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[[de:Miracle]]
[[de:Miracle]]
'''Miracle temperament''' is a regular temperament discovered by [[George Secor]] in 1974 which has as a generator an interval, called '''secor''', that serves as both [[15/14]] and [[16/15]] semitones. In terms of 13-limit extensions, it is discussed in [[Miracle extensions|miraculous, benediction, and manna]].
'''Miracle temperament''' is a regular temperament discovered by [[George Secor]] in 1974 which has as a generator an interval, called '''secor''', that serves as both [[15/14]] and [[16/15]] semitones. In terms of [[13-limit]] extensions, it is discussed in [[Miracle extensions|miraculous, benediction, and manna]].


Miracle is an exceptionally efficient linear temperament which is a member of both the [[Marvel_temperaments|marvel temperaments]] and the [[Gamelismic_clan|gamelismic clan]]. It is quite accurate, with TOP error only 0.63 cents/octave, meaning intervals of the 11-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 bold) in the following table:
Miracle is an exceptionally efficient linear temperament which is a member of both the [[marvel temperaments]] and the [[gamelismic clan]]. It is quite accurate, with [[TOP]] error only 0.63 [[cent]]s/octave, meaning intervals of the 11-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 bold) in the following table:


{| class="wikitable"
{| class="wikitable"
|-
|-
! | number of <br>secors
! number of <br>secors
! | cents value* <br>(octave-reduced)
! cents value<ref>in 11-limit [[POTE tuning]]</ref> <br>(octave-reduced)
! | JI intervals <br>represented
! JI intervals <br>represented
|-
|-
| | 0
| 0
| | 0.00
| 0.00
| | '''1/1'''
| '''1/1'''
|-
|-
| | 1
| 1
| | 116.63
| 116.63
| | 16/15, 15/14
| 16/15, 15/14
|-
|-
| | 2
| 2
| | 233.27
| 233.27
| | '''8/7'''
| '''8/7'''
|-
|-
| | 3
| 3
| | 349.90
| 349.90
| | '''11/9'''
| '''11/9'''
|-
|-
| | 4
| 4
| | 466.53
| 466.53
| | 21/16
| 21/16
|-
|-
| | 5
| 5
| | 583.16
| 583.16
| | '''7/5'''
| '''7/5'''
|-
|-
| | 6
| 6
| | 699.80
| 699.80
| | '''3/2'''
| '''3/2'''
|-
|-
| | 7
| 7
| | 816.43
| 816.43
| | '''8/5'''
| '''8/5'''
|-
|-
| | 8
| 8
| | 933.06
| 933.06
| | '''12/7'''
| '''12/7'''
|-
|-
| | 9
| 9
| | 1049.69
| 1049.69
| | '''11/6'''
| '''11/6'''
|-
|-
| | 10
| 10
| | 1166.33
| 1166.33
| | 88/45, 96/49, 49/25, <br>108/55, 55/28, 63/32
| 88/45, 96/49, 49/25, <br>108/55, 55/28, 63/32
|-
|-
| | 11
| 11
| | 82.96
| 82.96
| | 22/21, 21/20
| 22/21, 21/20
|-
|-
| | 12
| 12
| | 199.59
| 199.59
| | '''9/8'''
| '''9/8'''
|-
|-
| | 13
| 13
| | 316.23
| 316.23
| | '''6/5'''
| '''6/5'''
|-
|-
| | 14
| 14
| | 432.86
| 432.86
| | '''9/7'''
| '''9/7'''
|-
|-
| | 15
| 15
| | 549.49
| 549.49
| | '''11/8'''
| '''11/8'''
|-
|-
| | 16
| 16
| | 666.12
| 666.12
| | 22/15
| 22/15
|-
|-
| | 17
| 17
| | 782.76
| 782.76
| | '''11/7'''
| '''11/7'''
|-
|-
| | 18
| 18
| | 899.39
| 899.39
| | 42/25, 27/16
| 42/25, 27/16
|-
|-
| | 19
| 19
| | 1016.02
| 1016.02
| | '''9/5'''
| '''9/5'''
|-
|-
| | 20
| 20
| | 1132.65
| 1132.65
| | 48/25, 27/14
| 48/25, 27/14
|-
|-
| | 21
| 21
| | 49.29
| 49.29
| | 36/35, 33/32
| 36/35, 33/32
|-
|-
| | 22
| 22
| | 165.92
| 165.92
| | '''11/10'''
| '''11/10'''
|-
|-
| | 23
| 23
| | 282.55
| 282.55
| | 33/28
| 33/28
|-
|-
| | 24
| 24
| | 399.19
| 399.19
| | 44/35
| 44/35
|-
|-
| | 25
| 25
| | 515.82
| 515.82
| | 27/20
| 27/20
|-
|-
| | 26
| 26
| | 632.45
| 632.45
| | 36/25
| 36/25
|-
|-
| | 27
| 27
| | 749.08
| 749.08
| | 54/35, 77/50
| 54/35, 77/50
|-
|-
| | 28
| 28
| | 865.72
| 865.72
| | 33/20
| 33/20
|-
|-
| | 29
| 29
| | 982.35
| 982.35
| | 44/25
| 44/25
|-
|-
| | 30
| 30
| | 1098.98
| 1098.98
| | 66/35
| 66/35
|-
|-
| | 31
| 31
| | 15.62
| 15.62
| | 81/80
| 81/80
|}
|}
<span style="font-family: Arial,Helvetica,sans-serif;">*in 11-limit POTE tuning</span>
<references/>


