Xenharmonic Wiki

Garibaldi: Difference between revisions

← Older editNewer edit →
VisualWikitext
m Improve categories
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,016 edits
No edit summary
Line 1: Line 1:
'''Garibaldi temperament''' is a 7-limit (and higher) temperament of the [[Schismatic family #Garibaldi|schismatic family]]. It is an extension of [[helmholtz]] temperament beyond the 5-limit but with the same simple chain-of-fifths structure (so that standard notation may be used). As in helmholtz temperament, [[5/4]] is mapped to the diminished fourth (e.g. A-Db), and the new mapping specific to garibaldi is that [[7/4]] is mapped to the double diminished octave (e.g. A-Abb). This makes garibaldi a [[Marvel temperaments|marvel temperament]].  
'''Garibaldi temperament''' is a 7-limit (and higher) temperament of the [[Schismatic family #Garibaldi|schismatic family]]. It is an extension of [[helmholtz]] temperament beyond the 5-limit but with the same simple chain-of-fifths structure (so that standard notation may be used). As in helmholtz temperament, [[5/4]] is mapped to the diminished fourth (e.g. A-Db), and the new mapping specific to garibaldi is that [[7/4]] is mapped to the double diminished octave (e.g. A-Abb). This makes garibaldi a [[Marvel temperaments|marvel temperament]].  


Immediate 11-limit extensions include ''cassandra'' (41&53), mapping 11/8 to +23 steps, ''andromeda'' (29&41), mapping 11/8 to -18 steps, and ''helenus'' (53&65d), mapping 11/8 to -30 steps.  
Immediate 11-limit extensions include ''cassandra'' (41&53), mapping 11/8 to +23 steps, ''andromeda'' (29&41), mapping 11/8 to -18 steps, and ''helenus'' (53&65d), mapping 11/8 to -30 steps.  


== Interval chain ==
== Interval chain ==
Line 7: Line 7:


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
! rowspan="2" | #
! rowspan="3" | Fifth <br>generator
! rowspan="2" | Cents*
! rowspan="3" | Cents*
! colspan="4" | Approximate Ratios
! colspan="4" | Approximate Ratios
|-
|-
!7-limit
! rowspan="2" | 7-limit
!13-limit Extension<br>Cassandra Mapping
! colspan="3" | 13-limit Extension
!13-limit Extension<br>Andromeda Mapping
|-
!13-limit Extension<br>Helenus Mapping
! Cassandra
! Andromeda
! Helenus
|-
|-
| 0
| 0
| 0.0
| 0.00
| 1/1
| 1/1
|
|
Line 24: Line 26:
|-
|-
| 1
| 1
| 702.1
| 702.09
| '''3/2'''
| '''3/2'''
|
|
Line 31: Line 33:
|-
|-
| 2
| 2
| 204.2
| 204.17
| 9/8
| 9/8
|
|
Line 38: Line 40:
|-
|-
| 3
| 3
| 906.3
| 906.26
| 27/16, 42/25
| 27/16, 42/25
| 22/13
| 22/13
Line 45: Line 47:
|-
|-
| 4
| 4
| 408.3
| 408.34
| 63/50, 80/63
| 63/50, 80/63
|
|
Line 52: Line 54:
|-
|-
| 5
| 5
| 1110.4
| 1110.43
| 40/21
| 40/21
|
|
Line 59: Line 61:
|-
|-
| 6
| 6
| 612.5
| 612.51
| 10/7
| 10/7
|
|
Line 66: Line 68:
|-
|-
| 7
| 7
| 114.6
| 114.60
| 15/14, 16/15
| 15/14, 16/15
|
|
Line 73: Line 75:
|-
|-
| 8
| 8
| 816.7
| 816.68
| '''8/5'''
| '''8/5'''
|
|
Line 80: Line 82:
|-
|-
| 9
| 9
| 318.8
| 318.77
| 6/5
| 6/5
|
|
Line 87: Line 89:
|-
|-
| 10
| 10
| 1020.9
| 1020.85
| 9/5
| 9/5
|
|
Line 94: Line 96:
|-
|-
| 11
| 11
| 522.9
| 522.94
| 27/20
| 27/20
|
|
Line 101: Line 103:
|-
|-
| 12
| 12
| 25.0
| 25.02
| 50/49, 64/63, 81/80
| 50/49, 64/63, 81/80
|
|
Line 108: Line 110:
|-
|-
| 13
| 13
| 727.1
| 727.11
| 32/21
| 32/21
|
|
Line 115: Line 117:
|-
|-
| 14
| 14
| 229.2
| 229.19
| '''8/7'''
| '''8/7'''
|
|
Line 122: Line 124:
|-
|-
| 15
| 15
| 931.3
| 931.28
| 12/7
| 12/7
|
|
Line 129: Line 131:
|-
|-
| 16
| 16
| 433.4
| 433.36
| 9/7
| 9/7
|
|
Line 136: Line 138:
|-
|-
| 17
| 17
| 1135.4
| 1135.45
| 27/14, 48/25
| 27/14, 48/25
| 52/27
| 52/27
Line 143: Line 145:
|-
|-
| 18
| 18
| 637.5
| 637.53
| 36/25, 81/56
| 36/25, 81/56
| 13/9
| 13/9
Line 150: Line 152:
|-
|-
| 19
| 19
| 139.6
| 139.62
| 27/25
| 27/25
| 13/12
| 13/12
Line 157: Line 159:
|-
|-
| 20
| 20
| 841.7
| 841.70
| 80/49, 81/50
| 80/49, 81/50
| '''13/8''', 44/27
| '''13/8''', 44/27
Line 164: Line 166:
|-
|-
| 21
| 21
| 343.8
| 343.79
| 60/49
| 60/49
| 11/9, 39/32
| 11/9, 39/32
Line 171: Line 173:
|-
|-
| 22
| 22
| 1045.9
| 1045.87
| 64/35
| 64/35
| 11/6
| 11/6
Line 178: Line 180:
|-
|-
| 23
| 23
| 548.0
| 547.96
| 48/35
| 48/35
| '''11/8'''
| '''11/8'''
Line 185: Line 187:
|-
|-
| 24
| 24
| 50.0
| 50.04
| 36/35
| 36/35
| 33/32
| 33/32
Line 192: Line 194:
|-
|-
| 25
| 25
| 752.1
| 752.13
| 54/35
| 54/35
|
|
Line 199: Line 201:
|-
|-
| 26
| 26
| 254.2
| 254.21
| 81/70, 144/125
| 81/70, 144/125
|
|
Line 206: Line 208:
|-
|-
| 27
| 27
| 956.3
| 956.30
| 216/125, 256/147
| 216/125, 256/147
| 26/15
| 26/15
Line 213: Line 215:
|-
|-
| 28
| 28
| 458.4
| 458.38
| 64/49
| 64/49
| 13/10
| 13/10
Line 220: Line 222:
|-
|-
| 29
| 29
| 1160.5
| 1160.47
| 96/49
| 96/49
| 39/20, 88/45
| 39/20, 88/45
|
|
| 64/33
| 64/33
|-
| 30
| 662.55
|
|
|
| '''16/11'''
|-
| 31
| 164.64
|
|
|
| 12/11
|-
| 32
| 866.72
|
|
|
| 18/11
|-
| 33
| 368.81
|
|
|
| '''16/13''', 27/22
|-
| 34
| 1070.90
|
|
|
| 24/13
|-
| 35
| 572.98
|
|
|
| 18/13
|}
|}
<nowiki>*</nowiki> in 7-limit POTE tuning
<nowiki>*</nowiki> in 7-limit POTE tuning
Line 234: Line 278:


== Spectrum of garibaldi tunings by eigenmonzos ==
== Spectrum of garibaldi tunings by eigenmonzos ==
=== Cassandra mapping ===
Gencom: [2 4/3; 225/224 275/273 325/324 385/384]
Gencom map: [{{val|1 2 -1 -3 13 12}}, {{val|0 -1 8 14 -23 -20}}]


