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'''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]].  
{{Infobox regtemp
| Title = Garibaldi
| Subgroups = 2.3.5.7, 2.3.5.7.19
| Comma basis = [[225/224]], [[3125/3087]] (7-limit); <br>[[190/189]], [[225/224]], [[361/360]] (2.3.5.7.19)
| Mapping = 1; 1 -8 -14 -3
| Edo join 1 = 41 | Edo join 2 = 53
| Generators = 3/2
| Generators tuning = 702.10
| Optimization method = CWE
| Pergen = (P8, P5)
| MOS scales = [[5L&nbsp;2s]], [[5L&nbsp;7s]], [[12L&nbsp;5s]], [[12L 17s]]
| Odd limit 1 = 9 | Mistuning 1 = 4.33 | Complexity 1 = 17
| Odd limit 2 = 2.3.5.7.19 21 | Mistuning 2 = 4.65 | Complexity 2 = 17
}}
'''Garibaldi''' is a [[7-limit]] (and higher) [[regular temperament|temperament]] of the [[schismatic family #Garibaldi|schismatic family]]. It is an [[extension]] of [[helmholtz (temperament)|helmholtz]] temperament beyond the 5-limit but with the same simple [[chain of fifths|chain-of-fifths]] structure (so that [[chain-of-fifths notation|standard notation]] may be used). The garibaldi temperament tempers together the Pythagorean, syntonic, and archytas commas into a jack-of-all-trades "generic comma", which can be used to reach intervals of 3, 5, and 7. As in helmholtz temperament, [[5/4]] is mapped to the diminished fourth (e.g. C–F♭; a comma-flat major third), and the new mapping specific to garibaldi is that [[7/4]] is mapped to the double-diminished octave (e.g. C–C𝄫; a comma-flat minor seventh). This makes garibaldi a [[marvel temperaments|marvel]] and [[hemifamity temperaments|hemifamity]] temperament. Tuning the fifth a fraction of a cent sharp gives the best tunings.  


Immediate 11-limit extensions include ''cassandra'' (41&amp;53), mapping 11/8 to +23 steps, ''andromeda'' (29&amp;41), mapping 11/8 to -18 steps, and ''helenus'' (53&amp;65d), mapping 11/8 to -30 steps.  
Immediate 11-limit extensions include '''cassandra''' ({{nowrap| 41 & 53 }}), mapping 11/8 to +23 fifths, '''andromeda''' ({{nowrap| 29 & 41 }}), mapping 11/8 to −18 fifths, and '''helenus''' ({{nowrap| 53 & 65d }}), mapping 11/8 to −30 fifths. Garibaldi is most naturally a 2.3.5.7.19-[[subgroup]] temperament due to its immediate availability of [[19/16]] at the minor third (C–E♭). This is sometimes known as ''garibaldi nestoria.''
 
Garibaldi was named in honor of [[Eduardo Sábat-Garibaldi]], who developed the [[dinarra]], a 53-tone [[microtonal guitar]] in the 1/9-schisma tuning.
 
See [[Schismatic family #Garibaldi]] for technical data.


== Interval chain ==
== Interval chain ==
In the following table, prime harmonics are in '''bold'''.  
In the following table, odd harmonics 1–21 and their inverses are in '''bold'''.  


