Miracle: Difference between revisions

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[[de:Miracle]]

'''Miracle''' is a [[regular temperament]] discovered by [[George Secor]] in 1974 which ~~has as~~ a [[generator]] ~~an interval~~, called a ''[[secor]]'' (after George), that ~~serves~~ as both [[15/14]] and [[16/15]] semitones.

'''Miracle''' is a [[regular temperament]] discovered by [[George Secor]] in 1974 which splits a tempered [[3/2]] into six [[generator]]s, called ''[[secor]]s'' (after George), that serve as both [[15/14]] and [[16/15]] semitones. A stack of two generators represents [[8/7]], and a stack of seven generators represents [[8/5]]. It is a member of both the [[marvel temperaments]], by [[tempering out]] [[225/224]], and the [[gamelismic clan]], by tempering out [[1029/1024]]. It extends naturally to the [[11-limit]] by treating the neutral third from three generators as [[11/9]], tempering out [[243/242]], [[385/384]], [[441/440]], and [[540/539]].

Miracle is an exceptionally efficient linear temperament ~~which is a member of both the [[marvel temperaments]], by [[tempering out]] [[225/224]], and the [[gamelismic clan]], by tempering out [[1029/1024]]~~. It is quite accurate, with [[TOP]] error only 0.63 [[cent]]s/[[octave]], meaning intervals of the [[11-odd-limit]] [[tonality diamond]] are represented with only one or two cents of error. Yet it is also very low-complexity (efficient), as evidenced by the high density of [[11-odd-limit]] ratios in the [[#Interval chain]]. At least one inversion of every interval in the 11-odd-limit tonality diamond is represented within 22 secors of the starting value.

Miracle is an exceptionally efficient linear temperament. It is quite accurate, with [[TOP]] error only 0.63 [[cent]]s/[[octave]], meaning intervals of the [[11-odd-limit]] [[tonality diamond]] are represented with only one or two cents of error. Yet it is also very low-complexity (efficient), as evidenced by the high density of 11-odd-limit ratios in the [[#Interval chain]]. At least one inversion of every interval in the 11-odd-limit tonality diamond is represented within 22 secors of the starting value.

Some temperaments have [[11/9]] as a "neutral third", meaning it is exactly half of a [[3/2]] (tempering out [[243/242]]), and other temperaments ~~(→ [[Gamelismic clan]])~~ have [[8/7]] as exactly a third of [[3/2]]. Miracle is distinguished by doing both of these things at the same time, so 3/2 is divided into six equal parts~~. This is in fact the generator of miracle temperament, called a ''secor'', and it represents both [[16/15]] and [[15/14]]~~.

Some temperaments have 11/9 as a neutral third, meaning it is exactly half of a 3/2 (tempering out 243/242), and other temperaments have 8/7 as exactly a third of 3/2. Miracle is distinguished by doing both of these things at the same time, so 3/2 is divided into six equal parts.

Miracle can also be thought of as a [[cluster temperament]] with 10 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing [[45/44]] ~ [[49/48]] ~ [[50/49]] ~ [[55/54]] ~ [[56/55]] ~ [[64/63]] all [[tempered]] together.

~~In terms of~~ [[13-limit]] ~~extensions, it is discussed in~~ [[~~Miracle extensions~~]]. See [[Gamelismic clan #Miracle]] for technical data.

See [[Miracle extensions]] for [[13-limit]] and [[17-limit]] extensions. See [[Gamelismic clan #Miracle]] for technical data.

== Interval chain ==

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Displayed on the right is a chart of the tuning spectrum of miracle by how the odd harmonics up to 11 are tuned, showing the minimax generator, i.e. the secor.

=== Prime-optimized tunings ===

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 7-limit Prime-optimized tunings

|-

! rowspan="2" |

! colspan="2" | Euclidean

|-

! Constrained

! Constrained & skewed

|-

! Equilateral

| CEE: ~15/14 = 116.516{{c}}

| CSEE: ~15/14 = 116.561{{c}}

|-

! Tenney

| CTE: ~15/14 = 116.677{{c}}

| CWE: ~15/14 = 116.676{{c}}

|-

! Benedetti, Wilson

| CBE: ~15/14 = 116.730{{c}}

| CSBE: ~15/14 = 116.714{{c}}

|}

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 11-limit prime-optimized tunings

