Garibaldi: Difference between revisions

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'''Garibaldi''' is a [[7-limit]] (and higher) [[regular temperament|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|chain-of-fifths]] structure (so that [[chain-of-fifths notation|standard notation]] may be used). As in helmholtz temperament, [[5/4]] is mapped to the diminished fourth (e.g. C–F♭), and the new mapping specific to garibaldi is that [[7/4]] is mapped to the doubly-diminished octave (e.g. C–C𝄫). 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 fifths, ''andromeda'' {{nowrap|(29 &amp; 41)}}, mapping 11/8 to &minus;18 fifths, and ''helenus'' {{nowrap|(53 &amp; 65d)}}, mapping 11/8 to &minus;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''.
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.  
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 ==
Line 9: Line 25:


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
! rowspan="3" | #
! rowspan="3" | #
! rowspan="3" | Cents*
! rowspan="3" | Cents*
Line 14: Line 31:
|-
|-
! rowspan="2" | 2.3.5.7.19 subgroup
! rowspan="2" | 2.3.5.7.19 subgroup
! colspan="3" | 13-limit extension
! colspan="3" | 13-limit extensions
|-
|-
! Cassandra
! Cassandra
Line 28: Line 45:
|-
|-
| 1
| 1
| 702.06
| 702.10
| '''3/2'''
| '''3/2'''
|
|
Line 35: Line 52:
|-
|-
| 2
| 2
| 204.12
| 204.20
| '''9/8'''
| '''9/8'''
|
|
Line 42: Line 59:
|-
|-
| 3
| 3
| 906.18
| 906.30
| 27/16, '''32/19''', 42/25
| 27/16, '''32/19''', 42/25
| 22/13
| 22/13
Line 49: Line 66:
|-
|-
| 4
| 4
| 408.24
| 408.40
| 19/15, 24/19
| 19/15, 24/19
|
|
Line 56: Line 73:
|-
|-
| 5
| 5
| 1110.29
| 1110.50
| 19/10, 36/19, 40/21
| 19/10, 36/19, 40/21
|
|
Line 63: Line 80:
|-
|-
| 6
| 6
| 612.35
| 612.60
| 10/7
| 10/7
|
|
Line 70: Line 87:
|-
|-
| 7
| 7
| 114.41
| 114.70
| 15/14, '''16/15'''
| 15/14, '''16/15'''
|
|
Line 77: Line 94:
|-
|-
| 8
| 8
| 816.47
| 816.80
| '''8/5'''
| '''8/5'''
|
|
Line 84: Line 101:
|-
|-
| 9
| 9
| 318.53
| 318.90
| 6/5
| 6/5
|
|
Line 91: Line 108:
|-
|-
| 10
| 10
| 1020.59
| 1021.00
| 9/5, 38/21
| 9/5, 38/21
|
|
Line 98: Line 115:
|-
|-
| 11
| 11
| 522.65
| 523.09
| 19/14, 27/20
| 19/14, 27/20
|
|
Line 105: Line 122:
|-
|-
| 12
| 12
| 24.71
| 25.19
| 50/49, 57/56, 64/63, 81/80
| 50/49, 57/56, 64/63, 81/80
|
|
Line 112: Line 129:
|-
|-
| 13
| 13
| 726.77
| 727.29
| '''32/21'''
| '''32/21'''
|
|
Line 119: Line 136:
|-
|-
| 14
| 14
| 228.82
| 229.39
| '''8/7'''
| '''8/7'''
|
|
Line 126: Line 143:
|-
|-
| 15
| 15
| 930.88
| 931.49
| 12/7
| 12/7
|
|
Line 133: Line 150:
|-
|-
| 16
| 16
| 432.94
| 433.59
| 9/7
| 9/7
|
|
Line 140: Line 157:
|-
|-
| 17
| 17
| 1135.00
| 1135.69
| 27/14, 48/25
| 27/14, 48/25
| 52/27
| 52/27
Line 147: Line 164:
|-
|-
| 18
| 18
| 637.06
| 637.79
| 36/25, 81/56
| 36/25, 81/56
| 13/9
| 13/9
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|-
|-
| 19
| 19
| 139.12
| 139.89
| 27/25
| 27/25
| 13/12
| 13/12
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|-
|-
| 20
| 20
| 841.18
| 841.99
| 57/35, 80/49
| 57/35, 80/49
| '''13/8''', 44/27
| '''13/8''', 44/27
Line 168: Line 185:
|-
|-
| 21
| 21
| 343.24
| 344.09
| 60/49
| 60/49
| 11/9, 39/32
| 11/9, 39/32
Line 175: Line 192:
|-
|-
| 22
| 22
| 1045.30
| 1046.19
| 64/35
| 64/35
| 11/6
| 11/6
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|-
|-
| 23
| 23
| 547.35
| 548.29
| 48/35
| 48/35
| '''11/8''', 26/19
| '''11/8''', 26/19
Line 189: Line 206:
|-
|-
| 24
| 24
| 49.41
| 50.39
| 36/35
| 36/35
| 33/32
| 33/32
Line 196: Line 213:
|-
|-
| 25
| 25
| 751.47
| 752.49
| 54/35
| 54/35
|
|
Line 203: Line 220:
|-
|-
| 26
| 26
| 253.53
| 254.59
| 57/49, 81/70, 144/125
| 57/49, 81/70, 144/125
| 22/19
| 22/19
Line 210: Line 227:
|-
|-
| 27
| 27
| 955.59
| 956.69
| 171/98, 216/125, 256/147
| 171/98, 216/125, 256/147
| 26/15
| 26/15
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|-
|-
| 28
| 28
| 457.65
| 458.79
| 64/49
| 64/49
| 13/10
| 13/10
Line 224: Line 241:
|-
|-
| 29
| 29
| 1159.71
| 1160.89
| 96/49
| 96/49
| 39/20, 88/45
| 39/20, 88/45
Line 231: Line 248:
|-
|-
| 30
| 30
| 661.77
| 662.99
| 72/49
| 72/49
| 22/15
| 22/15
Line 238: Line 255:
|-
|-
| 31
| 31
| 163.83
| 165.08
| 54/49
| 54/49
| 11/10
| 11/10
Line 245: Line 262:
|-
|-
| 32
| 32
| 865.88
| 867.18
| 81/49
| 81/49
| 33/20
| 33/20
Line 252: Line 269:
|-
|-
| 33
| 33
| 367.94
| 369.28
| 216/175
| 216/175
| 26/21
| 26/21
Line 259: Line 276:
|-
|-
| 34
| 34
| 1070.00
| 1071.38
| 324/175
| 324/175
| 13/7
| 13/7
Line 266: Line 283:
|-
|-
| 35
| 35
| 572.06
| 573.48
| 243/175
| 243/175
|  
|  
Line 273: Line 290:
|-
|-
| 36
| 36
| 74.12
| 75.58
| 256/245
| 256/245
| 22/21
| 22/21
Line 280: Line 297:
|-
|-
| 37
| 37
| 776.18
| 777.68
| 384/245
| 384/245
| 11/7
| 11/7
Line 287: Line 304:
|-
|-
| 38
| 38
| 278.24
| 279.78
| 288/245
| 288/245
|  
|  
Line 294: Line 311:
|-
|-
| 39
| 39
| 980.30
| 981.88
| 432/245
| 432/245
|  
|  
Line 301: Line 318:
|-
|-
| 40
| 40
| 482.36
| 483.98
| 324/245
| 324/245
|  
|  
Line 308: Line 325:
|-
|-
| 41
| 41
| 1184.41
| 1186.08
| 486/245
| 486/245
|  
|  
Line 314: Line 331:
|  
|  
|}
|}
<nowiki />* In 7-limit CTE 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 ==
== Notation ==
Using garibaldi can be a challenge because it defies the tradition of tertian harmony in [[chain-of-fifths notation]]. The just major triad on C is C–Fb–G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C–vE–G.  
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.  


