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'''Garibaldi temperament''' is a [[7-limit]] (and higher) 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 double diminished octave (e.g. C-Cbb). 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). It tempers together the [[Pythagorean comma|Pythagorean]], [[81/80|syntonic]], and [[64/63|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, [[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 a [[septischismic clan|septischismic]] one, where [[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]], [[gariboh clan|gariboh]], [[octagar temperaments|octagar]] and [[aberschismic temperaments|aberschismic]] 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'' (29 &amp; 41), mapping 11/8 to -18 fifths, and ''helenus'' (53 &amp; 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-Eb).  
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]] ('''not''' Giuseppe 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 ==
In the following table, odd harmonics 1–21 are in '''bold'''.  
In the following table, odd harmonics 1–21 and their inverses are in '''bold'''.  


{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
! rowspan="3" | #
! rowspan="3" | #
! rowspan="3" | Cents*
! rowspan="3" | Cents*
! colspan="4" | Approximate Ratios
! colspan="4" | Approximate ratios
|-
|-
! 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 20: Line 39:
| 0
| 0
| 0.00
| 0.00
| 1/1
| '''1/1'''
|
|
|
|
Line 26: Line 45:
|-
|-
| 1
| 1
| 702.06
| 702.10
| '''3/2'''
| '''3/2'''
|
|
Line 33: Line 52:
|-
|-
| 2
| 2
| 204.12
| 204.20
| '''9/8'''
| '''9/8'''
|
|
Line 40: 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 47: Line 66:
|-
|-
| 4
| 4
| 408.24
| 408.40
| 19/15, 24/19, 63/50, 80/63
| 19/15, 24/19
|
|
| 14/11
| 14/11
Line 54: Line 73:
|-
|-
| 5
| 5
| 1110.29
| 1110.50
| 19/10, 36/19, 40/21
| 19/10, 36/19, 40/21
|
|
Line 61: Line 80:
|-
|-
| 6
| 6
| 612.35
| 612.60
| 10/7
| 10/7
|
|
Line 68: Line 87:
|-
|-
| 7
| 7
| 114.41
| 114.70
| 15/14, '''16/15'''
| 15/14, '''16/15'''
|
|
Line 75: Line 94:
|-
|-
| 8
| 8
| 816.47
| 816.80
| '''8/5'''
| '''8/5'''
|
|
Line 82: Line 101:
|-
|-
| 9
| 9
| 318.53
| 318.90
| 6/5
| 6/5
|
|
Line 89: Line 108:
|-
|-
| 10
| 10
| 1020.59
| 1021.00
| 9/5, 38/21
| 9/5, 38/21
|
|
Line 96: Line 115:
|-
|-
| 11
| 11
| 522.65
| 523.09
| 19/14, 27/20
| 19/14, 27/20
|
|
Line 103: 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 110: Line 129:
|-
|-
| 13
| 13
| 726.77
| 727.29
| '''32/21'''
| '''32/21'''
|
|
Line 117: Line 136:
|-
|-
| 14
| 14
| 228.82
| 229.39
| '''8/7'''
| '''8/7'''
|
|
Line 124: Line 143:
|-
|-
| 15
| 15
| 930.88
| 931.49
| 12/7
| 12/7
|
|
Line 131: Line 150:
|-
|-
| 16
| 16
| 432.94
| 433.59
| 9/7
| 9/7
|
|
Line 138: Line 157:
|-
|-
| 17
| 17
| 1135.00
| 1135.69
| 27/14, 48/25
| 27/14, 48/25
| 52/27
| 52/27
Line 145: Line 164:
|-
|-
| 18
| 18
| 637.06
| 637.79
| 36/25, 81/56
| 36/25, 81/56
| 13/9
| 13/9
