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| {{Breadcrumb|Miracle}}
| | This page is an alternative to [[Miracle extensions]], describing the extensions as ways to collapse [[mirage]] back into a rank-2. |
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| The [[11-limit]] [[miracle]] temperament has various [[extension]]s to the [[13-limit]]. The following temperaments are discussed in this article:
| | {{Editable user page|I might need help with formatting/organization/style but content edits are welcome too}} |
| * '''Miraculous''' ({{nowrap| 31 & 41 }}) – tempering out 105/104, 144/143, 196/195, and 243/242;
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| * '''Benediction''' ({{nowrap| 31 & 41f }}) – tempering out 225/224, 243/242, 351/350, and 385/384;
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| * '''Manna''' ({{nowrap| 31f & 41 }}) – tempering out 225/224, 243/242, 325/324, and 385/384.
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| As we will see in [[#Interval chain]], miraculous is the only extension whose complexity is at about the same level as the 11-limit. It is [[support]]ed by [[72edo|72f]]. The generator, representing [[15/14]], and [[16/15]], goes one step further to stand in for [[~]][[14/13]], and you can find [[11/9]]~[[16/13]] just three generator steps away. Benediction and manna are available if we want to use the more accurately tuned [[patent val]] mapping of prime [[13/1|13]] in 72edo, in which they merge into one. However, benediction benefits from a flatter tuning such as [[103edo]] whereas manna benefits from a sharper tuning such as [[113edo]].
| | * '''Miraculous''' (31 & 41) |
| | * '''Benediction''' (31 & 72) |
| | * '''Manna''' (41 & 72) |
| | * '''Semimiracle''' (10 & 72) |
| | * '''Hemisecordite''' (41 & 62) |
| | * '''31 & 82''' |
| | * '''Phicordial''' (103 & 113) |
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| Another possible path which relates a sense of compromise is to temper out [[169/168]], leading to [[semimiracle]]. This has the effect of slicing the period in two, and is supported by [[62edo|62]], 72, and [[82edo|82]]. Finally, there is [[mirage]], the [[rank-3 temperament|rank-3]] [[expansion]] of miracle with the addition of an independent generator for prime 13.
| | {{stub}} |
| | |
| All of these can be extended to the [[17-limit]] by recognizing [[21/16]]~[[17/13]], tempering out [[273/272]]. For miraculous it implies the generator also represents [[17/16]], which is supported by 72fg. For semimiracle it implies the half-octave period represents [[17/12]]~[[24/17]].
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| [[225/224]] factors into ([[400/399]])·([[513/512]]) in the [[19-limit]], suggesting that miracle can be extended to include prime 19 by tempering out both commas. However, this means 31edo is no longer in the valid [[diamond monotone]] range. There is also a natural extension to the [[23-limit]] that tempers out [[300/299]] and [[392/391]].
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| For technical information see [[Gamelismic clan #Miracle]].
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| | |
| == Interval chain ==
| |
| In the following table, odd harmonics and subharmonics 1–21 are labeled in '''bold'''.
