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| Title = Garibaldi
| Title = Garibaldi
| Subgroups = 2.3.5.7, 2.3.5.7.19
| 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]] (L7.19)
| Comma basis = [[225/224]], [[3125/3087]] (7-limit); <br>[[190/189]], [[225/224]], [[361/360]] (2.3.5.7.19)
| Generator = 3/2
| Mapping = 1; 1 -8 -14 -3
| Mapping = 1; 1 -8 -14 -3
| Pergen = (P8, P5)
| Edo join 1 = 41 | Edo join 2 = 53
| Edo join 1 = 41 | Edo join 2 = 53
| Generators = 3/2
| Generators tuning = 702.10
| Optimization method = CWE
| Optimization method = CWE
| Generator tuning = 702.10
| Pergen = (P8, P5)
| MOS scales = [[5L&nbsp;2s]], [[5L&nbsp;7s]], [[12L&nbsp;5s]], [[12L 17s]]
| 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 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
| 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 singular generalized "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 temperament]] and a [[hemifamity temperaments|hemifamity temperament]]. Tuning the fifth a fraction of a cent sharp gives the best tunings.  
'''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 ({{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''.
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.
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[[File:Garibaldi-cassandra 12et Detempering.png|thumb|Garibaldi/cassandra as a 53-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]]. The table below shows a 53-tone detempered scale, with a generator range of -26 to +26. Each interval category of the 12 equal temperament is further divided into "double-sub", "sub", "plain", "super" and "double-super" qualities, separated by an [[diesis (scale theory)|enharmonic diesis]], which represents the syntonic~septimal comma; the "plain" type here consists of a [[5L 7s]] scale in 6|5 mode. Combining this division with the minor and major qualities of the 12 equal temperament, and calling the "double-sub major" and "double-super minor" qualities ''artoneutral'' and ''tendoneutral'', respectively, garibaldi gives us at least ''eight'' qualities for each diatonic category: subminor, minor, supraminor, artoneutral, tendoneutral, submajor, major, and supermajor.
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:


Notice also the little comma between artoneutral and tendoneutral. This interval spans 41 generator steps. 41edo tempers it out so that it merges artoneutral and tendoneutral into one neutral interval whereas 53edo exaggerates it to the size of the syntonic~septimal comma. 94edo tunes it to one half the size of the syntonic~septimal comma, which can be seen as a good compromise.
In 12tet:


{| class="wikitable center-all mw-collapsible mw-collapsed"
* 7/6~19/16~6/5 (minor)
|-
* 5/4~19/15~9/7 (major)
! rowspan="2" | #
 
! rowspan="2" | Interval<br>category
In garibaldi (cassandra)
! colspan="3" style="border-left: double;" | "Double-Sub"
 
! colspan="3" style="border-left: double;" | "Sub"
* ~[[7/6]] (subminor)
! colspan="3" style="border-left: double;" | "Plain"
* '''~[[19/16]] (minor)'''
! colspan="3" style="border-left: double;" | "Super"
* ~[[6/5]] (superminor)
! colspan="3" style="border-left: double;" | "Double-super"
* ~[[11/9]] (artoneutral)
|-
* ~[[27/22]] (tendoneutral)
! style="border-left: double;" | Gen. || Cents* || Ratios
* ~[[5/4]] (submajor)
! style="border-left: double;" | Gen. || Cents* || Ratios
* '''~[[19/15]] (major)'''
! style="border-left: double;" | Gen. || Cents* || Ratios
* ~[[9/7]] (supermajor)
! style="border-left: double;" | Gen. || Cents* || Ratios
 