|-

! rowspan="2" |

! colspan="2" | Euclidean

|-

! Constrained

! Constrained & skewed

|-

! Equilateral

| CEE: ~15/14 = 116.687{{c}}

| CSEE: ~15/14 = 116.630{{c}}

|-

! Tenney

| CTE: ~15/14 = 116.711{{c}}

| CWE: ~15/14 = 116.647{{c}}

|-

! Benedetti, Wilson

| CBE: ~15/14 = 116.736{{c}}

| CSBE: ~15/14 = 116.677{{c}}

|}

=== Target tunings ===

{| class="wikitable center-all mw-collapsible mw-collapsed"

|+ style="white-space: nowrap;" | Minimax tunings

|-

! Target

! Generator

! Eigenmonzo*

|-

| 5-odd-limit

| ~16/15 = 116.588{{c}}

| 5/3

|-

| 7-odd-limit

| ~15/14 = 116.588{{c}}

| 5/3

|-

| 9-odd-limit

| ~15/14 = 116.716{{c}}

| 9/5

|-

| 11-odd-limit

| ~15/14 = 116.716{{c}}

| 9/5

|}

{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"

|+ style="white-space: nowrap;" | Least squares tunings

|-

! Target

! Generator

! Eigenmonzo*

|-

| 5-odd-limit

| ~16/15 = 116.578{{c}}

| {{Monzo| 0 -19 20 }}

|-

| 7-odd-limit

| ~15/14 = 116.573{{c}}

| {{Monzo| 0 -27 25 5 }}

|-

| 9-odd-limit

| ~15/14 = 116.721{{c}}

| {{Monzo| 0 117 -44 -19 }}

|-

| 11-odd-limit

| ~15/14 = 116.672{{c}}

| {{Monzo| 0 17 -11 -6 11 }}

|}

=== Tuning spectrum ===

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|-

! Edo generator

! [[Eigenmonzo|~~Eigenmonzo~~ (~~unchanged-interval~~)]]

! [[Eigenmonzo|Unchanged interval (eigenmonzo)]]*

! Generator (¢)

! Comments

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|

|-

| [[103edo|10\103]]

|

~~| {{monzo| 0 -27 25 5 }}~~

| 116.505

| 116.~~573~~

~~| 7-odd-limit least squares~~

|-

|

~~| {{monzo| 0 -19 20 }}~~

~~| 116.578~~

~~| 5-odd-limit least squares~~

|-

|

Line 271:

Line 360:

| 116.667

|

|-

|

~~| {{monzo| 0 17 -11 -6 11 }}~~

~~| 116.672~~

~~| 11-odd-limit least squares~~

|-

|

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Line 365:

| 116.716

| 9- and 11-odd-limit minimax, Secor's definition of secor

|-

|

~~| {{monzo| 0 117 -44 -19 }}~~

~~| 116.721~~

~~| 9-odd-limit least squares~~

|-

|

Line 300:

Line 379:

| 9/7

| 116.792

|

|-

| [[113edo|11\113]]

|

| 116.814

|

|-

Line 327:

Line 411:

| Upper bound of 7- and 9-odd-limit diamond monotone

|}

<nowiki/>* Besides the octave

== Music ==

; [[Herman Miller]]

* [https://soundcloud.com/morphosyntax-1/realm-of-possibility ''Realm of Possibility''] (2021) – Miracle[31] with a 116.72-cent generator and 1200.53-cent octave

* [https://soundcloud.com/morphosyntax-1/realm-of-possibility ''Realm of Possibility''] (2021) – in Miracle[31] with a 116.72-cent generator and 1200.53-cent octave

; [[Joseph Pehrson]]

* ''Blackjack'' (2001) – [https://web.archive.org/web/20201127013023/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blackjack.mp3 play] | [https://soundclick.com/share.cfm?id=706344 SoundClick] – [[Blackjack|Blackjack (Miracle{{lbrack}}21{{rbrack}})]]

* ''Blackjack'' (2001) – [https://web.archive.org/web/20201127013023/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blackjack.mp3 play] | [https://soundclick.com/share.cfm?id=706344 SoundClick] – in [[Blackjack|Blackjack (Miracle{{lbrack}}21{{rbrack}})]]

* ''Blacklight'' (2002) – [https://web.archive.org/web/20201127015033/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blacklight.mp3 play] | [https://soundclick.com/share.cfm?id=710783 SoundClick] – Blackjack (Miracle[21])

* ''Blacklight'' (2002) – [https://web.archive.org/web/20201127015033/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blacklight.mp3 play] | [https://soundclick.com/share.cfm?id=710783 SoundClick] – in Blackjack (Miracle[21])

* ''Black and Jill'' (2003) – Blackjack (Miracle[21])