| |
| | |
| {| class="wikitable center-1 right-2"
| |
| |-
| |
| ! rowspan="3" | #
| |
| ! rowspan="3" | Cents*
| |
| ! colspan="4" | Approximate ratios
| |
| |-
| |
| ! rowspan="2" | 11-limit
| |
| ! colspan="3" | 17-limit extensions
| |
| |-
| |
| ! Miraculous
| |
| ! Benediction
| |
| ! Manna
| |
| |-
| |
| | 0
| |
| | 0.0
| |
| | '''1/1'''
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| | 1
| |
| | 116.6
| |
| | 15/14, '''16/15'''
| |
| | 14/13, '''17/16'''
| |
| |
| |
| |
| |
| |-
| |
| | 2
| |
| | 233.3
| |
| | '''8/7'''
| |
| | 15/13, 17/15
| |
| |
| |
| |
| |
| |-
| |
| | 3
| |
| | 349.9
| |
| | 11/9
| |
| | '''16/13''', 17/14, 21/17
| |
| |
| |
| |
| |
| |-
| |
| | 4
| |
| | 466.6
| |
| | '''21/16'''
| |
| | 13/10, 17/13
| |
| | 17/13
| |
| | 17/13
| |
| |-
| |
| | 5
| |
| | 583.2
| |
| | 7/5
| |
| | 24/17
| |
| |
| |
| |
| |
| |-
| |
| | 6
| |
| | 699.9
| |
| | '''3/2'''
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| | 7
| |
| | 816.5
| |
| | '''8/5'''
| |
| | 21/13
| |
| |
| |
| |
| |
| |-
| |
| | 8
| |
| | 933.2
| |
| | 12/7
| |
| | 17/10
| |
| |
| |
| |
| |
| |-
| |
| | 9
| |
| | 1049.8
| |
| | 11/6
| |
| | 24/13
| |
| |
| |
| |
| |
| |-
| |
| | 10
| |
| | 1166.5
| |
| | 49/25, 55/28, 63/32, <br>88/45, 96/49, 108/55
| |
| | 39/20, 51/26, 77/39, <br>128/65, 135/68, 168/85
| |
| | 51/26, 100/51
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| | 51/26, 65/33
| |
| |-
| |
| | 11
| |
| | 83.1
| |
| | 21/20, 22/21
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| | 18/17, 26/25
| |
| |
| |
| |
| |
| |-
| |
| | 12
| |
| | 199.8
| |
| | '''9/8'''
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| | 13
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| | 316.4
| |
| | 6/5
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| | 14
| |
| | 433.1
| |
| | 9/7
| |
| | 22/17
| |
| |
| |
| |
| |
| |-
| |
| | 15
| |
| | 549.7
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| | '''11/8'''
| |
| | 18/13
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| |
| |
| |
| |
| |-
| |
| | 16
| |
| | 666.3
| |
| | 22/15
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| | 17
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| | 783.0
| |
| | 11/7
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| | 18
| |
| | 899.6
| |
| | 27/16, 42/25
| |
| | 22/13
| |
| |
| |
| |
| |
| |-
| |
| | 19
| |
| | 1016.3
| |
| | 9/5
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| | 20
| |
| | 1132.9
| |
| | 27/14, 48/25
| |
| |
| |
| |
| |
| | 52/27
| |
| |-
| |
| | 21
| |
| | 49.6
| |
| | 33/32, 36/35
| |
| | 27/26
| |
| | 35/34, 40/39
| |
| | 34/33
| |
| |-
| |
| | 22
| |
| | 166.2
| |
| | 11/10
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| | 23
| |
| | 282.9
| |
| | 33/28
| |
| |
| |
| | 20/17
| |
| | 13/11
| |
| |-
| |
| | 24
| |
| | 399.5
| |
| | 44/35