{| class="wikitable center-1 center-3"
The following table shows how to notate 2.3.5.7.11.13.19 intervals in each extension of garibaldi.
|+ style="font-size: 105%; white-space: nowrap;" | Cassandra nomenclature for<br />selected intervals
 
|-
{| class="wikitable" style="text-align:center; vertical-align:middle;"
! Ratio
|+Nomenclature of selected intervals
! Nominal
|- style="font-weight:bold;"
! Example
! rowspan="2" | Ratio
! colspan="3" | Example
|- style="font-weight:bold;"
| Cassandra
| Andromeda
| Helenus
|-
|-
| 3/2
| 3/2
| Perfect fifth
| colspan="3" | C–G (perfect fifth)
| C–G
|-
|-
| 5/4
| 5/4
| Downmajor third
| colspan="3" | C–↓E (downmajor third)
| C–vE
|-
|-
| 7/4
| 7/4
| Downminor seventh
| colspan="3" | C–↓Bb (downminor seventh)
| C–vBb
|-
|-
| 11/8
| 11/8
| Double-up fourth
| C–↑↑F (dupfourth)
| C–^^F
| C–↓↓F#* (dudtritone)
| C–↓3F#* (trudtritone)
|-
|-
| 13/8
| 13/8
| Double-up minor sixth
| C–↑↑Ab (dupminor sixth)
| C–^^Ab
| C–↓↓A (dudmajor sixth)
| C–↓3A (trudmajor sixth)
|-
|-
| 19/16
| 19/16
| Minor third
| colspan="3" | C–Eb (minor third)
| C–Eb
|}
|}