Some temperaments have 11/9 as a "neutral third", meaning it's exactly half of a 3/2 (tempering out 243/242), and other temperaments (in the [[Gamelismic_clan|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's exactly half of a [[3/2]] (tempering out [[243/242]]), and other temperaments (in the [[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]].


Miracle can also be thought of as a [[Cluster_temperament|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.
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.


=Spectrum of Miracle Tunings by Eigenmonzos=
=Spectrum of Miracle Tunings by Eigenmonzos=
Line 148: Line 148:
{| class="wikitable"
{| class="wikitable"
|-
|-
! | Eigenmonzo
! Eigenmonzo
! | Secor
! Secor
|-
|-
| | 8/7
| 8/7
| | 115.587
| 115.587
|-
|-
| | 11/9
| 11/9
| | 115.803
| 115.803
|-
|-
| | 5/4
| 5/4
| | 116.241
| 116.241
|-
|-
| | 7/5
| 7/5
| | 116.502
| 116.502
|-
|-
| | |0 -27 25 5&gt;
| |0 -27 25 5&gt;
| | 116.573 (7 limit least squares)
| 116.573 (7 limit least squares)
|-
|-
| | |0 -19 20&gt;
| |0 -19 20&gt;
| | 116.578 (5 limit least squares)
| 116.578 (5 limit least squares)
|-
|-
| | 6/5
| 6/5
| | 116.588 (5 and 7 limit minimax)
| 116.588 (5 and 7 limit minimax)
|-
|-
| | 11/10
| 11/10
| | 116.591
| 116.591
|-
|-
| | 12/11
| 12/11
| | 116.596
| 116.596
|-
|-
| | 14/11
| 14/11
| | 116.617
| 116.617
|-
|-
| | 7/6
| 7/6
| | 116.641
| 116.641
|-
|-
| | |0 17 -11 -6 11&gt;
| |0 17 -11 -6 11&gt;
| | 116.672 (11 limit least squares)
| 116.672 (11 limit least squares)
|-
|-
| | 10/9
| 10/9
| | 116.716 (9 and 11 limit minimax, <br>Secor's definition of secor)
| 116.716 (9 and 11 limit minimax, <br>Secor's definition of secor)
|-
|-
| | |0 117 -44 -19&gt;
| |0 117 -44 -19&gt;
| | 116.721 (9 limit least squares)
| 116.721 (9 limit least squares)
|-
|-
| | 11/8
| 11/8
| | 116.755
| 116.755
|-
|-
| | 9/7
| 9/7
| | 116.792
| 116.792
|-
|-
| | 4/3
| 4/3
| | 116.993
| 116.993
|}
|}


[[Category:miracle]]
[[Category:Miracle]]
[[Category:temperament]]
[[Category:Temperament]]
Retrieved from "https://en.xen.wiki/w/Miracle"