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| | 25
| |
| | 516.2
| |
| | 27/20
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| | 26
| |
| | 632.8
| |
| | 36/25
| |
| |
| |
| |
| |
| | 13/9
| |
| |-
| |
| | 27
| |
| | 749.5
| |
| | 54/35, 77/50
| |
| |
| |
| | 20/13
| |
| | 17/11
| |
| |-
| |
| | 28
| |
| | 866.1
| |
| | 33/20
| |
| |
| |
| | 28/17
| |
| |
| |
| |-
| |
| | 29
| |
| | 982.8
| |
| | 44/25
| |
| |
| |
| | 30/17
| |
| |
| |
| |-
| |
| | 30
| |
| | 1099.4
| |
| | 66/35
| |
| |
| |
| | '''32/17'''
| |
| | 17/9
| |
| |-
| |
| | 31
| |
| | 16.1
| |
| | 81/80, 99/98, 121/120
| |
| | 66/65
| |
| | 105/104, 120/119, 136/135, <br>144/143, 154/153, 170/169
| |
| | 65/64, 78/77, <br>85/84, 91/90
| |
| |-
| |
| | 32
| |
| | 132.7
| |
| | 27/25
| |
| |
| |
| | 14/13
| |
| | 13/12
| |
| |-
| |
| | 33
| |
| | 249.3
| |
| | 81/70
| |
| |
| |
| | 15/13
| |
| | 52/45
| |
| |-
| |
| | 34
| |
| | 366.0
| |
| | 99/80
| |
| |
| |
| | '''16/13''', 21/17
| |
| | 26/21
| |
| |-
| |
| | 35
| |
| | 482.6
| |
| | 33/25
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| | 36
| |
| | 599.3
| |
| | 99/70
| |
| |
| |
| | 24/17
| |
| | 17/12
| |
| |-
| |
| | 37
| |
| | 715.9
| |
| | 121/80
| |
| |
| |
| |
| |
| |
| |
| |-
| |
| | 38
| |
| | 832.6
| |
| | 121/75
| |
| |
| |
| | 21/13
| |
| | '''13/8''', 34/21
| |
| |-
| |
| | 39
| |
| | 949.2
| |
| | 121/70
| |
| |
| |
| | 45/26
| |
| | 26/15
| |
| |-
| |
| | 40
| |
| | 1065.9
| |
| | 231/125
| |
| |
| |
| | 24/13
| |
| | 13/7
| |
| |-
| |
| | 41
| |
| | 1182.5
| |
| | 99/50
| |
| |
| |
| | 77/39, 128/65, <br>168/85, 180/91
| |
| | 119/60, 143/72, 135/68, <br>153/77, 169/85, 195/98
| |
| |}
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| <nowiki/>* In 11-limit [[CWE tuning]], octave reduced
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| | |
| == Tunings ==
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| === Norm-based tunings ===
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| * 5-limit POTE: ~16/15 = 116.673{{c}}
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| * 7-limit POTE: ~15/14 = 116.675{{c}}
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| * 11-limit POTE
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| ** Miracle: ~15/14 = 116.633{{c}}
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| ** Revelation: ~15/14 = 116.277{{c}}
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| * 13-limit POTE
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| ** Miraculous: ~15/14 = 116.747{{c}}
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| ** Benediction: ~15/14 = 116.574{{c}}
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| ** Manna: ~15/14 = 116.739{{c}}
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| ** Revelation: ~15/14 = 116.268{{c}}
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| | |
| === Target tunings ===