{| class="wikitable center-1 center-3 mw-collapsible mw-collapsed"
<nowiki/>*Can also be spelt ↓Gb and ↓↓Gb respectively, since F# = ↑Gb.
|+ style="font-size: 105%; white-space: nowrap;" | Andromeda nomenclature for selected intervals
|-
! Ratio
! Nominal
! Example
|-
| 11/8
| Down-diminished fifth<br>Double-down augmented fourth
| C–vGb<br>C–vvF#
|-
| 13/8
| Double downmajor sixth
| C–vvA
|}
 
{| class="wikitable center-1 center-3 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Helenus nomenclature for selected intervals
|-
! Ratio
! Nominal
! Example
|-
| 11/8
| Double-down diminished fifth<br>Triple-down augmented fourth
| C–vvGb<br>C–v<sup>3</sup>F#
|-
| 13/8
| Triple-down major sixth
| C–v<sup>3</sup>A
|}


== Chords and harmony ==
== Chords and harmony ==
Line 391: Line 409:


If a warm, sweet, laid-back sound is desired, the thirds can be inflected inwards by a comma to yield
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–vE–G)
* 1–5/4–3/2 (C–↓E–G)
* 1–6/5–3/2 (C–^Eb–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
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)
* 1–9/7–3/2 (C–↑E-G)
* 1–7/6–3/2 (C–vEb-G)
* 1–7/6–3/2 (C–↓Eb-G)


== Scales ==
== Scales ==
Line 406: Line 424:


== Tunings ==
== 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 }}
|}
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (andromeda)
! 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
|}
{| 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
|}
=== Tuning spectra ===
=== Tuning spectra ===
==== Garibaldi ====
==== Garibaldi ====
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
! Edo<br>generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
| 7\12
| '''[[12edo|7\12]]'''
|  
|  
| 700.0000
| '''700.0000'''
| Lower bound of 9-odd-limit, <br>2.3.5.7.19 subgroup 19- and 21-odd-limit diamond monotone
| '''Lower bound of 9-odd-limit, <br>2.3.5.7.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|-
|  
|  
Line 429: Line 577:
| 1/4 undevicesimal schisma
| 1/4 undevicesimal schisma
|-
|-
| 38\65
| [[65edo|38\65]]
|  
|  
| 701.5385
| 701.5385
|  
| 65d val
|-
|-
|  
|  
| 15/8
| 15/8
| 701.676
| 701.6759
| 1/7 schisma
| 1/7 schisma
|-
|-
|  
|  
| 5/4
| 5/4
| 701.711
| 701.7108
| 1/8 schisma
| 1/8 schisma
|-
|-
Line 451: Line 599:
|  
|  
| 5/3
| 5/3
| 701.738
| 701.7379
| 5-odd-limit minimax, 1/9 schisma
| 5-odd-limit minimax, 1/9 schisma
|-
|-
|  
|  
| 9/5
| 9/5
| 701.760
| 701.7596
| 1/10 schisma
| 1/10 schisma
|-
|-
Line 464: Line 612:
| 1/12 schisma
| 1/12 schisma
|-
|-
| 31\53
| [[53edo|31\53]]
|  
|  
| 701.8868
| 701.8868
Line 477: Line 625:
| 36/35
| 36/35
| 702.0321
| 702.0321
|
|-
| [[94edo|55\94]]
|
| 702.1277
|  
|  
|-
|-
|  
|  
| 9/7
| 9/7
| 702.193
| 702.1928
| 9-odd-limit minimax, 1/16 septimal schisma
| 9-odd-limit minimax, 1/16 septimal schisma
|-
|-
|  
|  
| 7/6
| 7/6
| 702.209
| 702.2086
| 7-odd-limit minimax, 1/15 septimal schisma
| 7-odd-limit minimax, 1/15 septimal schisma
|-
|-
Line 496: Line 649:
|  
|  
| 7/4
| 7/4
| 702.227
| 702.2267
| 1/14 septimal schisma
| 1/14 septimal schisma
|-
|-
Line 519: Line 672:
|  
|  
|-
|-
| 24\41
| [[41edo|24\41]]
|  
|  
| 702.4390
| 702.4390
Line 536: Line 689:
|  
|  
| 15/14
| 15/14
| 702.778
| 702.7775
|  
|  
|-
|-
|  
|  
| 7/5
| 7/5
| 702.915
| 702.9146
|  
|  
|-
|-
Line 549: Line 702:
|  
|  
|-
|-
| 17\29
| '''[[29edo|17\29]]'''
|  
|  
| 703.4483
| '''703.4483'''
| Upper bound of 9-odd-limit, <br />2.3.5.7.19 subgroup 19- and 21-odd-limit diamond monotone
| '''Upper bound of 9-odd-limit, <br>2.3.5.7.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|-
|  
|  
| 13/11
| 13/11
| 703.597
| 703.5968
|  
|  
|}
|}
Line 563: Line 716:
{| class="wikitable mw-collapsible mw-collapsed center-all left-4"
{| class="wikitable mw-collapsible mw-collapsed center-all left-4"
! Edo<br>generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
| 7\12
| '''[[12edo|7\12]]'''
|  
|  
| 700.0000
| '''700.0000'''
| Lower bound of 9-odd-limit diamond monotone
| '''Lower bound of 9-odd-limit diamond monotone'''
|-
|-
|  
|  
Line 582: Line 735:
| 1/4 undevicesimal schisma
| 1/4 undevicesimal schisma
|-
|-
| 38\65
| [[65edo|38\65]]
|  
|  
| 701.5385
| 701.5385
|  
| 65def val
|-
|-
|  
|  
| 15/8
| 15/8
| 701.676
| 701.6759
| 1/7 schisma
| 1/7 schisma
|-
|-
|  
|  
| 5/4
| 5/4
| 701.711
| 701.7108
| 1/8 schisma
| 1/8 schisma
|-
|-
Line 601: Line 754:
| 701.7252
| 701.7252
| 2/17 schisma
| 2/17 schisma
|-
|
| {{monzo| 0 -10 17 }}
| 701.728
| 5-odd-limit least squares
|-
|-
|  
|  
| 5/3
| 5/3
| 701.738
| 701.7379
| 5-odd-limit minimax, 1/9 schisma
| 5-odd-limit minimax, 1/9 schisma
|-
|-
|  
|  
| 9/5
| 9/5
| 701.760
| 701.7596
| 1/10 schisma
| 1/10 schisma
|-
|-
Line 627: Line 775:
|  
|  
|-
|-
| 31\53
| '''[[53edo|31\53]]'''
|  
|  
| 701.8868
| '''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
| '''Lower bound of 11-, 13-, 15-odd-limit, <br>2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|-
|  
|  
Line 649: Line 797:
|  
|  
| 13/8
| 13/8
| 702.026
| 702.0264
|  
|  
|-
|-
|  
|  
| 13/12
| 13/12
| 702.030
| 702.0301
|  
|  
|-
|-
Line 664: Line 812:
|  
|  
| 13/9
| 13/9
| 702.034
| 702.0343
|  
|  
|-
|-
Line 674: Line 822:
|  
|  
| 11/10
| 11/10
| 702.097
| 702.0969
|  
