Garibaldi

Schismic (or helmholtz) is a 5-limit temperament which takes a roughly justly tuned 4/3 and stacks it eight times to reach 5/4, thus finding the 5th harmonic at the diminished fourth (e.g. C–F♭). This can be respelled as a major third flattened by one Pythagorean comma, and thus, the Pythagorean and syntonic commas are equated into a generalized "comma", and the octave can be split into two diatonic major thirds and one downmajor third representing 5/4. It is one of the most basic examples of a microtemperament, as the 4/3 generator can be detuned by a fraction of a cent from just, or left untouched entirely (as the difference between 8192/6561 and 5/4, the schisma being tempered out, is approximately 2 cents, which is unnoticeable to most people). Technically, the best tuning in the 5-limit is to flatten the fifth by a fraction of a cent, though tunings on both sides of the just interval work fine.

To reach intervals of 7, a reasonable choice is to further equate the Pythagorean-syntonic comma with the archytas comma of 64/63 (as in hemifamity), reaching the primary 7-limit extension called garibaldi. Like with hemifamity, the best tunings involve sharpening the fifth, but in this case only slightly, as the size of the comma is determined by the fifth itself. Thus, tuning the fifth a fraction of a cent sharp gives the best tunings. The new mapping specific to garibaldi is that 7/4 is mapped to the double-diminished octave (e.g. C–C𝄫; a comma-flat Pythagorean minor seventh). This makes garibaldi a marvel temperament and a hemifamity temperament. 41edo and 53edo make for good tunings.

It is useful to introduce a second kind of accidental to notate garibaldi, representing the comma interval, so that 5/4 does not have to be spelled as a fourth (and 7/4 does not have to be spelled as an octave).

Immediate 11-limit extensions to garibaldi 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.

Alternatively to garibaldi, there is another, extremely complex extension to schismic called pontiac, which finds 7/4 at +39 fifths, and is supported alongside garibaldi by 53edo.

See Schismatic family #Garibaldi and Schismatic family #Schismic for technical data.

Interval chain

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

#	Cents*	Approximate ratios
		Schismic (2.3.5 subgroup)	Garibaldi (2.3.5.7.19 subgroup)	13-limit extension
				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	81/64	19/15, 24/19		14/11
5	1110.50	243/128, 256/135	19/10, 36/19, 40/21		21/11
6	612.60	64/45	10/7
7	114.70	16/15	15/14		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	27/20	19/14		15/11
12	25.19	81/80	50/49, 57/56, 64/63		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	48/25	27/14	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	144/125	57/49, 81/70	22/19		15/13
27	956.69	216/125	171/98, 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

Schismic (and thus garibaldi) is very naturally considered as a detemperament of the 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 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.

Notice also the little comma between artoneutral and tendoaneutral. 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.

#	Interval category	"Double-Sub"			"Sub"			"Plain"			"Super"			"Double-super"
		Gen.	Cents*	Ratios	Gen.	Cents*	Ratios	Gen.	Cents*	Ratios	Gen.	Cents*	Ratios	Gen.	Cents*	Ratios
0	P1							0	0.0	1/1	12	25.2	64/63~81/80	24	50.5	33/32~36/35
1	m2				−17	64.2	27/26~28/27	−5	89.5	20/19~21/20	7	114.7	15/14~16/15	19	140.0	13/12
2	M2	−22	153.7	12/11	−10	178.9	10/9	2	204.2	9/8	14	229.5	8/7	26	254.7	22/19
3	m3				−15	268.4	7/6	−3	293.6	13/11~19/16	9	318.9	6/5	21	344.2	11/9
4	M3	−20	357.9	16/13	−8	383.2	5/4	4	408.4	19/15~24/19	16	433.7	9/7
5	P4	−25	447.4	35/27	−13	472.6	21/16	−1	497.9	4/3	11	523.2	19/14	23	548.4	11/8
6	A4, d5	−18	562.1	18/13	−6	587.4	7/5	6	612.6	10/7	18	637.9	13/9
7	P5	−23	651.6	16/11	−11	676.8	28/19	1	702.1	3/2	13	727.4	32/21	25	752.6	54/35
8	m6				−16	766.3	14/9	−4	791.6	19/12~30/19	8	816.8	8/5	20	842.1	13/8
9	M6	−21	855.8	18/11	−9	881.1	5/3	3	906.3	22/13~27/16	15	931.6	12/7
10	m7	−26	945.3	19/11	−14	970.5	7/4	−2	995.8	16/9	10	1021.1	9/5	22	1046.3	11/6
11	M7	−19	1060.0	24/13	−7	1085.3	15/8~28/15	5	1110.5	19/10~40/21	17	1135.8	27/14~52/27
12	P8	−24	1149.5	35/18~64/33	−12	1174.7	63/32~65/33	0	1200.0	2/1

See the diagrams on the right for isomorphic versions.

Notation

Using schismic 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. Due to the generalized comma of schismic, a natural choice is 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.