| |
| {| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
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| |+ style="white-space: nowrap;" | Target tunings (miracle)
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| |-
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| ! rowspan="2" | Target
| |
| ! colspan="2" | Minimax
| |
| ! colspan="2" | Least squares
| |
| |-
| |
| ! Generator
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| ! Eigenmonzo*
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| ! Generator
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| ! Eigenmonzo*
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| |-
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| | 5-odd-limit
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| | ~16/15 = 116.588{{c}}
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| | 5/3
| |
| | ~16/15 = 116.578{{c}}
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| | {{Monzo| 0 -19 20 }}
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| |-
| |
| | 7-odd-limit
| |
| | ~15/14 = 116.588{{c}}
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| | 5/3
| |
| | ~15/14 = 116.573{{c}}
| |
| | {{Monzo| 0 -27 25 5 }}
| |
| |-
| |
| | 9-odd-limit
| |
| | ~15/14 = 116.716{{c}}
| |
| | 9/5
| |
| | ~15/14 = 116.721{{c}}
| |
| | {{Monzo| 0 117 -44 -19 }}
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| |-
| |
| | 11-odd-limit
| |
| | ~15/14 = 116.716{{c}}
| |
| | 9/5
| |
| | ~15/14 = 116.672{{c}}
| |
| | {{Monzo| 0 17 -11 -6 11 }}
| |
| |}
| |
| | |
| {| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
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| |+ style="white-space: nowrap;" | Target tunings (miraculous)
| |
| |-
| |
| ! rowspan="2" | Target
| |
| ! colspan="2" | Minimax
| |
| ! colspan="2" | Least squares
| |
| |-
| |
| ! Generator
| |
| ! Eigenmonzo*
| |
| ! Generator
| |
| ! Eigenmonzo*
| |
| |-
| |
| | 13-odd-limit
| |
| | ~15/14 = 116.716{{c}}
| |
| | 9/5
| |
| | ~15/14 = 116.846{{c}}
| |
| | {{Monzo| 0 141 -70 -35 84 -42 }}
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| |-
| |
| | 15-odd-limit
| |
| | ~15/14 = 116.993{{c}}
| |
| | 3/2
| |
| | ~15/14 = 116.820{{c}}
| |
| | {{Monzo| 0 127 -84 -36 100 -44 }}
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| |}
| |
| | |
| {| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
| |
| |+ style="white-space: nowrap;" | Target tunings (benediction)
| |
| |-
| |
| ! rowspan="2" | Target
| |
| ! colspan="2" | Minimax
| |
| ! colspan="2" | Least squares
| |
| |-
| |
| ! Generator
| |
| ! Eigenmonzo*
| |
| ! Generator
| |
| ! Eigenmonzo*
| |
| |-
| |
| | 13-odd-limit
| |
| | ~15/14 = 116.595{{c}}
| |
| | 13/9
| |
| | ~15/14 = 116.56309{{c}}
| |
| | {{Monzo| 0 -234 39 4 -115 228 }}
| |
| |-
| |
| | 15-odd-limit
| |
| | ~15/14 = 116.588{{c}}
| |
| | 5/3
| |