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
! rowspan="3" | Fifth <br>generator
|-
! rowspan="3" | #
! rowspan="3" | Cents*
! rowspan="3" | Cents*
! colspan="4" | Approximate Ratios
! colspan="4" | Approximate ratios
|-
|-
! rowspan="2" | 7-limit
! rowspan="2" | 2.3.5.7.19 subgroup
! colspan="3" | 13-limit Extension
! colspan="3" | 13-limit extensions
|-
|-
! Cassandra
! Cassandra
Line 20: Line 39:
| 0
| 0
| 0.00
| 0.00
| 1/1
| '''1/1'''
|
|
|
|
Line 26: Line 45:
|-
|-
| 1
| 1
| 702.09
| 702.10
| '''3/2'''
| '''3/2'''
|
|
Line 33: Line 52:
|-
|-
| 2
| 2
| 204.17
| 204.20
| 9/8
| '''9/8'''
|
|
|
|
Line 40: Line 59:
|-
|-
| 3
| 3
| 906.26
| 906.30
| 27/16, 42/25
| 27/16, '''32/19''', 42/25
| 22/13
| 22/13
| 22/13
| 22/13
Line 47: Line 66:
|-
|-
| 4
| 4
| 408.34
| 408.40
| 63/50, 80/63
| 19/15, 24/19
|
|
| 14/11
| 14/11
Line 54: Line 73:
|-
|-
| 5
| 5
| 1110.43
| 1110.50
| 40/21
| 19/10, 36/19, 40/21
|
|
| 21/11
| 21/11
Line 61: Line 80:
|-
|-
| 6
| 6
| 612.51
| 612.60
| 10/7
| 10/7
|
|
Line 68: Line 87:
|-
|-
| 7
| 7
| 114.60
| 114.70
| 15/14, 16/15
| 15/14, '''16/15'''
|
|
| 14/13
| 14/13
Line 75: Line 94:
|-
|-
| 8
| 8
| 816.68
| 816.80
| '''8/5'''
| '''8/5'''
|
|
Line 82: Line 101:
|-
|-
| 9
| 9
| 318.77
| 318.90
| 6/5
| 6/5
|
|
Line 89: Line 108:
|-
|-
| 10
| 10
| 1020.85
| 1021.00
| 9/5
| 9/5, 38/21
|
|
| 20/11
| 20/11
Line 96: Line 115:
|-
|-
| 11
| 11
| 522.94
| 523.09
| 27/20
| 19/14, 27/20
|
|
| 15/11
| 15/11
Line 103: Line 122:
|-
|-
| 12
| 12
| 25.02
| 25.19
| 50/49, 64/63, 81/80
| 50/49, 57/56, 64/63, 81/80
|
|
| 40/39, 45/44
| 40/39, 45/44
Line 110: Line 129:
|-
|-
| 13
| 13
| 727.11
| 727.29
| 32/21
| '''32/21'''
|
|
| 20/13
| 20/13
Line 117: Line 136:
|-
|-
| 14
| 14
| 229.19
| 229.39
| '''8/7'''
| '''8/7'''
|
|
Line 124: Line 143:
|-
|-
| 15
| 15
| 931.28
| 931.49
| 12/7
| 12/7
|
|
|
| 19/11
|
|
|-
|-
| 16
| 16
| 433.36
| 433.59
| 9/7
| 9/7
|
|
Line 138: Line 157:
|-
|-
| 17
| 17
| 1135.45
| 1135.69
| 27/14, 48/25
| 27/14, 48/25
| 52/27
| 52/27
Line 145: Line 164:
|-
|-
| 18
| 18
| 637.53
| 637.79
| 36/25, 81/56
| 36/25, 81/56
| 13/9
| 13/9
| '''16/11'''
| '''16/11''', 19/13
|
|
|-
|-
| 19
| 19
| 139.62
| 139.89
| 27/25
| 27/25
| 13/12
| 13/12
Line 159: Line 178:
|-
|-
| 20
| 20
| 841.70
| 841.99
| 80/49, 81/50
| 57/35, 80/49
| '''13/8''', 44/27
| '''13/8''', 44/27
| 18/11, 64/39
| 18/11, 64/39
Line 166: Line 185:
|-
|-
| 21
| 21
| 343.79
| 344.09
| 60/49
| 60/49
| 11/9, 39/32
| 11/9, 39/32
Line 173: Line 192:
|-
|-
| 22
| 22
| 1045.87
| 1046.19
| 64/35
| 64/35
| 11/6
| 11/6
Line 180: Line 199:
|-
|-
| 23
| 23
| 547.96
| 548.29
| 48/35
| 48/35
| '''11/8'''
| '''11/8''', 26/19
| 18/13
| 18/13
| 15/11
| 15/11
|-
|-
| 24
| 24
| 50.04
| 50.39
| 36/35
| 36/35
| 33/32
| 33/32
Line 194: Line 213:
|-
|-
| 25
| 25
| 752.13
| 752.49
| 54/35
| 54/35
|
|
Line 201: Line 220:
|-
|-
| 26
| 26
| 254.21
| 254.59
| 81/70, 144/125
| 57/49, 81/70, 144/125
|
| 22/19
|
|
| 15/13
| 15/13
|-
|-
| 27
| 27
| 956.30
| 956.69
| 216/125, 256/147
| 171/98, 216/125, 256/147
| 26/15
| 26/15
|
|
|
| 19/11
|-
|-
| 28
| 28
| 458.38
| 458.79
| 64/49
| 64/49
| 13/10
| 13/10
Line 222: Line 241:
|-
|-
| 29
| 29
| 1160.47
| 1160.89
| 96/49
| 96/49
| 39/20, 88/45
| 39/20, 88/45
Line 229: Line 248:
|-
|-
| 30
| 30
| 662.55
| 662.99
| 72/49
| 22/15
|  
|  
| '''16/11''', 19/13
|-
| 31
| 165.08
| 54/49
| 11/10
|  
|  
| 12/11
|-
| 32
| 867.18
| 81/49
| 33/20
|  
|  
| '''16/11'''
| 18/11, 64/39
|-
|-
| 31
| 33
| 164.64
| 369.28
| 216/175
| 26/21
|
| '''16/13''', 27/22
|-
| 34
| 1071.38
| 324/175
| 13/7
|
| 24/13
|-
| 35
| 573.48
| 243/175
|  
|  
|
| 18/13
|-
| 36
| 75.58
| 256/245
| 22/21
|
| 27/26
|-
| 37
| 777.68
| 384/245
| 11/7
|  
|  
|  
|  
| 12/11
|-
|-
| 32
| 38
| 866.72
| 279.78
| 288/245
|  
|  
|  
|  
|  
|  
| 18/11, 64/39
|-
|-
| 33
| 39
| 368.81
| 981.88
| 432/245
|  
|  
|  
|  
|  
|  
| '''16/13''', 27/22
|-
|-
| 34
| 40
| 1070.90
| 483.98
| 324/245
|  
|  
|  
|  
|  
|  
| 24/13
|-
|-
| 35
| 41
| 572.98
| 1186.08
| 486/245
|  
|  
|  
|  
|  
|  
| 18/13
|}
|}
<nowiki>*</nowiki> in 7-limit POTE tuning
<nowiki/>* In 2.3.5.7.19-subgroup CWE tuning
 