Line 152: Line 171:
|-
|-
| 19
| 19
| 139.12
| 139.89
| 27/25
| 27/25
| 13/12
| 13/12
Line 159: Line 178:
|-
|-
| 20
| 20
| 841.18
| 841.99
| 80/49, 81/50
| 57/35, 80/49
| '''13/8''', 44/27
| '''13/8''', 44/27
| 18/11, 64/39
| 18/11, 64/39
Line 166: Line 185:
|-
|-
| 21
| 21
| 343.24
| 344.09
| 60/49
| 60/49
| 11/9, 39/32
| 11/9, 39/32
Line 173: Line 192:
|-
|-
| 22
| 22
| 1045.30
| 1046.19
| 64/35
| 64/35
| 11/6
| 11/6
Line 180: Line 199:
|-
|-
| 23
| 23
| 547.35
| 548.29
| 48/35
| 48/35
| '''11/8''', 26/19
| '''11/8''', 26/19
Line 187: Line 206:
|-
|-
| 24
| 24
| 49.41
| 50.39
| 36/35
| 36/35
| 33/32
| 33/32
Line 194: Line 213:
|-
|-
| 25
| 25
| 751.47
| 752.49
| 54/35
| 54/35
|
|
Line 201: Line 220:
|-
|-
| 26
| 26
| 253.53
| 254.59
| 81/70, 144/125
| 57/49, 81/70, 144/125
| 19/11
| 22/19
|
|
| 15/13
| 15/13
|-
|-
| 27
| 27
| 955.59
| 956.69
| 216/125, 256/147
| 171/98, 216/125, 256/147
| 26/15
| 26/15
|
|
Line 215: Line 234:
|-
|-
| 28
| 28
| 457.65
| 458.79
| 64/49
| 64/49
| 13/10
| 13/10
Line 222: Line 241:
|-
|-
| 29
| 29
| 1159.71
| 1160.89
| 96/49
| 96/49
| 39/20, 88/45
| 39/20, 88/45
Line 229: Line 248:
|-
|-
| 30
| 30
| 661.77
| 662.99
| 72/49
| 72/49
| 22/15
| 22/15
Line 236: Line 255:
|-
|-
| 31
| 31
| 163.83
| 165.08
| 54/49
| 54/49
| 11/10
| 11/10
Line 243: Line 262:
|-
|-
| 32
| 32
| 865.88
| 867.18
| 81/49
| 81/49
| 33/20
| 33/20
Line 250: Line 269:
|-
|-
| 33
| 33
| 367.94
| 369.28
| 216/175
| 216/175
| 26/21
| 26/21
Line 257: Line 276:
|-
|-
| 34
| 34
| 1070.00
| 1071.38
| 324/175
| 324/175
| 13/7
| 13/7
Line 264: Line 283:
|-
|-
| 35
| 35
| 572.06
| 573.48
| 243/175
| 243/175
| 39/28
|  
|  
|  
| 18/13
| 18/13
|-
|-
| 36
| 36
| 74.12
| 75.58
| 256/245
| 256/245
|  
| 22/21
|  
|  
| 27/26
| 27/26
|-
|-
| 37
| 37
| 776.18
| 777.68
| 384/245
| 384/245
|  
| 11/7
|  
|  
|  
|  
|-
|-
| 38
| 38
| 278.24
| 279.78
| 288/245
| 288/245
|  
|  
Line 292: Line 311:
|-
|-
| 39
| 39
| 980.30
| 981.88
| 432/245
| 432/245
|  
|  
Line 299: Line 318:
|-
|-
| 40
| 40
| 482.36
| 483.98
| 324/245
| 324/245
|  
|  
Line 306: Line 325:
|-
|-
| 41
| 41
| 1184.41
| 1186.08
| 486/245
| 486/245
|  
|  
Line 312: Line 331:
|  
|  
|}
|}
<nowiki>*</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:
* Minor: 7/6~19/16~6/5
* Major: 5/4~19/15~9/7
 
In garibaldi/cassandra:
* Subminor: [[7/6]]
* Minor: [[13/11]]~[[19/16]]
* Superminor: [[6/5]]
* Hyperminor{{idio}}: [[11/9]]~[[39/32]]
* Hypomajor{{idio}}: [[16/13]]~[[27/22]]
* Submajor: [[5/4]]
* Major: [[19/15]]~[[24/19]]
* Supermajor: [[9/7]]
 
The choice of nomenclature is due to the fact that cassandra is a relabeling of the 3-limit – it does not split primes in half, so it has no truly neutral intervals. Notice also the little interval between hyperminor and hypomajor, ~[[243/242]]. This interval spans 41 generator steps. 41edo tempers it out so that it merges hyperminor and hypominor into a true [[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.