| | ~15/14 = 116.56348{{c}}
| |
| | {{Monzo| 0 -251 22 5 -131 261 }}
| |
| |}
| |
| | |
| {| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
| |
| |+ style="white-space: nowrap;" | Target tunings (manna)
| |
| |-
| |
| ! rowspan="2" | Target
| |
| ! colspan="2" | Minimax
| |
| ! colspan="2" | Least squares
| |
| |-
| |
| ! Generator
| |
| ! Eigenmonzo*
| |
| ! Generator
| |
| ! Eigenmonzo*
| |
| |-
| |
| | 13-odd-limit
| |
| | ~15/14 = 116.760{{c}}
| |
| | 13/10
| |
| | ~15/14 = 116.780{{c}}
| |
| | {{Monzo| 0 18 -111 -76 43 204 }}
| |
| |-
| |
| | 15-odd-limit
| |
| | ~15/14 = 116.725{{c}}
| |
| | 15/13
| |
| | ~15/14 = 116.764{{c}}
| |
| | {{Monzo| 0 -37 -166 -77 59 243 }}
| |
| |}
| |
| | |
| {| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
| |
| |+ style="white-space: nowrap;" | Target tunings (revelation)
| |
| |-
| |
| ! rowspan="2" | Target
| |
| ! colspan="2" | Minimax
| |
| ! colspan="2" | Least squares
| |
| |-
| |
| ! Generator
| |
| ! Eigenmonzo*
| |
| ! Generator
| |
| ! Eigenmonzo*
| |
| |-
| |
| | 11-odd-limit
| |
| | ~15/14 = 116.164{{c}}
| |
| | 11/9
| |
| | ~15/14 = 116.198{{c}}
| |
| | {{Monzo| 0 -195 35 5 89 }}
| |
| |-
| |
| | 13-odd-limit
| |
| | ~15/14 = 116.164{{c}}
| |
| | 11/9
| |
| | ~15/14 = 116.249{{c}}
| |
| | {{Monzo| 0 -234 39 4 102 11 }}
| |
| |-
| |
| | 15-odd-limit
| |
| | ~15/14 = 116.164{{c}}
| |
| | 11/9
| |
| | ~15/14 = 116.229{{c}}
| |
| | {{Monzo| 0 -251 22 5 117 13 }}
| |
| |}
| |
| | |
| === Tuning spectra ===
| |
| ==== Miraculous ====
| |
| {| class="wikitable center-all left-4"
| |
| |-
| |
| ! Edo<br>generator
| |
| ! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
| |
| ! Generator (¢)
| |
| ! Comments
| |
| |-
| |
| |
| |
| | 17/16
| |
| | 104.955
| |
| |
| |
| |-
| |
| |
| |
| | 17/15
| |
| | 108.343
| |
| |
| |
| |-
| |
| |
| |
| | 15/8
| |
| | 111.731
| |
| |
| |
| |-
| |
| |
| |
| | 17/14
| |
| | 112.043
| |
| |
| |
| |-
| |
| |
| |
| | 13/10
| |
| | 113.553
| |
| |
| |
| |-
| |
| |
| |
| | 17/10
| |
| | 114.830
| |
| |
| |
| |-
| |
| |
| |
| | 7/4
| |
| | 115.587
| |
| |
| |
| |-
| |
| |
| |
| | 11/9
| |
| | 115.803
| |
| |
| |
| |-
| |
| |
| |
| | 17/13
| |
| | 116.107
| |
| |
| |
| |-
| |
| | 3\31
| |
| |
| |
| | 116.129
| |
| | Lower bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
| |
| |-
| |
| |
| |
| | 5/4
| |
| | 116.241
| |
| |
| |
| |-
| |
| |
| |
| | 15/11
| |
| | 116.441
| |
| |
| |
| |-
| |
| |
| |
| | 7/5
| |
| | 116.502
| |
| |
| |
| |-
| |
| | 10\103
| |
| |
| |
| | 116.505
| |
| | 103fg val
| |
| |-
| |
| | 17\175
| |
| |
| |
| | 116.571
| |
| | 175ffggg val
| |
| |-
| |
| |
| |
| | 5/3
| |
| | 116.588
| |
| | 5- and 7-odd-limit minimax
| |
| |-
| |
| |
| |
| | 11/10
| |
| | 116.591
| |
| |
| |
| |-
| |
| |
| |
| | 11/6
| |
| | 116.596
| |
| |
| |
| |-
| |
| |
| |
| | 11/7
| |
| | 116.617
| |
| |
| |
| |-
| |
| |
| |
| | 7/6