* ''Black and Jill'' (2003) – in Blackjack (Miracle[21])

** Soprano version – [https://web.archive.org/web/20201127012730/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/blackandjill.mp3 play] | [https://soundclick.com/share.cfm?id=2373400 SoundClick]

** [https://soundclick.com/share.cfm?id=9583778 Udderbot version]

* [https://soundclick.com/share.cfm?id=2623155 ''Inner Voices''] (2005) – Blackjack (Miracle[21])

* [https://soundclick.com/share.cfm?id=2623155 ''Inner Voices''] (2005) – in Blackjack (Miracle[21])

* [https://soundclick.com/share.cfm?id=6593353 ''Transpian''] (2006) – Blackjack (Miracle[21])

* [https://soundclick.com/share.cfm?id=6593353 ''Transpian''] (2006) – in Blackjack (Miracle[21])

* [https://soundclick.com/share.cfm?id=5049231 ''microproj''] (2007) – Blackjack (Miracle[21])

* [https://soundclick.com/share.cfm?id=5049231 ''microproj''] (2007) – in Blackjack (Miracle[21])

; [[Gene Ward Smith]]

* ''Rachmaninoff Plays Blackjack'' (archived 2010) – [http://www.archive.org/details/RachmaninoffPlaysBlackjack detail] | [http://www.archive.org/download/RachmaninoffPlaysBlackjack/rachman.mp3 play] – Blackjack (Miracle[21] in 175edo tuning

* ''Rachmaninoff Plays Blackjack'' (archived 2010) – [http://www.archive.org/details/RachmaninoffPlaysBlackjack detail] | [http://www.archive.org/download/RachmaninoffPlaysBlackjack/rachman.mp3 play] – in Blackjack (Miracle[21]), 175edo tuning

~~[[Category:Temperaments]]~~

[[Category:Miracle| ]]

[[Category:Rank-2 temperaments]]

[[Category:Marvel temperaments]]

[[Category:Gamelismic clan]]

