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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)|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 Pythagorean 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 Pythagorean minor seventh). This makes garibaldi a [[marvel temperaments|marvel temperament]] and a [[hemifamity temperaments|hemifamity temperament]].  
'''Schismic''' (or '''helmholtz''') is a 5-limit temperament which takes a roughly justly tuned [[3/2|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 comma|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).


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''.  
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 temperaments|marvel temperament]] and a [[hemifamity temperaments|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'' ({{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.
Alternatively to garibaldi, there is another, extremely complex extension to schismic called ''schism'', 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 ==
== Interval chain ==
Line 14: Line 20:
! rowspan="3" | #
! rowspan="3" | #
! rowspan="3" | Cents*
! rowspan="3" | Cents*
! colspan="4" | Approximate ratios
! colspan="5" |Approximate ratios
|-
|-
! rowspan="2" | 2.3.5.7.19 subgroup
! rowspan="2" |Schismic (2.3.5 subgroup)
! rowspan="2" | Garibaldi (2.3.5.7.19 subgroup)
! colspan="3" | 13-limit extension
! colspan="3" | 13-limit extension
|-
|-
Line 25: Line 32:
| 0
| 0
| 0.00
| 0.00
| '''1/1'''
|'''1/1'''
|
|
|
|
|
Line 32: Line 40:
| 1
| 1
| 702.10
| 702.10
| '''3/2'''
|'''3/2'''
|
|
|
|
|
Line 39: Line 48:
| 2
| 2
| 204.20
| 204.20
| '''9/8'''
|'''9/8'''
|
|
|
|
|
Line 46: Line 56:
| 3
| 3
| 906.30
| 906.30
| 27/16, '''32/19''', 42/25
|27/16
| '''32/19''', 42/25
| 22/13
| 22/13
| 22/13
| 22/13
Line 53: Line 64:
| 4
| 4
| 408.40
| 408.40
|81/64
| 19/15, 24/19
| 19/15, 24/19
|
|
Line 60: Line 72:
| 5
| 5
| 1110.50
| 1110.50
|243/128, 256/135
| 19/10, 36/19, 40/21
| 19/10, 36/19, 40/21
|
|
Line 67: Line 80:
| 6
| 6
| 612.60
| 612.60
|64/45
| 10/7
| 10/7
|
|
Line 74: Line 88:
| 7
| 7
| 114.70
| 114.70
| 15/14, '''16/15'''
|'''16/15'''
| 15/14
|
|
| 14/13
| 14/13
Line 81: Line 96:
| 8
| 8
| 816.80
| 816.80
| '''8/5'''
|'''8/5'''
|
|
|
| 21/13
| 21/13
Line 88: Line 104:
| 9
| 9
| 318.90
| 318.90
| 6/5
|6/5
|
|
|
| 40/33
| 40/33
Line 95: Line 112:
| 10
| 10
| 1021.00
| 1021.00
| 9/5, 38/21
|9/5
| 38/21
|
|
| 20/11
| 20/11
Line 102: Line 120:
| 11
| 11
| 523.09
| 523.09
| 19/14, 27/20
|27/20
| 19/14
|
|
| 15/11
| 15/11
Line 109: Line 128:
| 12
| 12
| 25.19
| 25.19
| 50/49, 57/56, 64/63, 81/80
|81/80
| 50/49, 57/56, 64/63
|
|
| 40/39, 45/44
| 40/39, 45/44
Line 116: Line 136:
| 13
| 13
| 727.29
| 727.29
|
| '''32/21'''
| '''32/21'''
|
|
Line 123: Line 144:
| 14
| 14
| 229.39
| 229.39
|
| '''8/7'''
| '''8/7'''
|
|
Line 130: Line 152:
| 15
| 15
| 931.49
| 931.49
|
| 12/7
| 12/7
|
|
Line 137: Line 160:
| 16
| 16
| 433.59
| 433.59
|
| 9/7
| 9/7
|
|
Line 144: Line 168:
| 17
| 17
| 1135.69
| 1135.69
| 27/14, 48/25
|48/25
| 27/14
| 52/27
| 52/27
| 64/33
| 64/33
Line 151: Line 176:
| 18
| 18
| 637.79
| 637.79
| 36/25, 81/56
|36/25
| 81/56
| 13/9
| 13/9