=== As a detemperament of 12et ===
[[File:Garibaldi 12et Detempering.png|thumb|Garibaldi as a 41-tone 12et detempering]]
[[File:Garibaldi-cassandra 12et Detempering.png|thumb|Garibaldi/cassandra as a 53-tone 12et detempering]]
 
Garibaldi is very naturally considered as a [[detemperament]] of the [[12edo|12 equal temperament]] (12et), where the chromatic scale becomes a near-equal [[5L 7s]]. The diagram on the right shows a 53-tone detempered scale, with a generator range of -26 to +26. 53 is the largest number of tones for a mos where the 12 categories never overlap. 
 
Each pitch category of 12et is further divided into four or five qualities, separated by a [[pythagorean comma]], which represents the syntonic~septimal comma. Combining this division with the minor and major diatonic qualities of 12et, garibaldi can give up to ''eight'' qualities for each diatonic category. Taking thirds as an example: 
 
In 12tet:
 
* 7/6~19/16~6/5 (minor)
* 5/4~19/15~9/7 (major)
 
In garibaldi (cassandra)
 
* ~[[7/6]] (subminor)
* '''~[[19/16]] (minor)'''
* ~[[6/5]] (superminor)
* ~[[11/9]] (artoneutral)
* ~[[27/22]] (tendoneutral)
* ~[[5/4]] (submajor)
* '''~[[19/15]] (major)'''
* ~[[9/7]] (supermajor)
 
Notice also the little interval between artoneutral and tendoneutral, ~[[243/242]]. This interval spans 41 generator steps. 41edo tempers it out so that it merges artoneutral and tendoneutral into a [[Sqrt(3/2)|hemififth]] whereas 53edo exaggerates it to the size of the generic comma. 94edo tunes it to one half the size of the general comma, which can be seen as a good compromise.
 
On another note, excluding 41edo, the two neutral intervals also have natural 13-limit interpretations in cassandra: 11/9~[[39/32]] and 27/22~[[16/13]], tempering out [[352/351]]. This also means the minor third is ~[[13/11]].
 
== Notation ==
Like in [[schismic]], it is recommended to adopt an additional module of accidentals such as arrows to represent the comma step. Garibaldi further benefits from this as the arrow also stands in for the septimal comma, so that the same inflection can be used to reach classical and septimal intervals alike.
 
The following table shows how to notate 2.3.5.7.11.13.19 intervals in each extension of garibaldi.
 
{| class="wikitable" style="text-align:center; vertical-align:middle;"
|+Nomenclature of selected intervals
|- style="font-weight:bold;"
! rowspan="2" | Ratio
! colspan="3" | Example
|- style="font-weight:bold;"
| Cassandra
| Andromeda
| Helenus
|-
| 3/2
| colspan="3" | C–G (perfect fifth)
|-
| 5/4
| colspan="3" | C–↓E (downmajor third)
|-
| 7/4
| colspan="3" | C–↓Bb (downminor seventh)
|-
| 11/8
| C–↑↑F (dupfourth)
| C–↓↓F#* (dudtritone)
| C–↓3F#* (trudtritone)
|-
| 13/8
| C–↑↑Ab (dupminor sixth)
| C–↓↓A (dudmajor sixth)
| C–↓3A (trudmajor sixth)
|-
| 19/16
| colspan="3" | C–Eb (minor third)
|}
 
<nowiki/>*Can also be spelt ↓Gb and ↓↓Gb respectively, since F# = ↑Gb.
 