@@ Line 1: / Line 1: @@
 [[de:Miracle]]
 {{Wikipedia|Miracle temperament}}
-'''Miracle''' is a [[regular temperament]] discovered by [[George Secor]] in 1974 which has as a [[generator]] an interval, called a ''[[secor]]'' (after George), that serves as both [[15/14]] and [[16/15]] semitones.
+'''Miracle''' is a [[regular temperament]] discovered by [[George Secor]] in 1974 which splits a tempered [[3/2]] into six [[generator]]s, called ''[[secor]]s'' (after George), that serve as both [[15/14]] and [[16/15]] semitones. A stack of two generators represents [[8/7]], and a stack of seven generators represents [[8/5]]. It is a member of both the [[marvel temperaments]], by [[tempering out]] [[225/224]], and the [[gamelismic clan]], by tempering out [[1029/1024]]. It extends naturally to the [[11-limit]] by treating the neutral third from three generators as [[11/9]], tempering out [[243/242]], [[385/384]], [[441/440]], and [[540/539]].
-Miracle is an exceptionally efficient linear temperament which is a member of both the [[marvel temperaments]], by [[tempering out]] [[225/224]], and the [[gamelismic clan]], by tempering out [[1029/1024]]. It is quite accurate, with [[TOP]] error only 0.63 [[cent]]s/[[octave]], meaning intervals of the [[11-odd-limit]] [[tonality diamond]] are represented with only one or two cents of error. Yet it is also very low-complexity (efficient), as evidenced by the high density of [[11-odd-limit]] ratios in the [[#Interval chain]]. At least one inversion of every interval in the 11-odd-limit tonality diamond is represented within 22 secors of the starting value.
+Miracle is an exceptionally efficient linear temperament. It is quite accurate, with [[TOP]] error only 0.63 [[cent]]s/[[octave]], meaning intervals of the [[11-odd-limit]] [[tonality diamond]] are represented with only one or two cents of error. Yet it is also very low-complexity (efficient), as evidenced by the high density of 11-odd-limit ratios in the [[#Interval chain]]. At least one inversion of every interval in the 11-odd-limit tonality diamond is represented within 22 secors of the starting value.
-Some temperaments have [[11/9]] as a "neutral third", meaning it is exactly half of a [[3/2]] (tempering out [[243/242]]), and other temperaments (&rarr; [[Gamelismic clan]]) have [[8/7]] as exactly a third of [[3/2]]. Miracle is distinguished by doing both of these things at the same time, so 3/2 is divided into six equal parts. This is in fact the generator of miracle temperament, called a ''secor'', and it represents both [[16/15]] and [[15/14]].
+Some temperaments have 11/9 as a neutral third, meaning it is exactly half of a 3/2 (tempering out 243/242), and other temperaments have 8/7 as exactly a third of 3/2. Miracle is distinguished by doing both of these things at the same time, so 3/2 is divided into six equal parts.
 Miracle can also be thought of as a [[cluster temperament]] with 10 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing [[45/44]] ~ [[49/48]] ~ [[50/49]] ~ [[55/54]] ~ [[56/55]] ~ [[64/63]] all [[tempered]] together.
-In terms of [[13-limit]] extensions, it is discussed in [[Miracle extensions]]. See [[Gamelismic clan #Miracle]] for technical data.
+See [[Miracle extensions]] for [[13-limit]] and [[17-limit]] extensions. See [[Gamelismic clan #Miracle]] for technical data.
 == Interval chain ==
@@ Line 178: / Line 178: @@
 Displayed on the right is a chart of the tuning spectrum of miracle by how the odd harmonics up to 11 are tuned, showing the minimax generator, i.e. the secor.
+=== Prime-optimized tunings ===
+{| class="wikitable mw-collapsible mw-collapsed"
+|+ style="font-size: 105%; white-space: nowrap;" | 7-limit Prime-optimized tunings
+|-
+! rowspan="2" |
+! colspan="2" | Euclidean
+|-
+! Constrained
+! Constrained & skewed
+|-
+! Equilateral
+| CEE: ~15/14 = 116.516{{c}}
+| CSEE: ~15/14 = 116.561{{c}}
+|-
+! Tenney
+| CTE: ~15/14 = 116.677{{c}}
+| CWE: ~15/14 = 116.676{{c}}
+|-
+! Benedetti, <br>Wilson
+| CBE: ~15/14 = 116.730{{c}}
+| CSBE: ~15/14 = 116.714{{c}}
+|}
+{| class="wikitable mw-collapsible mw-collapsed"
+|+ style="font-size: 105%; white-space: nowrap;" | 11-limit prime-optimized tunings
+|-
+! rowspan="2" |
+! colspan="2" | Euclidean
+|-
+! Constrained
+! Constrained & skewed
+|-
+! Equilateral
+| CEE: ~15/14 = 116.687{{c}}
+| CSEE: ~15/14 = 116.630{{c}}
+|-
+! Tenney
+| CTE: ~15/14 = 116.711{{c}}
+| CWE: ~15/14 = 116.647{{c}}
+|-
+! Benedetti, <br>Wilson
+| CBE: ~15/14 = 116.736{{c}}
+| CSBE: ~15/14 = 116.677{{c}}
+|}
+=== Target tunings ===
+{| class="wikitable center-all mw-collapsible mw-collapsed"
+|+ style="white-space: nowrap;" | Minimax tunings
+|-