| |
| | 116.641
| |
| |
| |
| |-
| |
| | 7\72
| |
| |
| |
| | 116.667
| |
| | 72fg val
| |
| |-
| |
| |
| |
| | 9/5
| |
| | 116.716
| |
| | 9-, 11- and 13-odd-limit minimax
| |
| |-
| |
| |
| |
| | 11/8
| |
| | 116.755
| |
| |
| |
| |-
| |
| | 18\185
| |
| |
| |
| | 116.757
| |
| | 185cffggg val
| |
| |-
| |
| |
| |
| | 9/7
| |
| | 116.792
| |
| |
| |
| |-
| |
| | 11\113
| |
| |
| |
| | 116.814
| |
| | 113fgg val
| |
| |-
| |
| |
| |
| | 3/2
| |
| | 116.993
| |
| | 15-odd-limit minimax
| |
| |-
| |
| | 4\41
| |
| |
| |
| | 117.073
| |
| | Upper bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
| |
| |-
| |
| |
| |
| | 13/11
| |
| | 117.266
| |
| |
| |
| |-
| |
| |
| |
| | 13/9
| |
| | 117.559
| |
| |
| |
| |-
| |
| |
| |
| | 17/11
| |
| | 117.597
| |
| |
| |
| |-
| |
| |
| |
| | 13/12
| |
| | 117.936
| |
| |
| |
| |-
| |
| |
| |
| | 17/9
| |
| | 118.087
| |
| |
| |
| |-
| |
| |
| |
| | 17/12
| |
| | 119.400
| |
| |
| |
| |-
| |
| |
| |
| | 15/14
| |
| | 119.443
| |
| |
| |
| |-
| |
| |
| |
| | 13/8
| |
| | 119.824
| |
| |
| |
| |-
| |
| |
| |
| | 21/17
| |
| | 121.942
| |
| |
| |
| |-
| |
| |
| |
| | 15/13
| |
| | 123.871
| |
| |
| |
| |-
| |
| |
| |
| | 13/7
| |
| | 128.298
| |
| |
| |
| |}
| |
| | |
| ==== Benediction ====
| |
| {| class="wikitable center-all left-4"
| |
| |-
| |
| ! Edo<br>generator
| |
| ! Unchanged interval<br>(eigenmonzo)
| |
| ! Generator (¢)
| |
| ! Comments
| |
| |-
| |
| |
| |
| | 15/8
| |
| | 111.731
| |
| |
| |
| |-
| |
| |
| |
| | 7/4
| |
| | 115.587
| |
| |
| |
| |-
| |
| |
| |
| | 11/9
| |
| | 115.803
| |
| |
| |
| |-
| |
| |
| |
| | 17/13
| |
| | 116.107
| |
| |
| |
| |-
| |
| | 3\31
| |
| |
| |
| | 116.129
| |
| | Lower bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
| |
| |-
| |
| |
| |
| | 5/4
| |
| | 116.241
| |
| |
| |
| |-
| |
| |
| |
| | 15/11
| |
| | 116.441
| |
| |
| |
| |-
| |
| |
| |
| | 13/8
| |
| | 116.455
| |
| |
| |
| |-
| |
| |
| |
| | 17/16
| |
| | 116.501
| |
| |
| |
| |-
| |
| |
| |
| | 7/5
| |
| | 116.502
| |
| |
| |
| |-
| |
| | 10\103
| |
| |
| |
| | 116.505
| |
| |
| |
| |-
| |
| |
| |
| | 13/7
| |
| | 116.509
| |
| |
| |
| |-
| |
| |
| |
| | 13/10
| |
| | 116.511
| |
| |
| |
| |-
| |
| |
| |
| | 13/12
| |
| | 116.536
| |
| |
| |
| |-
| |
| |
| |
| | 13/11
| |
| | 116.547
| |
| |
| |
| |-
| |
| |
| |
| | 17/14
| |
| | 116.567
| |
| |
| |
| |-
| |
| | 17\175
| |
| |
| |
| | 116.571
| |
| | 175f val
| |
| |-
| |
| |
| |
| | 17/10
| |
| | 116.581
| |
| |
| |
| |-
| |
| |
| |
| | 17/12
| |
| | 116.583
| |
| |
| |
| |-
| |
| |
| |
| | 17/11
| |
| | 116.586
| |
| |
| |
| |-
| |
| |
| |
| | 5/3
| |
| | 116.588
| |
| | 5-, 7- and 15-odd-limit minimax
| |
| |-
| |
| |
| |
| | 11/10
| |
| | 116.591
| |
| |
| |
| |-
| |
| |
| |
| | 13/9
| |
| | 116.595
| |
| | 13-odd-limit minimax
| |
| |-
| |
| |
| |
| | 11/6
| |
| | 116.596
| |
| |
| |
| |-
| |
| |
| |