Cassandra nomenclature for
selected intervals

Ratio	Nominal	Example
3/2	Perfect fifth	C–G
5/4	Downmajor third	C–vE
7/4	Downminor seventh	C–vBb
11/8	Double-up fourth	C–^^F
13/8	Double-up minor sixth	C–^^Ab
19/16	Minor third	C–Eb

Andromeda nomenclature for selected intervals

Ratio	Nominal	Example
11/8	Down-diminished fifth Double-down augmented fourth	C–vGb C–vvF#
13/8	Double downmajor sixth	C–vvA

Helenus nomenclature for selected intervals

Ratio	Nominal	Example
11/8	Double-down diminished fifth Triple-down augmented fourth	C–vvGb C–v3F#
13/8	Triple-down major sixth	C–v3A

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–vE–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–vEb-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

Tuning spectra

Garibaldi

Edo generator	Eigenmonzo (unchanged-interval)*	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
	15/8	701.676	1/7 schisma
	5/4	701.711	1/8 schisma
	25/24	701.7252	2/17 schisma
	5/3	701.738	5-odd-limit minimax, 1/9 schisma
	9/5	701.760	1/10 schisma
	81/80	701.7922	1/12 schisma
31\53		701.8868
	3/2	701.9550	Pythagorean tuning
	36/35	702.0321
	9/7	702.193	9-odd-limit minimax, 1/16 septimal schisma
	7/6	702.209	7-odd-limit minimax, 1/15 septimal schisma
	49/48	702.2174	2/29 septimal schisma
	7/4	702.227	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	702.778
	7/5	702.915
	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	703.597

Cassandra

Edo generator	Eigenmonzo (unchanged-interval)*	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
	15/8	701.676	1/7 schisma
	5/4	701.711	1/8 schisma
	25/24	701.7252	2/17 schisma
	[0 -10 17⟩	701.728	5-odd-limit least squares
	5/3	701.738	5-odd-limit minimax, 1/9 schisma
	9/5	701.760	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	702.026
	13/12	702.030
	36/35	702.0321
	13/9	702.034
	19/11	702.0694
	11/10	702.097
	15/11	702.102
	13/7	702.109	13- and 15-odd-limit minimax
	[0 -95 -137 -129 167 143⟩	702.112	15-odd-limit least squares
	21/13	702.1135
	[0 -27 7 17⟩	702.114	9-odd-limit least squares
	[0 -38 -80 -122 137 116⟩	702.128	13-odd-limit least squares
	[0 -25 11 35⟩	702.140	7-odd-limit least squares
	[0 17 -52 -88 134⟩	702.183	11-odd-limit least squares
	9/7	702.193	9- and 11-odd-limit minimax, 1/16 septimal schisma
	7/6	702.209	7-odd-limit minimax, 1/15 septimal schisma
	49/48	702.2174	2/29 septimal schisma
	7/4	702.227	1/14 septimal schisma
	11/7	702.230
	11/8	702.231
	21/11	702.2371
	19/10	702.2399
	11/6	702.244
	21/16	702.2476	1/13 septimal schisma
	11/9	702.258
	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	702.778
	7/5	702.915
	21/20	703.1066
17\29		703.4483	Upper bound of 9-odd-limit diamond monotone
	13/11	703.597

Andromeda

Edo generator	Eigenmonzo (unchanged-interval)*	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
	15/8	701.676	1/7 schisma
	5/4	701.711	1/8 schisma
	25/24	701.7252	2/17 schisma
	5/3	701.738	5-odd-limit minimax, 1/9 schisma
	9/5	701.760	1/10 schisma
	81/80	701.7922	1/12 schisma
31\53		701.8868
	3/2	701.9550	Pythagorean tuning
	36/35	702.0321
	9/7	702.193	9-odd-limit minimax, 1/16 septimal schisma
	7/6	702.209	7-odd-limit minimax, 1/15 septimal schisma
	49/48	702.2174	2/29 septimal schisma
	7/4	702.227	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	702.630	11-odd-limit minimax
	11/6	702.665
	21/19	702.6732
	11/8	702.705
	13/9	702.756	13- and 15-odd-limit minimax
	15/14	702.778
	13/12	702.792
	13/8	702.832
	7/5	702.915
	19/11	703.0797
	21/20	703.1066
	19/13	703.1659
	15/11	703.359
	15/13	703.410
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	703.500
	13/10	703.522
	13/11	703.597
	21/13	701.7817
	19/10	702.2399
	21/11	703.8926
	13/7	704.043
	11/7	704.377

Helenus

Edo generator	Eigenmonzo (unchanged-interval)*	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	701.094
	19/12	701.1105	1/4 undevicesimal schisma
	21/11	701.1149
	13/7	701.489
	21/13	701.5127
38\65		701.5385	Lower bound of 13-, 15-odd-limit, 2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone
	11/10	701.591
	15/11	701.607
	11/8	701.623
	11/6	701.633
	11/9	701.644	11-, 13-, and 15-odd-limit minimax
	15/8	701.676	1/7 schisma
	19/11	701.7109
	5/4	701.711	1/8 schisma
	25/24	701.7252	2/17 schisma
	5/3	701.738	5-odd-limit minimax, 1/9 schisma
	9/5	701.760	1/10 schisma
	81/80	701.7922	1/12 schisma
	13/8	701.802
	13/12	701.807
	13/9	701.811
	13/10	701.831
	15/13	701.836
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	702.193	9-odd-limit minimax, 1/16 septimal schisma
	7/6	702.209	7-odd-limit minimax, 1/15 septimal schisma
	49/48	702.2174	2/29 septimal schisma
	7/4	702.227	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	702.778
	7/5	702.915
	21/20	703.1066
17\29		703.4483	Upper bound of 9-odd-limit diamond monotone
	13/11	703.597

* Besides the octave