+! Target
+! Generator
+! Eigenmonzo*
+|-
+| 5-odd-limit
+| ~16/15 = 116.588{{c}}
+| 5/3
+|-
+| 7-odd-limit
+| ~15/14 = 116.588{{c}}
+| 5/3
+|-
+| 9-odd-limit
+| ~15/14 = 116.716{{c}}
+| 9/5
+|-
+| 11-odd-limit
+| ~15/14 = 116.716{{c}}
+| 9/5
+|}
+{| class="wikitable center-all left-3 mw-collapsible mw-collapsed"
+|+ style="white-space: nowrap;" | Least squares tunings
+|-
+! Target
+! Generator
+! Eigenmonzo*
+|-
+| 5-odd-limit
+| ~16/15 = 116.578{{c}}
+| {{Monzo| 0 -19 20 }}
+|-
+| 7-odd-limit
+| ~15/14 = 116.573{{c}}
+| {{Monzo| 0 -27 25 5 }}
+|-
+| 9-odd-limit
+| ~15/14 = 116.721{{c}}
+| {{Monzo| 0 117 -44 -19 }}
+|-
+| 11-odd-limit
+| ~15/14 = 116.672{{c}}
+| {{Monzo| 0 17 -11 -6 11 }}
+|}
 === Tuning spectrum ===
@@ Line 183: / Line 277: @@
 |-
 ! Edo<br>generator
-! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
+! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
 ! Generator (¢)
 ! Comments
@@ Line 232: / Line 326: @@
 |
 |-
+| [[103edo|10\103]]
 |
-| {{monzo| 0 -27 25 5 }}
+| 116.505
-| 116.573
-| 7-odd-limit least squares
-|-
 |
-| {{monzo| 0 -19 20 }}
-| 116.578
-| 5-odd-limit least squares
 |-
 |
@@ Line 271: / Line 360: @@
 | 116.667
 |
-|-
-|
-| {{monzo| 0 17 -11 -6 11 }}
-| 116.672
-| 11-odd-limit least squares
 |-
 |
@@ Line 281: / Line 365: @@
 | 116.716
 | 9- and 11-odd-limit minimax, <br>Secor's definition of secor
-|-
-|
-| {{monzo| 0 117 -44 -19 }}
-| 116.721
-| 9-odd-limit least squares
 |-
 |
@@ Line 300: / Line 379: @@
 | 9/7
 | 116.792
+|
+|-
+| [[113edo|11\113]]
+|
+| 116.814
 |
 |-
@@ Line 327: / Line 411: @@
 | Upper bound of 7- and 9-odd-limit diamond monotone
 |}
+<nowiki/>* Besides the octave
 == Music ==
 ; [[Herman Miller]]
-* [https://soundcloud.com/morphosyntax-1/realm-of-possibility ''Realm of Possibility''] (2021) – Miracle[31] with a 116.72-cent generator and 1200.53-cent octave
+* [https://soundcloud.com/morphosyntax-1/realm-of-possibility ''Realm of Possibility''] (2021) – in Miracle[31] with a 116.72-cent generator and 1200.53-cent octave
 ; [[Joseph Pehrson]]
-* ''Blackjack'' (2001) – [https://web.archive.org/web/20201127013023/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blackjack.mp3 play] | [https://soundclick.com/share.cfm?id=706344 SoundClick] – [[Blackjack|Blackjack (Miracle{{lbrack}}21{{rbrack}})]]
+* ''Blackjack'' (2001) – [https://web.archive.org/web/20201127013023/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blackjack.mp3 play] | [https://soundclick.com/share.cfm?id=706344 SoundClick] – in [[Blackjack|Blackjack (Miracle{{lbrack}}21{{rbrack}})]]
-* ''Blacklight'' (2002) – [https://web.archive.org/web/20201127015033/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blacklight.mp3 play] | [https://soundclick.com/share.cfm?id=710783 SoundClick] – Blackjack (Miracle[21])
+* ''Blacklight'' (2002) – [https://web.archive.org/web/20201127015033/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blacklight.mp3 play] | [https://soundclick.com/share.cfm?id=710783 SoundClick] – in Blackjack (Miracle[21])
-* ''Black and Jill'' (2003) – Blackjack (Miracle[21])
+* ''Black and Jill'' (2003) – in Blackjack (Miracle[21])
 ** Soprano version – [https://web.archive.org/web/20201127012730/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/blackandjill.mp3 play] | [https://soundclick.com/share.cfm?id=2373400 SoundClick]
 ** [https://soundclick.com/share.cfm?id=9583778 Udderbot version]
-* [https://soundclick.com/share.cfm?id=2623155 ''Inner Voices''] (2005) – Blackjack (Miracle[21])
+* [https://soundclick.com/share.cfm?id=2623155 ''Inner Voices''] (2005) – in Blackjack (Miracle[21])
-* [https://soundclick.com/share.cfm?id=6593353 ''Transpian''] (2006) – Blackjack (Miracle[21])
+* [https://soundclick.com/share.cfm?id=6593353 ''Transpian''] (2006) – in Blackjack (Miracle[21])
-* [https://soundclick.com/share.cfm?id=5049231 ''microproj''] (2007) – Blackjack (Miracle[21])
+* [https://soundclick.com/share.cfm?id=5049231 ''microproj''] (2007) – in Blackjack (Miracle[21])
 ; [[Gene Ward Smith]]
-* ''Rachmaninoff Plays Blackjack'' (archived 2010) – [http://www.archive.org/details/RachmaninoffPlaysBlackjack detail] | [http://www.archive.org/download/RachmaninoffPlaysBlackjack/rachman.mp3 play] – Blackjack (Miracle[21] in 175edo tuning
+* ''Rachmaninoff Plays Blackjack'' (archived 2010) – [http://www.archive.org/details/RachmaninoffPlaysBlackjack detail] | [http://www.archive.org/download/RachmaninoffPlaysBlackjack/rachman.mp3 play] – in Blackjack (Miracle[21]), 175edo tuning
-[[Category:Temperaments]]
 [[Category:Miracle| ]] <!-- main article -->
+[[Category:Rank-2 temperaments]]
 [[Category:Marvel temperaments]]
 [[Category:Gamelismic clan]]