Garibaldi: Difference between revisions

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m →As a detemperament of 12et: I don't know about linking "MOS" as MOSDiagrams. Also generally lowercase is preferred.
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New selected interval table to go with Cassaschisimsicd article
 
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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.
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]].
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 ==
Line 438: 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 ==

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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