|  
|-
|-
|  
|  
| 15/11
| 15/11
| 702.102
| 702.1016
|  
|  
|-
|-
|  
|  
| 13/7
| 13/7
| 702.109
| 702.1089
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
|-
|
| <span style="font-size:0.75em">{{monzo| 0 -95 -137 -129 167 143 }}</span>
| 702.112
| 15-odd-limit least squares
|-
|-
|  
|  
Line 697: Line 840:
|  
|  
|-
|-
| [[94edo|55\94]]
|  
|  
| {{monzo| 0 -27 7 17 }}
| 702.1277
| 702.114
| 9-odd-limit least squares
|-
|  
|  
| <span style="font-size:0.75em">{{monzo| 0 -38 -80 -122 137 116 }}</span>
| 702.128
| 13-odd-limit least squares
|-
|
| {{monzo| 0 -25 11 35 }}
| 702.140
| 7-odd-limit least squares
|-
|
| <span style="font-size:0.9em">{{monzo| 0 17 -52 -88 134 }}</span>
| 702.183
| 11-odd-limit least squares
|-
|-
|  
|  
| 9/7
| 9/7
| 702.193
| 702.1928
| 9- and 11-odd-limit minimax, 1/16 septimal schisma
| 9- and 11-odd-limit minimax, 1/16 septimal schisma
|-
|-
|  
|  
| 7/6
| 7/6
| 702.209
| 702.2086
| 7-odd-limit minimax, 1/15 septimal schisma
| 7-odd-limit minimax, 1/15 septimal schisma
|-
|-
Line 734: Line 862:
|  
|  
| 7/4
| 7/4
| 702.227
| 702.2267
| 1/14 septimal schisma
| 1/14 septimal schisma
|-
|-
|  
|  
| 11/7
| 11/7
| 702.230
| 702.2295
|  
|  
|-
|-
|  
|  
| 11/8
| 11/8
| 702.231
| 702.2312
|  
|  
|-
|-
Line 759: Line 887:
|  
|  
| 11/6
| 11/6
| 702.244
| 702.2438
|  
|  
|-
|-
Line 769: Line 897:
|  
|  
| 11/9
| 11/9
| 702.258
| 702.2575
|  
|  
|-
|-
Line 782: Line 910:
|  
|  
|-
|-
| 24\41
| '''[[41edo|24\41]]'''
|  
|  
| 702.4390
| '''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
| '''Upper bound of 11-, 13-, 15-odd-limit, <br>2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|-
|  
|  
Line 799: Line 927:
|  
|  
| 15/14
| 15/14
| 702.778
| 702.7775
|  
|  
|-
|-
|  
|  
| 7/5
| 7/5
| 702.915
| 702.9146
|  
|  
|-
|-
Line 812: Line 940:
|  
|  
|-
|-
| 17\29
| '''[[29edo|17\29]]'''
|  
|  
| 703.4483
| '''703.4483'''
| Upper bound of 9-odd-limit diamond monotone
| '''29ef val, upper bound of 9-odd-limit diamond monotone'''
|-
|-
|  
|  
| 13/11
| 13/11
| 703.597
| 703.5968
|  
|  
|}
|}
Line 826: Line 954:
{| class="wikitable mw-collapsible mw-collapsed center-all left-4"
{| class="wikitable mw-collapsible mw-collapsed center-all left-4"
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)*
! Unchanged interval<br>(eigenmonzo)*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
| 7\12
| '''[[12edo|7\12]]'''
|  
|  
| 700.0000
| '''700.0000'''
| Lower bound of 9- and 11-odd-limit diamond monotone
| '''Lower bound of 9- and 11-odd-limit diamond monotone'''
|-
|-
|  
|  
Line 845: Line 973:
| 1/4 undevicesimal schisma
| 1/4 undevicesimal schisma
|-
|-
| 38\65
| [[65edo|38\65]]
|  
|  
| 701.5385
| 701.5385
|  
| 65deeff val
|-
|-
|  
|  
| 15/8
| 15/8
| 701.676
| 701.6759
| 1/7 schisma
| 1/7 schisma
|-
|-
|  
|  
| 5/4
| 5/4
| 701.711
| 701.7108
| 1/8 schisma
| 1/8 schisma
|-
|-
Line 867: Line 995:
|  
|  
| 5/3
| 5/3
| 701.738
| 701.7379
| 5-odd-limit minimax, 1/9 schisma
| 5-odd-limit minimax, 1/9 schisma
|-
|-
|  
|  
| 9/5
| 9/5
| 701.760
| 701.7596
| 1/10 schisma
| 1/10 schisma
|-
|-
Line 880: Line 1,008:
| 1/12 schisma
| 1/12 schisma
|-
|-
| 31\53
| [[53edo|31\53]]
|  
|  
| 701.8868
| 701.8868
|  
| 53ef val
|-
|-
|  
|  
Line 897: Line 1,025:
|  
|  
| 9/7
| 9/7
| 702.193
| 702.1928
| 9-odd-limit minimax, 1/16 septimal schisma
| 9-odd-limit minimax, 1/16 septimal schisma
|-
|-
|  
|  
| 7/6
| 7/6
| 702.209
| 702.2086
| 7-odd-limit minimax, 1/15 septimal schisma
| 7-odd-limit minimax, 1/15 septimal schisma
|-
|-
Line 912: Line 1,040:
|  
|  
| 7/4
| 7/4
| 702.227
| 702.2267
| 1/14 septimal schisma
| 1/14 septimal schisma
|-
|-
Line 930: Line 1,058:
|  
|  
|-
|-
| 24\41
| '''[[41edo|24\41]]'''
|  
|  
| 702.4390
| '''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
| '''Lower bound of 13-, 15-odd-limit, <br>2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|-
|  
|  
Line 942: Line 1,070:
|  
|  
| 11/9
| 11/9
| 702.630
| 702.6296
| 11-odd-limit minimax
| 11-odd-limit minimax
|-
|-