! style="border-left: double;" | Gen. || Cents* || Ratios
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.
|-
| 0
| P1
| style="border-left: double;" |  ||  ||
| style="border-left: double;" |  ||  ||
| style="border-left: double;" | 0 || 0.0 || 1/1
| style="border-left: double;" | 12 || 25.2 || 64/63~81/80
| style="border-left: double;" | 24 || 50.5 || 33/32~36/35
|-
| 1
| m2
| style="border-left: double;" |  ||  ||
| style="border-left: double;" | −17 || 64.2 || 27/26~28/27
| style="border-left: double;" | −5 || 89.5 || 20/19~21/20
| style="border-left: double;" | 7 || 114.7 || 15/14~16/15
| style="border-left: double;" | 19 || 140.0 || 13/12
|-
| 2
| M2
| style="border-left: double;" | −22 || 153.7 || 12/11
| style="border-left: double;" | −10 || 178.9 || 10/9
| style="border-left: double;" | 2 || 204.2 || 9/8
| style="border-left: double;" | 14 || 229.5 || 8/7
| style="border-left: double;" | 26 || 254.7 || 22/19
|-
| 3
| m3
| style="border-left: double;" |  ||  ||
| style="border-left: double;" | −15 || 268.4 || 7/6
| style="border-left: double;" | −3 || 293.6 || 13/11~19/16
| style="border-left: double;" | 9 || 318.9 || 6/5
| style="border-left: double;" | 21 || 344.2 || 11/9
|-
| 4
| M3
| style="border-left: double;" | −20 || 357.9 || 16/13
| style="border-left: double;" | −8 || 383.2 || 5/4
| style="border-left: double;" | 4 || 408.4 || 19/15~24/19
| style="border-left: double;" | 16 || 433.7 || 9/7
| style="border-left: double;" |  ||  ||
|-
| 5
| P4
| style="border-left: double;" | −25 || 447.4 || 35/27
| style="border-left: double;" | −13 || 472.6 || 21/16
| style="border-left: double;" | −1 || 497.9 || 4/3
| style="border-left: double;" | 11 || 523.2 || 19/14
| style="border-left: double;" | 23 || 548.4 || 11/8
|-
| 6
| A4, d5
| style="border-left: double;" | −18 || 562.1 || 18/13
| style="border-left: double;" | −6 || 587.4 || 7/5
| style="border-left: double;" | 6 || 612.6 || 10/7
| style="border-left: double;" | 18 || 637.9 || 13/9
| style="border-left: double;" |  ||  ||
|-
| 7
| P5
| style="border-left: double;" | −23 || 651.6 || 16/11
| style="border-left: double;" | −11 || 676.8 || 28/19
| style="border-left: double;" | 1 || 702.1 || 3/2
| style="border-left: double;" | 13 || 727.4 || 32/21
| style="border-left: double;" | 25 || 752.6 || 54/35
|-
| 8
| m6
| style="border-left: double;" |  ||  ||
| style="border-left: double;" | −16 || 766.3 || 14/9
| style="border-left: double;" | −4 || 791.6 || 19/12~30/19
| style="border-left: double;" | 8 || 816.8 || 8/5
| style="border-left: double;" | 20 || 842.1 || 13/8
|-
| 9
| M6
| style="border-left: double;" | −21 || 855.8 || 18/11
| style="border-left: double;" | −9 || 881.1 || 5/3
| style="border-left: double;" | 3 || 906.3 || 22/13~27/16
| style="border-left: double;" | 15 || 931.6 || 12/7
| style="border-left: double;" |  ||  ||
|-
| 10
| m7
| style="border-left: double;" | −26 || 945.3 || 19/11
| style="border-left: double;" | −14 || 970.5 || 7/4
| style="border-left: double;" | −2 || 995.8 || 16/9
| style="border-left: double;" | 10 || 1021.1 || 9/5
| style="border-left: double;" | 22 || 1046.3 || 11/6
|-
| 11
| M7
| style="border-left: double;" | −19 || 1060.0 || 24/13
| style="border-left: double;" | −7 || 1085.3 || 15/8~28/15
| style="border-left: double;" | 5 || 1110.5 || 19/10~40/21
| style="border-left: double;" | 17 || 1135.8 || 27/14~52/27
| style="border-left: double;" |  ||  ||
|-
| 12
| P8
| style="border-left: double;" | −24 || 1149.5 || 35/18~64/33
| style="border-left: double;" | −12 || 1174.7 || 63/32~65/33
| style="border-left: double;" | 0 || 1200.0 || 2/1
| style="border-left: double;" |  ||  ||
| style="border-left: double;" |  ||  ||
|}


See the diagrams on the right for isomorphic versions.
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 ==
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.  
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 tables show how to notate 11- and 13-limit intervals in each extension of garibaldi.  
The following table shows how to notate 2.3.5.7.11.13.19 intervals in each extension of garibaldi.  


{| class="wikitable center-1 center-3"
{| class="wikitable" style="text-align:center; vertical-align:middle;"
|+ style="font-size: 105%; white-space: nowrap;" | Cassandra nomenclature for<br>selected intervals
|+Nomenclature of selected intervals
|-
|- style="font-weight:bold;"
! Ratio
! rowspan="2" | Ratio
! Nominal
! colspan="3" | Example
! 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"
|+ 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"
<nowiki/>*Can also be spelt ↓Gb and ↓↓Gb respectively, since F# = ↑Gb.
|+ 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 ==
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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 ==
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