== Chords and harmony ==
Traditional tertian harmony is effective. The default triads on the Pythagorean spine are undevicesimal in quality:
* 1–19/15–3/2 (C–E–G)
* 1–19/16–3/2 (C–Eb–G)
 
Note that the major third also represents [[24/19]], and the minor third, [[13/11]]. These chords are typically associated with a sort of coldness and metalness, like those in [[12edo]] if not more so.
 
If a warm, sweet, laid-back sound is desired, the thirds can be inflected inwards by a comma to yield
* 1–5/4–3/2 (C–↓E–G)
* 1–6/5–3/2 (C–↑Eb–G)
 
Contrarily, for a more sour and active sound, they can be inflected outwards by a comma to yield
* 1–9/7–3/2 (C–↑E-G)
* 1–7/6–3/2 (C–↓Eb-G)
 
== Scales ==
* [[Garibaldi5]] – proper [[2L 3s]]
* [[Garibaldi7]] – improper [[5L 2s]]
* [[Garibaldi12]] – proper [[5L 7s]]
* [[Garibaldi17]] – improper [[12L 5s]]
* [[Garibaldi24opt]] – optimized 24-note scale for 13-limit
 
== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 702.0589{{c}}
| CWE: ~3/2 = 702.0774{{c}}
| POTE: ~3/2 = 702.0852{{c}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings (cassandra)
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 702.1192{{c}}
| CWE: ~3/2 = 702.1135{{c}}
| POTE: ~3/2 = 702.1125{{c}}
|}
 
=== Target tunings ===
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (garibaldi)
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 7-odd-limit
| ~3/2 = 702.2086{{c}}
| 7/6
| ~3/2 = 702.140{{c}}
| {{Monzo| 0 -25 11 35 }}
|-
| 9-odd-limit
| ~3/2 = 702.1928{{c}}
| 9/7
| ~3/2 = 702.114{{c}}
| {{Monzo| 0 -27 7 17 }}
|}
 
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (cassandra)
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 11-odd-limit
| ~3/2 = 702.1928{{c}}
| 9/7
| ~3/2 = 702.183{{c}}
| {{Monzo| 0 17 -52 -88 134 }}
|-
| 13-odd-limit
| ~3/2 = 702.1089{{c}}
| 13/7
| ~3/2 = 702.128{{c}}
| {{Monzo| 0 -38 -80 -122 137 116 }}
|-
| 15-odd-limit
| ~3/2 = 702.1089{{c}}
| 13/7
| ~3/2 = 702.112{{c}}
| {{Monzo| 0 -95 -137 -129 167 143 }}
|}


== Spectrum of garibaldi tunings by eigenmonzos ==
{| class="wikitable center-all mw-collapsible mw-collapsed"
=== Cassandra mapping ===
|+ style="white-space: nowrap;" | Target tunings (andromeda)
Gencom: [2 4/3; 225/224 275/273 325/324 385/384]
! rowspan="2" | Target
! colspan="2" | Minimax
|-
! Generator
! Eigenmonzo*
|-
| 11-odd-limit
| ~3/2 = 702.6296{{c}}
| 11/9
|-
| 13-odd-limit
| ~3/2 = 702.7558{{c}}
| 13/9
|-
| 15-odd-limit
| ~3/2 = 702.7558{{c}}
| 13/9
|}


Gencom map: [{{val|1 2 -1 -3 13 12}}, {{val|0 -1 8 14 -23 -20}}]
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (helenus)
! rowspan="2" | Target
! colspan="2" | Minimax
|-
! Generator
! Eigenmonzo*
|-
| 11-odd-limit
| ~3/2 = 701.6435{{c}}
| 11/9
|-
| 13-odd-limit
| ~3/2 = 701.6435{{c}}
| 11/9
|-
| 15-odd-limit
| ~3/2 = 701.6435{{c}}
| 11/9
|}