== Notation ==
== Notation ==
Using garibaldi can be a challenge because it defies the tradition of tertian harmony in [[circle-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.
 
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 center-1 center-3"
|+Cassandra nomenclature<br>for selected intervals
|+ style="font-size: 105%; white-space: nowrap;" | Cassandra nomenclature for<br>selected intervals
|-
! Ratio
! Ratio
! Nominal
! Nominal
Line 325: Line 371:
| 3/2
| 3/2
| Perfect fifth
| Perfect fifth
| C-G
| C–G
|-
|-
| 5/4
| 5/4
| Down major third
| Submajor third
| C-vE
| C–↓E
|-
|-
| 7/4
| 7/4
| Down minor seventh
| Subminor seventh
| C-vBb
| C–↓B♭
|-
|-
| 11/8
| 11/8
| Double-up fourth
| Hyperfourth
| C-^^F
| C–↑↑F
|-
|-
| 13/8
| 13/8
| Double-up minor sixth
| Hyperminor sixth
| C-^^Ab
| C–↑↑A♭
|-
|-
| 19/16
| 19/16
| Minor third
| Minor third
| C-Eb
| C–E♭
|}
|}


{| class="wikitable center-1 center-3 mw-collapsible mw-collapsed"
{| class="wikitable center-1 center-3 mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | Andromeda nomenclature for selected intervals
|+ style="font-size: 105%; white-space: nowrap;" | Andromeda nomenclature for selected intervals
|-
! Ratio
! Ratio
! Nominal
! Nominal
Line 355: Line 402:
|-
|-
| 11/8
| 11/8
| Down diminished fifth<br>Double-down augmented fourth
| Subdiminished fifth<br>Hypo-augmented fourth
| C-vGb<br>C-vvF#
| C–↓G♭<br>C–↓↓F♯
|-
|-
| 13/8
| 13/8
| Double down major sixth
| Hypomajor sixth
| C-vvA
| C–↓↓A
|}
|}


{| class="wikitable center-1 center-3 mw-collapsible mw-collapsed"
{| class="wikitable center-1 center-3 mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | Helenus nomenclature for selected intervals
|+ style="font-size: 105%; white-space: nowrap;" | Helenus nomenclature for selected intervals
|-
! Ratio
! Ratio
! Nominal
! Nominal
Line 370: Line 418:
|-
|-
| 11/8
| 11/8
| Double-down diminished fifth<br>Triple-down augmented fourth
| Hypodiminished fifth<br>3-subaugmented fourth
| C-vvGb<br>C-v<sup>3</sup>F#
| C–↓↓G♭<br>C–↓<sup>3</sup>F♯
|-
|-
| 13/8
| 13/8
| Triple-down major sixth
| 3-submajor sixth
| C-v<sup>3</sup>A
| C–↓<sup>3</sup>A
|}
|}


== Chords and harmony ==
== Chords and harmony ==
Traditional tertian harmony is effective. The default triads on the Pythagorean spine are undevicesimal in quality:  
Traditional tertian harmony is effective. The default triads on the Pythagorean spine are undevicesimal in quality:  
* 1-19/15-3/2 (C-E-G)
* 1–19/15–3/2 (C–E–G)
* 1-19/16-3/2 (C-Eb-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.  
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, due to the [[hard]]<nowiki/>er fifths.  