|  
|  
| 11/6
| 11/6
| 702.665
| 702.6651
|  
|  
|-
|-
Line 957: Line 1,085:
|  
|  
| 11/8
| 11/8
| 702.705
| 702.7046
|  
|  
|-
|-
|  
|  
| 13/9
| 13/9
| 702.756
| 702.7558
| 13- and 15-odd-limit minimax
| 13- and 15-odd-limit minimax
|-
|-
|  
|  
| 15/14
| 15/14
| 702.778
| 702.7775
|  
|  
|-
|-
|  
|  
| 13/12
| 13/12
| 702.792
| 702.7922
|  
|  
|-
|-
|  
|  
| 13/8
| 13/8
| 702.832
| 702.8320
|  
|  
|-
|-
|  
|  
| 7/5
| 7/5
| 702.915
| 702.9146
|  
|  
|-
|-
Line 1,002: Line 1,130:
|  
|  
| 15/11
| 15/11
| 703.359
| 703.3592
|  
|  
|-
|-
|  
|  
| 15/13
| 15/13
| 703.410
| 703.4101
|  
|  
|-
|-
| 17\29
| '''[[29edo|17\29]]'''
|  
|  
| 703.4483
| '''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
| '''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
|  
|  
|-
|-
Line 1,047: Line 1,175:
|  
|  
| 13/7
| 13/7
| 704.043
| 704.0426
|  
|  
|-
|-
|  
|  
| 11/7
| 11/7
| 704.377
| 704.3770
|  
|  
|}
|}
Line 1,059: Line 1,187:
{| class="wikitable mw-collapsible mw-collapsed center-all left-4"
{| class="wikitable mw-collapsible mw-collapsed center-all left-4"
! Edo<br>generator
! Edo<br>generator
! Eigenmonzo<br>(unchanged-interval)*
! Unchanged interval<br>(eigenmonzo)*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
| 7\12
| '''[[12edo|7\12]]'''
|  
|  
| 700.0000
| '''700.0000'''
| Lower bound of 9- and 11-odd-limit diamond monotone
| '''Lower bound of 9- and 11-odd-limit diamond monotone'''
|-
|-
|  
|  
Line 1,075: Line 1,203:
|  
|  
| 11/7
| 11/7
| 701.094
| 701.0942
|  
|  
|-
|-
Line 1,090: Line 1,218:
|  
|  
| 13/7
| 13/7
| 701.489
| 701.4894
|  
|  
|-
|-
Line 1,098: Line 1,226:
|  
|  
|-
|-
| 38\65
| '''[[65edo|38\65]]'''
|  
|  
| 701.5385
| '''701.5385'''
| Lower bound of 13-, 15-odd-limit, <br />2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
| '''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
|  
|  
|-
|-
|  
|  
| 11/6
| 11/6
| 701.633
| 701.6335
|  
|  
|-
|-
|  
|  
| 11/9
| 11/9
| 701.644
| 701.6435
| 11-, 13-, and 15-odd-limit minimax
| 11-, 13-, and 15-odd-limit minimax
|-
|-
|  
|  
| 15/8
| 15/8
| 701.676
| 701.6759
| 1/7 schisma
| 1/7 schisma
|-
|-
Line 1,140: Line 1,268:
|  
|  
| 5/4
| 5/4
| 701.711
| 701.7108
| 1/8 schisma
| 1/8 schisma
|-
|-
Line 1,150: Line 1,278:
|  
|  
| 5/3
| 5/3
| 701.738
| 701.7379
| 5-odd-limit minimax, 1/9 schisma
| 5-odd-limit minimax, 1/9 schisma
|-
|-
|  
|  
| 9/5
| 9/5
| 701.760
| 701.7596
| 1/10 schisma
| 1/10 schisma
|-
|-
Line 1,165: Line 1,293:
|  
|  
| 13/8
| 13/8
| 701.802
| 701.8022
|  
|  
|-
|-
|  
|  
| 13/12
| 13/12
| 701.807
| 701.8067
|  
|  
|-
|-
|  
|  
| 13/9
| 13/9
| 701.811
| 701.8109
|  
|  
|-
|-
|  
|  
| 13/10
| 13/10
| 701.831
| 701.8314
|  
|  
|-
|-
|  
|  
| 15/13
| 15/13
| 701.836
| 701.8362
|  
|  
|-
|-
| 31\53
| '''[[53edo|31\53]]'''
|  
|  
| 701.8868
| '''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
| '''Upper bound of 11-, 13-, 15-odd-limit, <br>2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|-
|  
|  
Line 1,210: Line 1,338:
|  
|  
| 9/7
| 9/7
| 702.193
| 702.1928
| 9-odd-limit minimax, 1/16 septimal schisma
| 9-odd-limit minimax, 1/16 septimal schisma
|-
|-
|  
|  
| 7/6
| 7/6
| 702.209
| 702.2086
| 7-odd-limit minimax, 1/15 septimal schisma
| 7-odd-limit minimax, 1/15 septimal schisma
|-
|-
Line 1,225: Line 1,353:
|  
|  
| 7/4
| 7/4
| 702.227
| 702.2267
| 1/14 septimal schisma
| 1/14 septimal schisma
|-
|-
Line 1,248: Line 1,376:
|  
|  
|-
|-
| 24\41
| [[41edo|24\41]]
|  
|  
| 702.4390
| 702.4390
|  
| 41ef val
|-
|-
|  
|  
Line 1,265: Line 1,393:
|  
|  
| 15/14
| 15/14
| 702.778
| 702.7775
|  
|  
|-
|-
|  
|  
| 7/5
| 7/5
| 702.915
| 702.9146
|  
|  
|-
|-
Line 1,278: Line 1,406:
|  
|  
|-
|-
| 17\29
| '''[[29edo|17\29]]'''
|  
|  
| 703.4483
| '''703.4483'''
| Upper bound of 9-odd-limit diamond monotone
| '''29eeff val, upper bound of 9-odd-limit diamond monotone'''
|-
|-
|  
|  
| 13/11
| 13/11
| 703.597
| 703.5968
|  
|  
|}
|}
<nowiki>*</nowiki> Besides the octave
<nowiki/>* Besides the octave