{| class="wikitable center-1 right-2"
=== Tuning spectra ===
==== Garibaldi ====
{| class="wikitable center-all left-4"
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
| '''[[12edo|7\12]]'''
|
| '''700.0000'''
| '''Lower bound of 9-odd-limit, <br>2.3.5.7.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|
| 19/16
| 700.8290
| 1/3 undevicesimal schisma
|-
|
| 19/12
| 701.1105
| 1/4 undevicesimal schisma
|-
| [[65edo|38\65]]
|
| 701.5385
| 65d val
|-
|
| 15/8
| 701.6759
| 1/7 schisma
|-
|
| 5/4
| 701.7108
| 1/8 schisma
|-
|
| 25/24
| 701.7252
| 2/17 schisma
|-
|
| 5/3
| 701.7379
| 5-odd-limit minimax, 1/9 schisma
|-
|
| 9/5
| 701.7596
| 1/10 schisma
|-
|
| 81/80
| 701.7922
| 1/12 schisma
|-
| [[53edo|31\53]]
|
| 701.8868
|
|-
|
| 3/2
| 701.9550
| Pythagorean tuning
|-
|
| 36/35
| 702.0321
|
|-
| [[94edo|55\94]]
|
| 702.1277
|
|-
|
| 9/7
| 702.1928
| 9-odd-limit minimax, 1/16 septimal schisma
|-
|
| 7/6
| 702.2086
| 7-odd-limit minimax, 1/15 septimal schisma
|-
|
| 49/48
| 702.2174
| 2/29 septimal schisma
|-
|
| 7/4
| 702.2267
| 1/14 septimal schisma
|-
|
| 19/10
| 702.2399
|
|-
|
| 21/16
| 702.2476
| 1/13 septimal schisma
|-
|
| 64/63
| 702.2720
| 1/12 septimal schisma
|-
|
| 19/15
| 702.3111
|
|-
| [[41edo|24\41]]
|
| 702.4390
|
|-
|
| 19/14
| 702.6079
|
|-
|
| 21/19
| 702.6732
|
|-
|
| 15/14
| 702.7775
|
|-
|
| 7/5
| 702.9146
|
|-
|
| 21/20
| 703.1066
|
|-
| '''[[29edo|17\29]]'''
|
| '''703.4483'''
| '''Upper bound of 9-odd-limit, <br>2.3.5.7.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|-
! Eigenmonzo
|
! Fifth
| 13/11
| 703.5968
|
|}
 