| | 15/13
| |
| | 116.598
| |
| |
| |
| |-
| |
| |
| |
| | 11/7
| |
| | 116.617
| |
| |
| |
| |-
| |
| |
| |
| | 7/6
| |
| | 116.641
| |
| |
| |
| |-
| |
| |
| |
| | 17/9
| |
| | 116.642
| |
| |
| |
| |-
| |
| |
| |
| | 21/17
| |
| | 116.642
| |
| |
| |
| |-
| |
| |
| |
| | 17/15
| |
| | 116.666
| |
| |
| |
| |-
| |
| | 7\72
| |
| |
| |
| | 116.667
| |
| | Upper bound of 13- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
| |
| |-
| |
| |
| |
| | 9/5
| |
| | 116.716
| |
| | 9- and 11-odd-limit minimax
| |
| |-
| |
| |
| |
| | 11/8
| |
| | 116.755
| |
| |
| |
| |-
| |
| | 18\185
| |
| |
| |
| | 116.757
| |
| | 185cfffgg val
| |
| |-
| |
| |
| |
| | 9/7
| |
| | 116.792
| |
| |
| |
| |-
| |
| | 11\113
| |
| |
| |
| | 116.814
| |
| | 113ffg val
| |
| |-
| |
| |
| |
| | 3/2
| |
| | 116.993
| |
| |
| |
| |-
| |
| | 4\41
| |
| |
| |
| | 117.073
| |
| | 41fg val, upper bound of 11-odd-limit diamond monotone
| |
| |-
| |
| |
| |
| | 15/14
| |
| | 119.443
| |
| |
| |
| |}
| |
| | |
| ==== Manna ====
| |
| {| class="wikitable center-all left-4"
| |
| |-
| |
| ! Edo<br>generator
| |
| ! Unchanged interval<br>(eigenmonzo)
| |
| ! Generator (¢)
| |
| ! Comments
| |
| |-
| |
| |
| |
| | 15/8
| |
| | 111.731
| |
| |
| |
| |-
| |
| |
| |
| | 7/4
| |
| | 115.587
| |
| |
| |
| |-
| |
| |
| |
| | 11/9
| |
| | 115.803
| |
| |
| |
| |-
| |
| |
| |
| | 17/13
| |
| | 116.107
| |
| |
| |
| |-
| |
| | 3\31
| |
| |
| |
| | 116.129
| |
| | 31fg val, lower bound of 11-odd-limit diamond monotone
| |
| |-
| |
| |
| |
| | 5/4
| |
| | 116.241
| |
| |
| |
| |-
| |
| |
| |
| | 15/11
| |
| | 116.441
| |
| |
| |
| |-
| |
| |
| |
| | 7/5
| |
| | 116.502
| |
| |
| |
| |-
| |
| | 10\103
| |
| |
| |
| | 116.505
| |
| | 103ffgg val
| |
| |-
| |
| | 17\175
| |
| |
| |
| | 116.571
| |
| | 175fffgg val
| |
| |-
| |
| |
| |
| | 5/3
| |
| | 116.588
| |
| | 5- and 7-odd-limit minimax
| |
| |-
| |
| |
| |
| | 11/10
| |
| | 116.591
| |
| |
| |
| |-
| |
| |
| |
| | 11/6
| |
| | 116.596
| |
| |
| |
| |-
| |
| |
| |
| | 11/7
| |
| | 116.617
| |
| |
| |
| |-
| |
| |
| |
| | 7/6
| |
| | 116.641
| |
| |
| |
| |-
| |
| | 7\72
| |
| |
| |
| | 116.667
| |
| | Lower bound of 13- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
| |
| |-
| |
| |
| |
| | 17/15
| |
| | 116.667
| |
| |
| |
| |-
| |
| |
| |
| | 21/17
| |
| | 116.689
| |
| |
| |
| |-
| |
| |
| |
| | 17/9
| |
| | 116.702
| |
| |
| |
| |-
| |
| |
| |
| | 17/10
| |
| | 116.707
| |
| |
| |
| |-
| |
| |
| |
| | 9/5
| |
| | 116.716
| |
| | 9- and 11-odd-limit minimax
| |
| |-
| |
| |
| |
| | 15/13
| |
| | 116.725
| |
| | 15-odd-limit minimax
| |
| |-
| |
| |
| |
| | 17/14
| |
| | 116.730
| |
| |
| |
| |-
| |
| |
| |
| | 17/12
| |
| | 116.750
| |
| |
| |
| |-
| |
| |
| |
| | 11/8
| |
| | 116.755
| |
| |
| |
| |-
| |
| | 18\185
| |
| |
| |
| | 116.757
| |
| | 185cf val
| |
| |-
| |
| |
| |
| | 13/10
| |
| | 116.760
