [[Archytas clan #Quasisuper]] and [[Archytas clan #Supra|#Supra]].

== Interval chain ==

In the following ~~table~~, odd harmonics and subharmonics 1–11 are in '''bold'''.

In the following tables, odd harmonics and subharmonics 1–11 are in '''bold'''.

{| class="wikitable center-1 right-2 right-4"

|+ style="font-size: 105%;" | Supra (2.3.7.11)

|-

! # !! Cents* !! Approximate ratios

|-

| 0 || 0.0 || '''1/1'''

|-

| 1 || 707.5 || '''3/2'''

|-

| 2 || 215.0 || '''8/7''', '''9/8'''

|-

| 3 || 922.5 || 12/7

|-

| 4 || 430.0 || 9/7, 14/11

|-

| 5 || 1137.5 || 21/11, 27/14, 64/33

|-

| 6 || 645.0 || '''16/11'''

|-

| 7 || 152.5 || 12/11

|}

<nowiki/>* In 2.3.7.11-subgroup [[CWE]] tuning, <br>octave reduced

</div></div>

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

|+ style="font-size: 105%;" | Quasisuper/quasisupra

|-

! #

! Cents*

Line 100:

Line 128:

| 81/80, 56/55

|}

<nowiki/>* in 11-limit [[CWE]] tuning, octave reduced

<nowiki/>* in 11-limit CWE tuning, octave reduced

</div></div>

== Scales ==

@@ Line 1: / Line 1: @@
-{{Infobox Regtemp
+{{Infobox regtemp
 | Title = Quasisuper;&nbsp;quasisupra
 | Subgroups = 2.3.5.7, 2.3.5.7.11
@@ Line 5: / Line 5: @@
 | Edo join 1 = 17c | Edo join 2 = 22
 | Mapping = 1; 1 -13 -2 -6
-| Generator = 3/2
+| Generators = 3/2
-| Generator tuning = 708.3
+| Generators tuning = 708.3
 | Optimization method = CWE
-| MOS scales = [[5L 2s]], [[5L 7s]], [[5L 12s]]
+| MOS scales = [[5L&nbsp;2s]], [[5L&nbsp;7s]], [[5L&nbsp;12s]], [[17L&nbsp;5s]]
 | Pergen = (P8, P5)
 | Color name = Sasaguti
 | Odd limit 1 = 9 | Mistuning 1 = 13.7 | Complexity 1 = 17
-| Odd limit 2 = (11-limit) 15 | Mistuning 2 = 14.9 | Complexity 2 = 17
+| Odd limit 2 = 11-limit 15 | Mistuning 2 = 14.9 | Complexity 2 = 17
 }}
-'''Quasisuper''' is an alternative [[extension]] of the [[archy]] [[chain of fifths]] to [[superpyth]]. Like superpyth, it is a [[regular temperament|temperament]] generated by a perfect fifth, where stacking two of them reaches the interval of [[8/7]][[~]][[9/8]], [[tempering out]] [[64/63]]. The difference is that this extension maps [[prime interval|prime]] [[5/1|5]] to -13 [[generator]]s, as a double-diminished fifth (C–G𝄫). This extension works in the range [[17edo|17c-edo]] to [[22edo|22-edo]]. In contrast, full 7-limit [[superpyth]] does not work in this range, as tunings with a flatter fifth than 22edo swap the sizes of [[7/5]] and [[10/7]]. This extension may be preferred over superpyth due to having a softer [[5L 2s|diatonic]] scale, with a small step of around 60 [[cent]]s compared to about 50 cents in regular 7-limit superpyth.
-The best extension to the [[11-limit]], '''quasisupra''', uses the [[supra]] mapping of prime [[11/1|11]] to -6 generators, as a diminished fifth (C–G♭). This tempers out [[99/98]] as in supra, as well as [[121/120]] and [[540/539]].
+'''Quasisuper''' is an alternative [[extension]] of the [[archy]] [[chain of fifths]] to [[superpyth]]. Like superpyth, it is a [[regular temperament|temperament]] generated by a perfect fifth, where stacking two of them reaches the interval of [[8/7]][[~]][[9/8]], [[tempering out]] [[64/63]]. The difference is that this extension maps [[prime interval|prime]] [[5/1|5]] to −13 [[generator]]s, as a double-diminished fifth (C–G𝄫). This extension works in the range [[17edo|17c-edo]] to [[22edo|22-edo]]. In contrast, full 7-limit [[superpyth]] does not work in this range, as tunings with a flatter fifth than 22edo swap the sizes of [[7/5]] and [[10/7]]. This extension may be preferred over superpyth due to having a softer [[5L 2s|diatonic]] scale, with a small step of around 60 [[cent]]s compared to about 50 cents in regular 7-limit superpyth.
-For technical data see [[Archytas clan #Quasisuper]].
+The best extension to the [[11-limit]], '''quasisupra''', maps prime [[11/1|11]] to −6 generators as a diminished fifth (C–G♭), tempering out [[99/98]] as well as [[121/120]] and [[540/539]]. Removing prime 5 from quasisupra results in a 2.3.7.11-subgroup restriction, called '''supra''', which is notable for its simplicity. Finally, taking every other step of supra gives a 2.9.7.11-subgroup restriction, called [[machine]].
+For technical data see [[Archytas clan #Quasisuper]] and [[Archytas clan #Supra|#Supra]].
 == Interval chain ==
-In the following table, odd harmonics and subharmonics 1–11 are in '''bold'''.
+In the following tables, odd harmonics and subharmonics 1–11 are in '''bold'''.
+<div><div style="display: inline-grid; margin-right: 25px;">
+{| class="wikitable center-1 right-2 right-4"
+|+ style="font-size: 105%;" | Supra (2.3.7.11)
+|-
+! # !! Cents* !! Approximate ratios
+|-
+| 0 || 0.0 || '''1/1'''
+|-
+| 1 || 707.5 || '''3/2'''
+|-
+| 2 || 215.0 || '''8/7''', '''9/8'''
+|-
+| 3 || 922.5 || 12/7
+|-
+| 4 || 430.0 || 9/7, 14/11
+|-
+| 5 || 1137.5 || 21/11, 27/14, 64/33
+|-
+| 6 || 645.0 || '''16/11'''
+|-
+| 7 || 152.5 || 12/11
+|}
+<nowiki/>* In 2.3.7.11-subgroup [[CWE]] tuning, <br>octave reduced
+</div></div>
+<div><div style="display: inline-grid;">
 {| class="wikitable center-1 right-2"
+|+ style="font-size: 105%;" | Quasisuper/quasisupra
+|-
 ! #
 ! Cents*
@@ Line 100: / Line 128: @@
 | 81/80, 56/55
 |}
-<nowiki/>* in 11-limit [[CWE]] tuning, octave reduced
+<nowiki/>* in 11-limit CWE tuning, octave reduced
+</div></div>
 == Scales ==