==== Cassandra ====
{| class="wikitable mw-collapsible mw-collapsed center-all left-4"
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
! Comments
|-
|-
| 16/15
| '''[[12edo|7\12]]'''
| 701.676
|
| '''700.0000'''
| '''Lower bound of 9-odd-limit diamond monotone'''
|-
|
| 19/16
| 700.8290
| 1/3 undevicesimal schisma
|-
|
| 19/12
| 701.1105
| 1/4 undevicesimal schisma
|-
| [[65edo|38\65]]
|  
|  
| 701.5385
| 65def val
|-
|-
|
| 15/8
| 701.6759
| 1/7 schisma
|-
|
| 5/4
| 5/4
| 701.711
| 701.7108
| 1/8 schisma
|-
|  
|  
| 25/24
| 701.7252
| 2/17 schisma
|-
|-
| {{monzo| 0 -10 17 }}
|  
| 701.728
| 5/3
| 5-odd-limit least squares
| 701.7379
| 5-odd-limit minimax, 1/9 schisma
|-
|
| 9/5
| 701.7596
| 1/10 schisma
|-
|-
| 6/5
|  
| 701.738
| 81/80
| 5-odd-limit minimax
| 701.7922
| 1/12 schisma
|-
|
| 19/13
| 701.8702
|
|-
|-
| 10/9
| '''[[53edo|31\53]]'''
| 701.760
|  
|  
| '''701.8868'''
| '''Lower bound of 11-, 13-, 15-odd-limit, <br>2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|-
|
| 15/13
| 15/13
| 701.9355
| 701.9355
|  
|  
|-
|-
|
| 13/10
| 13/10
| 701.9362
| 701.9362
|  
|  
|-
|-
| 4/3
| 701.955
|  
|  
| 3/2
| 701.9550
| Pythagorean tuning
|-
|-
| 16/13
|  
| 702.026
| 13/8
| 702.0264
|  
|  
|-
|-
|
| 13/12
| 13/12
| 702.030
| 702.0301
|
|-
|
| 36/35
| 702.0321
|
|-
|
| 13/9
| 702.0343
|  
|  
|-
|-
| 18/13
|  
| 702.034
| 19/11
| 702.0694
|  
|  
|-
|-
|
| 11/10
| 11/10
| 702.097
| 702.0969
|  
|  
|-
|-
|
| 15/11
| 15/11
| 702.102
| 702.1016
|  
|  
|-
|-
| 14/13
|  
| 702.109
| 13/7
| 702.1089
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
|-
|-
| {{monzo| 0 -95 -137 -129 167 143 }}
|  
| 702.112
| 21/13
| 15-odd-limit least squares
| 702.1135
|  
|-
|-
| {{monzo| 0 -27 7 17 }}
| [[94edo|55\94]]
| 702.114
|  
| 9-odd-limit least squares
| 702.1277
|  
|-
|-
| {{monzo| 0 -38 -80 -122 137 116 }}
|  
| 702.128
| 9/7
| 13-odd-limit least squares
| 702.1928
| 9- and 11-odd-limit minimax, 1/16 septimal schisma
|-
|-
| {{monzo| 0 -25 11 35 }}
|  
| 702.140
| 7/6
| 7-odd-limit least squares
| 702.2086
| 7-odd-limit minimax, 1/15 septimal schisma
|-
|-
| {{monzo| 0 17 -52 -88 134 }}
|  
| 702.183
| 49/48
| 11-odd-limit least squares
| 702.2174
| 2/29 septimal schisma
|-
|
| 7/4
| 702.2267
| 1/14 septimal schisma
|-
|-
| 9/7
|  
| 702.193
| 11/7
| 9- and 11-odd-limit minimax
| 702.2295
|  
|-
|-
| 7/6
|  
| 702.209
| 11/8
| 7-odd-limit minimax
| 702.2312
|  
|-
|-
| 8/7
|  
| 702.227
| 21/11
| 702.2371
|  
|  
|-
|-
| 14/11
|  
| 702.230
| 19/10
| 702.2399
|  
|  
|-
|-
| 11/8
|
| 702.231
| 11/6
| 702.2438
|  
|  
|-
|-
| 12/11
| 702.244
|  
|  
| 21/16
| 702.2476
| 1/13 septimal schisma
|-
|-
|
| 11/9
| 11/9
| 702.258
| 702.2575
|  
|  
|-
|-
|
| 64/63
| 702.2720
| 1/12 septimal schisma
|-
|
| 19/15
| 702.3111
|
|-
| '''[[41edo|24\41]]'''
|
| '''702.4390'''
| '''Upper bound of 11-, 13-, 15-odd-limit, <br>2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|
| 19/14
| 702.6079
|
|-
|
| 21/19
| 702.6732
|
|-
|
| 15/14
| 15/14
| 702.778
| 702.7775
|  
|  
|-
|-
|
| 7/5
| 7/5
| 702.915
| 702.9146
|
|-
|
| 21/20
| 703.1066
|
|-
| '''[[29edo|17\29]]'''
|  
|  
| '''703.4483'''
| '''29ef val, upper bound of 9-odd-limit diamond monotone'''
|-
|-
|
| 13/11
| 13/11
| 703.597
| 703.5968
|  
|  
|}
|}


=== Andromeda mapping ===
==== Andromeda ====
Gencom: [2 4/3; 100/99 105/104 196/195 245/242]
{| class="wikitable mw-collapsible mw-collapsed center-all left-4"
 
! Edo<br>generator
Gencom map: [{{val|1 2 -1 -3 -4 -5}}, {{val|0 -1 8 14 18 21}}]
! Unchanged interval<br>(eigenmonzo)*
 