| |
| | 13-odd-limit minimax
| |
| |-
| |
| |
| |
| | 17/16
| |
| | 116.785
| |
| |
| |
| |-
| |
| |
| |
| | 9/7
| |
| | 116.792
| |
| |
| |
| |-
| |
| |
| |
| | 13/7
| |
| | 116.79254
| |
| |
| |
| |-
| |
| |
| |
| | 13/9
| |
| | 116.79299
| |
| |
| |
| |-
| |
| |
| |
| | 17/11
| |
| | 116.801
| |
| |
| |
| |-
| |
| | 11\113
| |
| |
| |
| | 116.814
| |
| |
| |
| |-
| |
| |
| |
| | 13/12
| |
| | 116.830
| |
| |
| |
| |-
| |
| |
| |
| | 13/8
| |
| | 116.856
| |
| |
| |
| |-
| |
| |
| |
| | 13/11
| |
| | 116.922
| |
| |
| |
| |-
| |
| |
| |
| | 3/2
| |
| | 116.993
| |
| |
| |
| |-
| |
| | 4\41
| |
| |
| |
| | 117.073
| |
| | Upper bound of 11- to 17-odd-limit, <br>and 17-limit 21-odd-limit diamond monotone
| |
| |-
| |
| |
| |
| | 15/14
| |
| | 119.443
| |
| |
| |
| |}
| |
| | |
| ==== Revelation ====
| |
| {| class="wikitable center-all left-4"
| |
| |-
| |
| ! Edo<br>generator
| |
| ! Unchanged interval<br>(eigenmonzo)
| |
| ! Generator (¢)
| |
| ! Comments
| |
| |-
| |
| |
| |
| | 15/8
| |
| | 111.731
| |
| |
| |
| |-
| |
| |
| |
| | 13/10
| |
| | 113.553
| |
| |
| |
| |-
| |
| | 2\21
| |
| |
| |
| | 114.286
| |
| |
| |
| |-
| |
| |
| |
| | 13/11
| |
| | 114.555
| |
| |
| |
| |-
| |
| |
| |
| | 11/10
| |
| | 115.000
| |
| |
| |
| |-
| |
| | 5\52
| |
| |
| |
| | 115.385
| |
| | 52f val
| |
| |-
| |
| |
| |
| | 11/7
| |
| | 115.536
| |
| |
| |
| |-
| |
| |
| |
| | 11/8
| |
| | 115.543
| |
| |
| |
| |-
| |
| |
| |
| | 7/4
| |
| | 115.587
| |
| |
| |
| |-
| |
| |
| |
| | 15/11
| |
| | 115.797
| |
| |
| |
| |-
| |
| |
| |
| | 11/6
| |
| | 115.938
| |
| |
| |
| |-
| |
| | 3\31
| |
| |
| |
| | 116.129
| |
| | 11- to 15-odd-limit, <br>and 13-limit 21-odd-limit diamond monotone (singleton)
| |
| |-
| |
| |
| |
| | 11/9
| |
| | 116.164
| |
| | 11-, 13- and 15-odd-limit minimax
| |
| |-
| |
| |
| |
| | 5/4
| |
| | 116.241
| |
| |
| |
| |-
| |
| |
| |
| | 7/5
| |
| | 116.502
| |
| |
| |
| |-
| |
| |
| |
| | 5/3
| |
| | 116.588
| |
| | 5- and 7-odd-limit minimax
| |
| |-
| |
| |
| |
| | 7/6
| |
| | 116.641
| |
| |
| |
| |-
| |
| | 7\72
| |
| |
| |
| | 116.667
| |
| | 72ee val
| |
| |-
| |
| |
| |
| | 9/5
| |
| | 116.716
| |
| | 9-odd-limit minimax
| |
| |-
| |
| |
| |
| | 9/7
| |
| | 116.792
| |
| |
| |
| |-
| |
| |
| |
| | 3/2
| |
| | 116.993
| |
| |
| |
| |-
| |
| | 4\41
| |
| |
| |
| | 117.073
| |
| | 41ef val
| |
| |-
| |
| |
| |
| | 13/9
| |
| | 117.559
| |
| |
| |
| |-
| |
| |
| |
| | 13/12
| |
| | 117.936
| |
| |
| |
| |-
| |
| |
| |
| | 15/14
| |
| | 119.443
| |
| |
| |
| |-
| |
| |
| |
| | 13/8
| |
| | 119.824
| |
| |
| |
| |-
| |
| |
| |
| | 15/13
| |
| | 123.871
| |
| |
| |
| |-
| |
| |
| |
| | 13/7
| |
| | 128.298
| |
| |
| |
| |}
| |
|
| |
|
| [[Category:Miracle]] | | [[Category:Miracle]] |
| [[Category:Temperament extensions]] | | [[Category:Temperament extensions]] |
| [[Category:Rank-2 temperaments]]
| |