[[Category:Garibaldi| ]] <!-- Main article -->
[[Category:Garibaldi| ]] <!-- Main article -->
[[Category:Temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Schismatic family]]
[[Category:Schismatic family]]
[[Category:Marvel temperaments]]
[[Category:Marvel temperaments]]
[[Category:Gariboh clan]]
[[Category:Gariboh clan]]
[[Category:Hemifamity temperaments]]
[[Category:Hemifamity temperaments]]

Latest revision as of 21:53, 30 May 2026

Garibaldi
Subgroups 2.3.5.7, 2.3.5.7.19
Comma basis 225/224, 3125/3087 (7-limit);
190/189, 225/224, 361/360 (2.3.5.7.19)
Reduced mapping ⟨1; 1 -8 -14 -3]
ET join 41 & 53
Generators (CWE) ~3/2 = 702.10 ¢
MOS scales 5L 2s, 5L 7s, 12L 5s, 12L 17s
Ploidacot monocot
Pergen (P8, P5)
Minimax error 9-odd-limit: 4.33 ¢;
2.3.5.7.19 21-odd-limit: 4.65 ¢
Target scale size 9-odd-limit: 17 notes;
2.3.5.7.19 21-odd-limit: 17 notes

Garibaldi is a 7-limit (and higher) temperament of the 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). 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 and hemifamity temperament. Tuning the fifth a fraction of a cent sharp gives the best tunings.

Immediate 11-limit extensions include cassandra (41 & 53), mapping 11/8 to +23 fifths, andromeda (29 & 41), mapping 11/8 to −18 fifths, and helenus (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

In the following table, odd harmonics 1–21 and their inverses are in bold.

# Cents* Approximate ratios
2.3.5.7.19 subgroup 13-limit extensions
Cassandra Andromeda Helenus
0 0.00 1/1
1 702.10 3/2
2 204.20 9/8
3 906.30 27/16, 32/19, 42/25 22/13 22/13 22/13
4 408.40 19/15, 24/19 14/11
5 1110.50 19/10, 36/19, 40/21 21/11
6 612.60 10/7
7 114.70 15/14, 16/15 14/13
8 816.80 8/5 21/13
9 318.90 6/5 40/33
10 1021.00 9/5, 38/21 20/11
11 523.09 19/14, 27/20 15/11
12 25.19 50/49, 57/56, 64/63, 81/80 40/39, 45/44
13 727.29 32/21 20/13
14 229.39 8/7 15/13
15 931.49 12/7 19/11
16 433.59 9/7 14/11
17 1135.69 27/14, 48/25 52/27 64/33 21/11
18 637.79 36/25, 81/56 13/9 16/11, 19/13
19 139.89 27/25 13/12 12/11 14/13
20 841.99 57/35, 80/49 13/8, 44/27 18/11, 64/39 21/13
21 344.09 60/49 11/9, 39/32 16/13, 27/22 40/33
22 1046.19 64/35 11/6 24/13 20/11
23 548.29 48/35 11/8, 26/19 18/13 15/11
24 50.39 36/35 33/32 27/26 40/39, 45/44
25 752.49 54/35 20/13
26 254.59 57/49, 81/70, 144/125 22/19 15/13
27 956.69 171/98, 216/125, 256/147 26/15 19/11
28 458.79 64/49 13/10
29 1160.89 96/49 39/20, 88/45 64/33
30 662.99 72/49 22/15 16/11, 19/13
31 165.08 54/49 11/10 12/11
32 867.18 81/49 33/20 18/11, 64/39
33 369.28 216/175 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
38 279.78 288/245
39 981.88 432/245
40 483.98 324/245
41 1186.08 486/245

* In 2.3.5.7.19-subgroup CWE tuning

As a detemperament of 12et

Garibaldi as a 41-tone 12et detempering
Garibaldi/cassandra as a 53-tone 12et detempering

Garibaldi is very naturally considered as a detemperament of the 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)

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 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.

Nomenclature of selected intervals
Ratio Example
Cassandra Andromeda Helenus
3/2 C–G (perfect fifth)
5/4 C–↓E (downmajor third)
7/4 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 C–Eb (minor third)

*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

Tunings

Norm-based tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 702.0589 ¢ CWE: ~3/2 = 702.0774 ¢ POTE: ~3/2 = 702.0852 ¢
13-limit norm-based tunings (cassandra)
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 702.1192 ¢ CWE: ~3/2 = 702.1135 ¢ POTE: ~3/2 = 702.1125 ¢