! Generator (¢)
{| class="wikitable center-1 right-2"
! Comments
|-
| '''[[12edo|7\12]]'''
|
| '''700.0000'''
| '''Lower bound of 9- and 11-odd-limit diamond monotone'''
|-
|
| 19/16
| 700.8290
| 1/3 undevicesimal schisma
|-
|
| 19/12
| 701.1105
| 1/4 undevicesimal schisma
|-
|-
! Eigenmonzo
| [[65edo|38\65]]
! Fifth
|
! Comments
| 701.5385
| 65deeff val
|-
|-
| 16/15
| 701.676
|  
|  
| 15/8
| 701.6759
| 1/7 schisma
|-
|-
|
| 5/4
| 5/4
| 701.711
| 701.7108
| 1/8 schisma
|-
|
| 25/24
| 701.7252
| 2/17 schisma
|-
|  
|  
| 5/3
| 701.7379
| 5-odd-limit minimax, 1/9 schisma
|-
|-
| 6/5
|  
| 701.738
| 9/5
| 5-odd-limit minimax
| 701.7596
| 1/10 schisma
|-
|
| 81/80
| 701.7922
| 1/12 schisma
|-
|-
| 10/9
| [[53edo|31\53]]
| 701.760
|  
|  
| 701.8868
| 53ef val
|-
|-
| 4/3
| 701.955
|  
|  
| 3/2
| 701.9550
| Pythagorean tuning
|-
|-
|
| 36/35
| 702.0321
|
|-
|
| 9/7
| 9/7
| 702.193
| 702.1928
| 9-odd-limit minimax
| 9-odd-limit minimax, 1/16 septimal schisma
|-
|-
|
| 7/6
| 7/6
| 702.209
| 702.2086
| 7-odd-limit minimax
| 7-odd-limit minimax, 1/15 septimal schisma
|-
|
| 49/48
| 702.2174
| 2/29 septimal schisma
|-
|
| 7/4
| 702.2267
| 1/14 septimal schisma
|-
|
| 21/16
| 702.2476
| 1/13 septimal schisma
|-
|-
| 8/7
| 702.227
|  
|  
| 64/63
| 702.2720
| 1/12 septimal schisma
|-
|-
|
| 19/15
| 702.3111
|
|-
| '''[[41edo|24\41]]'''
|
| '''702.4390'''
| '''Lower bound of 13-, 15-odd-limit, <br>2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|
| 19/14
| 702.6079
|
|-
|
| 11/9
| 11/9
| 702.630
| 702.6296
| 11-odd-limit minimax
| 11-odd-limit minimax
|-
|-
| 12/11
|  
| 702.665
| 11/6
| 702.6651
|
|-
|
| 21/19
| 702.6732
|  
|  
|-
|-
|
| 11/8
| 11/8
| 702.705
| 702.7046
|  
|  
|-
|-
| 18/13
| 702.756
|  
|  
| 13/9
| 702.7558
| 13- and 15-odd-limit minimax
|-
|-
|
| 15/14
| 15/14
| 702.778
| 702.7775
|  
|  
|-
|-
|
| 13/12
| 13/12
| 702.792
| 702.7922
|  
|  
|-
|-
| 16/13
|  
| 702.832
| 13/8
| 702.8320
|  
|  
|-
|-
|
| 7/5
| 7/5
| 702.915
| 702.9146
|
|-
|
| 19/11
| 703.0797
|
|-
|
| 21/20
| 703.1066
|  
|  
|-
|-
|
| 19/13
| 703.1659
|
|-
|
| 15/11
| 15/11
| 703.359
| 703.3592
|  
|  
|-
|-
|
| 15/13
| 15/13
| 703.410
| 703.4101
|
|-
| '''[[29edo|17\29]]'''
|  
|  
| '''703.4483'''
| '''Upper bound of 9-, 11-, 13-, 15-odd-limit, <br>2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|-
|
| 11/10
| 11/10
| 703.500
| 703.4996
|  
|  
|-
|-
|
| 13/10
| 13/10
| 703.522
| 703.5220
|  
|  
|-
|-
|
| 13/11
| 13/11
| 703.597
| 703.5968
|  
|  
|-
|-
| 14/13
|  
| 704.043
| 21/13
| 701.7817
|
|-
|
| 19/10
| 702.2399
|
|-
|
| 21/11
| 703.8926
|
|-
|
| 13/7
| 704.0426
|  
|  
|-
|-
| 14/11
|  
| 704.377
| 11/7
| 704.3770
|  
|  
|}
|}


=== Helenus mapping ===
==== Helenus ====
Gencom: [2 4/3; 99/98 176/175 275/273 847/845]
{| class="wikitable mw-collapsible mw-collapsed center-all left-4"
 
! Edo<br>generator
Gencom map: [{{val|1 2 -1 -3 -9 -10}}, {{val|0 -1 8 14 30 33}}]
! Unchanged interval<br>(eigenmonzo)*
 