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 401: Line 449:


== Tunings ==
== Tunings ==
=== Tuning spectra ===
=== Norm-based tunings ===
==== Cassandra ====
{| class="wikitable mw-collapsible mw-collapsed"
Gencom: [2 4/3; 225/224 275/273 325/324 385/384]
|+ 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 }}
|}


Gencom mapping: {{mapping| 1 2 -1 -3 13 12 | 0 -1 8 14 -23 -20 }}
{| 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 ===
==== Garibaldi ====
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
| '''[[12edo|7\12]]'''
|
| '''700.0000'''
| '''Lower bound of 9-odd-limit, <br>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
|-
| [[65edo|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
|-
| [[53edo|31\53]]
|
| 701.8868
|
|-
|
| 3/2
| 701.9550
| Pythagorean tuning
|-
|
| 36/35
| 702.0321
|
|-
| [[94edo|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
|
|-
| [[41edo|24\41]]
|
| 702.4390
|
|-
|
| 19/14
| 702.6079
|
|-
|-
! Edo<br>Generator
|
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
| 21/19
| 702.6732
|
|-
|
| 15/14
| 702.7775
|
|-
|
| 7/5
| 702.9146
|
|-
|
| 21/20
| 703.1066
|
|-
| '''[[29edo|17\29]]'''
|
| '''703.4483'''
| '''Upper bound of 9-odd-limit, <br>2.3.5.7.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|
| 13/11
| 703.5968
|
|}
 
==== Cassandra ====
{| class="wikitable mw-collapsible mw-collapsed center-all left-4"
! Edo<br>generator
! [[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'''
|-
|-
| 38\65
|
| 19/16
| 700.8290
| 1/3 undevicesimal schisma
|-
|
| 19/12
| 701.1105
| 1/4 undevicesimal schisma
|-
| [[65edo|38\65]]
|  
|  
| 701.5385
| 701.5385
|  
| 65def val
|-
|-
|  
|  
| 15/8
| 15/8
| 701.676
| 701.6759
|  
| 1/7 schisma
|-
|-
|  
|  
| 5/4
| 5/4
| 701.711
| 701.7108
|  
| 1/8 schisma
|-
|-
|  
|  
| {{monzo| 0 -10 17 }}
| 25/24
| 701.728
| 701.7252
| 5-odd-limit least squares
| 2/17 schisma
|-
|-
|  
|  
| 5/3
| 5/3
| 701.738
| 701.7379
| 5-odd-limit minimax
| 5-odd-limit minimax, 1/9 schisma
|-
|-
|  
|  
| 9/5
| 9/5
| 701.760
| 701.7596
| 1/10 schisma
|-
|
| 81/80
| 701.7922
| 1/12 schisma
|-
|
| 19/13
| 701.8702
|  
|  
|-
|-
| 31\53
| '''[[53edo|31\53]]'''
|  
|  
| 701.8868
| '''701.8868'''
| Lower bound of 11-, 13-, and 15-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 466: Line 817:
|  
|  
| 3/2
| 3/2
| 701.955
| 701.9550
|  
| Pythagorean tuning
|-
|-
|  
|  
| 13/8
| 13/8
| 702.026
| 702.0264
|  
|  
|-
|-
|  
|  
| 13/12
| 13/12
| 702.030
| 702.0301
|
|-
|
| 36/35
| 702.0321
|  
|  
|-
|-
|  
|  
| 13/9
| 13/9
| 702.034
| 702.0343
|
|-
|
| 19/11
| 702.0694
|  
|  
|-
|-
|  
|  
| 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>
| 21/13
| 702.112
| 702.1135
| 15-odd-limit least squares
|  
|-
|-
| [[94edo|55\94]]
|
| 702.1277
|  
|  
| {{monzo| 0 -27 7 17 }}
| 702.114
| 9-odd-limit least squares
|-
|-
|  
|  
| <span style="font-size:0.75em">{{monzo| 0 -38 -80 -122 137 116 }}</span>
| 9/7
| 702.128
| 702.1928
| 13-odd-limit least squares
| 9- and 11-odd-limit minimax, 1/16 septimal schisma
|-
|-
|  
|  
| {{monzo| 0 -25 11 35 }}
| 7/6
| 702.140
| 702.2086
| 7-odd-limit least squares
| 7-odd-limit minimax, 1/15 septimal schisma
|-
|-
|  
|  
| <span style="font-size:0.9em">{{monzo| 0 17 -52 -88 134 }}</span>
| 49/48
| 702.183
| 702.2174
| 11-odd-limit least squares
| 2/29 septimal schisma
|-
|-
|  
|  
| 9/7
| 7/4
| 702.193
| 702.2267
| 9- and 11-odd-limit minimax
| 1/14 septimal schisma
|-
|-
|  
|  
| 7/6
| 11/7
| 702.209
| 702.2295
| 7-odd-limit minimax
|  
|-
|-
|  
|  
| 7/4
| 11/8
| 702.227
| 702.2312
|  
|  
|-
|-
|  
|  
| 11/7
| 21/11
| 702.230
| 702.2371
|  
|  
|-
|-
|  
|  
| 11/8
| 19/10
| 702.231
| 702.2399
|  
|  
|-
|-
|  
|  
| 11/6
| 11/6
| 702.244
| 702.2438
|
|-
|  
|  
| 21/16
| 702.2476
| 1/13 septimal schisma
|-
|-
|  
|  
| 11/9
| 11/9
| 702.258
| 702.2575
|
|-
|
| 64/63
| 702.2720
| 1/12 septimal schisma
|-
|
| 19/15
| 702.3111
|
|-
| '''[[41edo|24\41]]'''
|  
|  
| '''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'''
|-
|-
| 24\41
|  
|  
| 702.4390
| 19/14
| Upper bound of 11-, 13-, and 15-odd-limit diamond monotone
| 702.6079
|
|-
|
| 21/19
| 702.6732
|
|-
|-
|  
|  
| 15/14
| 15/14
| 702.778
| 702.7775
|  
|  
|-
|-
|  
|  
| 7/5
| 7/5
| 702.915
| 702.9146
|  
|  
|-
|-
| 17\29
|  
|  
| 703.4483
| 21/20
| Upper bound of 9-odd-limit diamond monotone
| 703.1066
|
|-
| '''[[29edo|17\29]]'''
|
| '''703.4483'''
| '''29ef val, upper bound of 9-odd-limit diamond monotone'''
|-
|-
|  
|  
| 13/11
| 13/11
| 703.597
| 703.5968
|  
|  
|}
|}