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
Line 243: Line 291:
| 16/15
| 16/15
| 701.676
| 701.676
|
|-
| (69\118)
| 701.695
|  
|  
|-
|-
Line 260: Line 304:
| 701.738
| 701.738
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
| 100\171
| 701.754
|
|-
|-
| 10/9
| 10/9
| 701.760
| 701.760
|
|-
| (31\53)
| 701.887
|  
|  
|-
|-
Line 295: Line 331:
| 18/13
| 18/13
| 702.034
| 702.034
|
|-
| 86\147
| 702.041
|  
|  
|-
|-
Line 320: Line 352:
| 702.114
| 702.114
| 9-odd-limit least squares
| 9-odd-limit least squares
|-
| (55\94)
| 702.12766
|
|-
|-
| {{monzo| 0 -38 -80 -122 137 116 }}
| {{monzo| 0 -38 -80 -122 137 116 }}
| 702.12770
| 702.128
| 13-odd-limit least squares
| 13-odd-limit least squares
|-
|-
Line 344: Line 372:
| 702.209
| 702.209
| 7-odd-limit minimax
| 7-odd-limit minimax
|-
| (79\135)
| 702.222
|
|-
|-
| 8/7
| 8/7
Line 369: Line 393:
|  
|  
|-
|-
| (24\41)
| 15/14
| 702.439
| 702.778
|
|-
| 7/5
| 702.915
|
|-
| 13/11
| 703.597
|
|}
 
=== Andromeda mapping ===
Gencom: [2 4/3; 100/99 105/104 196/195 245/242]
 
Gencom map: [{{val|1 2 -1 -3 -4 -5}}, {{val|0 -1 8 14 18 21}}]
 
{| class="wikitable center-1 right-2"
|-
! Eigenmonzo
! Fifth
! Comments
|-
| 16/15
| 701.676
|
|-
| 5/4
| 701.711
|
|-
| 6/5
| 701.738
| 5-odd-limit minimax
|-
| 10/9
| 701.760
|
|-
| 4/3
| 701.955
|
|-
| 9/7
| 702.193
| 9-odd-limit minimax
|-
| 7/6
| 702.209
| 7-odd-limit minimax
|-
| 8/7
| 702.227
|
|-
| 11/9
| 702.630
|
|-
| 12/11
| 702.665
|
|-
| 11/8
| 702.705
|
|-
| 18/13
| 702.756
|  
|  
|-
|-
| 15/14
| 15/14
| 702.778
| 702.778
|
|-
| 13/12
| 702.792
|
|-
| 16/13
| 702.832
|  
|  
|-
|-
Line 381: Line 481:
|  
|  
|-
|-
| (17\29)
| 15/11
| 703.448
| 703.359
|
|-
| 15/13
| 703.410
|
|-
| 11/10
| 703.500
|
|-
| 13/10
| 703.522
|
|-
| 13/11
| 703.597
|
|-
| 14/13
| 704.043
|
|-
| 14/11
| 704.377
|
|}
 
=== Helenus mapping ===
Gencom: [2 4/3; 99/98 176/175 275/273 847/845]
 
Gencom map: [{{val|1 2 -1 -3 -9 -10}}, {{val|0 -1 8 14 30 33}}]
 
{| class="wikitable center-1 right-2"
|-
! Eigenmonzo
! Fifth
! Comments
|-
| 14/11
| 701.094
|
|-
| 14/13
| 701.489
|
|-
| 11/10
| 701.591
|
|-
| 15/11
| 701.607
|
|-
| 11/8
| 701.623
|
|-
| 12/11
| 701.633
|
|-
| 11/9
| 701.644
|
|-
| 16/15
| 701.676
|
|-
| 5/4
| 701.711
|
|-
| 6/5
| 701.738
| 5-odd-limit minimax
|-
| 10/9
| 701.760
|
|-
| 16/13
| 701.802
|
|-
| 13/12
| 701.807
|
|-
| 18/13
| 701.811
|
|-
| 13/10
| 701.831
|
|-
| 15/13
| 701.836
|
|-
| 4/3
| 701.955
|
|-
| 9/7
| 702.193
| 9-odd-limit minimax
|-
| 7/6
| 702.209
| 7-odd-limit minimax
|-
| 8/7
| 702.227
|
|-
| 15/14
| 702.778
|
|-
| 7/5
| 702.915
|  
|  
|-
|-
Retrieved from "https://en.xen.wiki/w/Garibaldi"