| '''16/11''', 19/13
| '''16/11''', 19/13
Line 158: Line 184:
| 19
| 19
| 139.89
| 139.89
| 27/25
|27/25
|
| 13/12
| 13/12
| 12/11
| 12/11
Line 165: Line 192:
| 20
| 20
| 841.99
| 841.99
|
| 57/35, 80/49
| 57/35, 80/49
| '''13/8''', 44/27
| '''13/8''', 44/27
Line 172: Line 200:
| 21
| 21
| 344.09
| 344.09
|
| 60/49
| 60/49
| 11/9, 39/32
| 11/9, 39/32
Line 179: Line 208:
| 22
| 22
| 1046.19
| 1046.19
|
| 64/35
| 64/35
| 11/6
| 11/6
Line 186: Line 216:
| 23
| 23
| 548.29
| 548.29
|
| 48/35
| 48/35
| '''11/8''', 26/19
| '''11/8''', 26/19
Line 193: Line 224:
| 24
| 24
| 50.39
| 50.39
|
| 36/35
| 36/35
| 33/32
| 33/32
Line 200: Line 232:
| 25
| 25
| 752.49
| 752.49
|
| 54/35
| 54/35
|
|
Line 207: Line 240:
| 26
| 26
| 254.59
| 254.59
| 57/49, 81/70, 144/125
|144/125
| 57/49, 81/70
| 22/19
| 22/19
|
|
Line 214: Line 248:
| 27
| 27
| 956.69
| 956.69
| 171/98, 216/125, 256/147
|216/125
| 171/98, 256/147
| 26/15
| 26/15
|
|
Line 221: Line 256:
| 28
| 28
| 458.79
| 458.79
|
| 64/49
| 64/49
| 13/10
| 13/10
Line 228: Line 264:
| 29
| 29
| 1160.89
| 1160.89
|
| 96/49
| 96/49
| 39/20, 88/45
| 39/20, 88/45
Line 235: Line 272:
| 30
| 30
| 662.99
| 662.99
|
| 72/49
| 72/49
| 22/15
| 22/15
Line 242: Line 280:
| 31
| 31
| 165.08
| 165.08
|
| 54/49
| 54/49
| 11/10
| 11/10
Line 249: Line 288:
| 32
| 32
| 867.18
| 867.18
|
| 81/49
| 81/49
| 33/20
| 33/20
Line 256: Line 296:
| 33
| 33
| 369.28
| 369.28
|
| 216/175
| 216/175
| 26/21
| 26/21
Line 263: Line 304:
| 34
| 34
| 1071.38
| 1071.38
|
| 324/175
| 324/175
| 13/7
| 13/7
Line 270: Line 312:
| 35
| 35
| 573.48
| 573.48
|
| 243/175
| 243/175
|  
|  
Line 277: Line 320:
| 36
| 36
| 75.58
| 75.58
|
| 256/245
| 256/245
| 22/21
| 22/21
Line 284: Line 328:
| 37
| 37
| 777.68
| 777.68
|
| 384/245
| 384/245
| 11/7
| 11/7
Line 291: Line 336:
| 38
| 38
| 279.78
| 279.78
|
| 288/245
| 288/245
|  
|  
Line 298: Line 344:
| 39
| 39
| 981.88
| 981.88
|
| 432/245
| 432/245
|  
|  
Line 305: Line 352:
| 40
| 40
| 483.98
| 483.98
|
| 324/245
| 324/245
|  
|  
Line 312: Line 360:
| 41
| 41
| 1186.08
| 1186.08
|
| 486/245
| 486/245
|  
|  
Line 323: Line 372:
[[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.  
Schismic (and thus 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.  


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 tendoaneutral 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.  
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.  


{| class="wikitable center-all mw-collapsible mw-collapsed"
{| class="wikitable center-all mw-collapsible mw-collapsed"
Line 451: Line 500:


== 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. Due to the generalized comma of garibaldi, 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.  
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.  


{| class="wikitable center-1 center-3"
{| class="wikitable center-1 center-3"

Revision as of 07:44, 19 June 2025

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).

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 schism, 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

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

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