Target tunings

Target tunings (garibaldi)
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
7-odd-limit ~3/2 = 702.2086 ¢ 7/6 ~3/2 = 702.140 ¢ [0 -25 11 35⟩
9-odd-limit ~3/2 = 702.1928 ¢ 9/7 ~3/2 = 702.114 ¢ [0 -27 7 17⟩
Target tunings (cassandra)
Target Minimax Least squares
Generator Eigenmonzo* Generator Eigenmonzo*
11-odd-limit ~3/2 = 702.1928 ¢ 9/7 ~3/2 = 702.183 ¢ [0 17 -52 -88 134⟩
13-odd-limit ~3/2 = 702.1089 ¢ 13/7 ~3/2 = 702.128 ¢ [0 -38 -80 -122 137 116⟩
15-odd-limit ~3/2 = 702.1089 ¢ 13/7 ~3/2 = 702.112 ¢ [0 -95 -137 -129 167 143⟩
Target tunings (andromeda)
Target Minimax
Generator Eigenmonzo*
11-odd-limit ~3/2 = 702.6296 ¢ 11/9
13-odd-limit ~3/2 = 702.7558 ¢ 13/9
15-odd-limit ~3/2 = 702.7558 ¢ 13/9
Target tunings (helenus)
Target Minimax
Generator Eigenmonzo*
11-odd-limit ~3/2 = 701.6435 ¢ 11/9
13-odd-limit ~3/2 = 701.6435 ¢ 11/9
15-odd-limit ~3/2 = 701.6435 ¢ 11/9

Tuning spectra

Garibaldi

Edo
generator 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
7\12 700.0000 Lower bound of 9-odd-limit,
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
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
31\53 701.8868
3/2 701.9550 Pythagorean tuning
36/35 702.0321
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
24\41 702.4390
19/14 702.6079
21/19 702.6732
15/14 702.7775
7/5 702.9146
21/20 703.1066
17\29 703.4483 Upper bound of 9-odd-limit,
2.3.5.7.19 subgroup 19- and 21-odd-limit diamond monotone
13/11 703.5968

Cassandra

Edo
generator 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
7\12 700.0000 Lower bound of 9-odd-limit diamond monotone
19/16 700.8290 1/3 undevicesimal schisma
19/12 701.1105 1/4 undevicesimal schisma
38\65 701.5385 65def 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
19/13 701.8702
31\53 701.8868 Lower bound of 11-, 13-, 15-odd-limit,
2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
15/13 701.9355
13/10 701.9362
3/2 701.9550 Pythagorean tuning
13/8 702.0264
13/12 702.0301
36/35 702.0321
13/9 702.0343
19/11 702.0694
11/10 702.0969
15/11 702.1016
13/7 702.1089 13- and 15-odd-limit minimax
21/13 702.1135
55\94 702.1277
9/7 702.1928 9- and 11-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
11/7 702.2295
11/8 702.2312
21/11 702.2371
19/10 702.2399
11/6 702.2438
21/16 702.2476 1/13 septimal schisma
11/9 702.2575
64/63 702.2720 1/12 septimal schisma
19/15 702.3111
24\41 702.4390 Upper bound of 11-, 13-, 15-odd-limit,
2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
19/14 702.6079
21/19 702.6732
15/14 702.7775
7/5 702.9146
21/20 703.1066
17\29 703.4483 29ef val, upper bound of 9-odd-limit diamond monotone
13/11 703.5968

Andromeda

Edo
generator 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
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
38\65 701.5385 65deeff 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
31\53 701.8868 53ef val
3/2 701.9550 Pythagorean tuning
36/35 702.0321
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
21/16 702.2476 1/13 septimal schisma
64/63 702.2720 1/12 septimal schisma
19/15 702.3111
24\41 702.4390 Lower bound of 13-, 15-odd-limit,
2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
19/14 702.6079
11/9 702.6296 11-odd-limit minimax
11/6 702.6651
21/19 702.6732
11/8 702.7046
13/9 702.7558 13- and 15-odd-limit minimax
15/14 702.7775
13/12 702.7922
13/8 702.8320
7/5 702.9146
19/11 703.0797
21/20 703.1066
19/13 703.1659
15/11 703.3592
15/13 703.4101
17\29 703.4483 Upper bound of 9-, 11-, 13-, 15-odd-limit,
2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
11/10 703.4996
13/10 703.5220
13/11 703.5968
21/13 701.7817
19/10 702.2399
21/11 703.8926
13/7 704.0426
11/7 704.3770

Helenus

Edo
generator 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
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
13/7 701.4894
21/13 701.5127
38\65 701.5385 65d val, lower bound of 13-, 15-odd-limit,
2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
11/10 701.5907
15/11 701.6066
11/8 701.6227
11/6 701.6335
11/9 701.6435 11-, 13-, and 15-odd-limit minimax
15/8 701.6759 1/7 schisma
19/11 701.7109
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
13/8 701.8022
13/12 701.8067
13/9 701.8109
13/10 701.8314
15/13 701.8362
31\53 701.8868 Upper bound of 11-, 13-, 15-odd-limit,
2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
19/13 701.8995
3/2 701.9550 Pythagorean tuning
36/35 702.0321
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
24\41 702.4390 41ef val
19/14 702.6079
21/19 702.6732
15/14 702.7775
7/5 702.9146
21/20 703.1066
17\29 703.4483 29eeff val, upper bound of 9-odd-limit diamond monotone
13/11 703.5968

* Besides the octave

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