! Generator (¢)
{| class="wikitable center-1 right-2"
! Comments
|-
| '''[[12edo|7\12]]'''
|
| '''700.0000'''
| '''Lower bound of 9- and 11-odd-limit diamond monotone'''
|-
|
| 19/16
| 700.8290
| 1/3 undevicesimal schisma
|-
|
| 11/7
| 701.0942
|
|-
|
| 19/12
| 701.1105
| 1/4 undevicesimal schisma
|-
|
| 21/11
| 701.1149
|
|-
|-
! Eigenmonzo
|
! Fifth
| 13/7
! Comments
| 701.4894
|
|-
|-
| 14/11
|  
| 701.094
| 21/13
| 701.5127
|  
|  
|-
|-
| 14/13
| '''[[65edo|38\65]]'''
| 701.489
|  
|  
| '''701.5385'''
| '''65d val, lower bound of 13-, 15-odd-limit, <br>2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|-
|
| 11/10
| 11/10
| 701.591
| 701.5907
|  
|  
|-
|-
|
| 15/11
| 15/11
| 701.607
| 701.6066
|  
|  
|-
|-
|
| 11/8
| 11/8
| 701.623
| 701.6227
|  
|  
|-
|-
| 12/11
|  
| 701.633
| 11/6
| 701.6335
|  
|  
|-
|-
|
| 11/9
| 11/9
| 701.644
| 701.6435
| 11-odd-limit minimax
| 11-, 13-, and 15-odd-limit minimax
|-
|
| 15/8
| 701.6759
| 1/7 schisma
|-
|-
| 16/15
|  
| 701.676
| 19/11
| 701.7109
|  
|  
|-
|-
|
| 5/4
| 5/4
| 701.711
| 701.7108
| 1/8 schisma
|-
|
| 25/24
| 701.7252
| 2/17 schisma
|-
|  
|  
| 5/3
| 701.7379
| 5-odd-limit minimax, 1/9 schisma
|-
|-
| 6/5
|  
| 701.738
| 9/5
| 5-odd-limit minimax
| 701.7596
| 1/10 schisma
|-
|-
| 10/9
| 701.760
|  
|  
| 81/80
| 701.7922
| 1/12 schisma
|-
|-
| 16/13
|  
| 701.802
| 13/8
| 701.8022
|  
|  
|-
|-
|
| 13/12
| 13/12
| 701.807
| 701.8067
|  
|  
|-
|-
| 18/13
|  
| 701.811
| 13/9
| 701.8109
|  
|  
|-
|-
|
| 13/10
| 13/10
| 701.831
| 701.8314
|  
|  
|-
|-
|
| 15/13
| 15/13
| 701.836
| 701.8362
|
|-
| '''[[53edo|31\53]]'''
|
| '''701.8868'''
| '''Upper bound of 11-, 13-, 15-odd-limit, <br>2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|
| 19/13
| 701.8995
|  
|  
|-
|-
| 4/3
|  
| 701.955
| 3/2
| 701.9550
| Pythagorean tuning
|-
|
| 36/35
| 702.0321
|  
|  
|-
|-
|
| 9/7
| 9/7
| 702.193
| 702.1928
| 9-odd-limit minimax
| 9-odd-limit minimax, 1/16 septimal schisma
|-
|-
|
| 7/6
| 7/6
| 702.209
| 702.2086
| 7-odd-limit minimax
| 7-odd-limit minimax, 1/15 septimal schisma
|-
|
| 49/48
| 702.2174
| 2/29 septimal schisma
|-
|
| 7/4
| 702.2267
| 1/14 septimal schisma
|-
|
| 19/10
| 702.2399
|
|-
|
| 21/16
| 702.2476
| 1/13 septimal schisma
|-
|
| 64/63
| 702.2720
| 1/12 septimal schisma
|-
|
| 19/15
| 702.3111
|
|-
| [[41edo|24\41]]
|
| 702.4390
| 41ef val
|-
|
| 19/14
| 702.6079
|
|-
|-
| 8/7
|  
| 702.227
| 21/19
| 702.6732
|  
|  
|-
|-
|
| 15/14
| 15/14
| 702.778
| 702.7775
|  
|  
|-
|-
|
| 7/5
| 7/5
| 702.915
| 702.9146
|
|-
|
| 21/20
| 703.1066
|
|-
| '''[[29edo|17\29]]'''
|  
|  
| '''703.4483'''
| '''29eeff val, upper bound of 9-odd-limit diamond monotone'''
|-
|-
|
| 13/11
| 13/11
| 703.597
| 703.5968
|  
|  
|}
|}
<nowiki/>* Besides the octave


== Scales ==
[[Category:Garibaldi| ]] <!-- Main article -->
* [[Garibaldi5]] - proper [[2L 3s]]
[[Category:Rank-2 temperaments]]
* [[Garibaldi7]] - improper [[5L 2s]]
[[Category:Schismatic family]]
* [[Garibaldi12]] - proper [[5L 7s]]
* [[Garibaldi17]] - improper [[12L 5s]]
* [[Garibaldi24opt]] - optimized 24-note scale for 13-limit
 
[[Category:Garibaldi| ]] <!-- main article -->
[[Category:Marvel temperaments]]
[[Category:Marvel temperaments]]
[[Category:Schismatic family]]
[[Category:Gariboh clan]]
{{IoT}}
[[Category:Hemifamity temperaments]]
Retrieved from "https://en.xen.wiki/w/Garibaldi"