==== Andromeda ====
==== Andromeda ====
Gencom: [2 4/3; 100/99 105/104 196/195 245/242]
{| class="wikitable mw-collapsible mw-collapsed center-all left-4"
 
! Edo<br>generator
Gencom mapping: {{mapping| 1 2 -1 -3 -4 -5 | 0 -1 8 14 18 21 }}
! Unchanged interval<br>(eigenmonzo)*
 
{| class="wikitable center-all left-4"
|-
! Edo<br>Generator
! Eigenmonzo<br>(Unchanged-interval)
! 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'''
|-
|-
| 38\65
|
| 19/16
| 700.8290
| 1/3 undevicesimal schisma
|-
|
| 19/12
| 701.1105
| 1/4 undevicesimal schisma
|-
| [[65edo|38\65]]
|  
|  
| 701.5385
| 701.5385
|  
| 65deeff val
|-
|-
|  
|  
| 15/8
| 15/8
| 701.676
| 701.6759
|  
| 1/7 schisma
|-
|-
|  
|  
| 5/4
| 5/4
| 701.711
| 701.7108
| 1/8 schisma
|-
|  
|  
| 25/24
| 701.7252
| 2/17 schisma
|-
|-
|  
|  
| 5/3
| 5/3
| 701.738
| 701.7379
| 5-odd-limit minimax
| 5-odd-limit minimax, 1/9 schisma
|-
|-
|  
|  
| 9/5
| 9/5
| 701.760
| 701.7596
| 1/10 schisma
|-
|  
|  
| 81/80
| 701.7922
| 1/12 schisma
|-
|-
| 31\53
| [[53edo|31\53]]
|  
|  
| 701.8868
| 701.8868
| 53ef val
|-
|  
|  
| 3/2
| 701.9550
| Pythagorean tuning
|-
|-
|  
|  
| 3/2
| 36/35
| 701.955
| 702.0321
|  
|  
|-
|-
|  
|  
| 9/7
| 9/7
| 702.193
| 702.1928
| 9-odd-limit minimax
| 9-odd-limit minimax, 1/16 septimal schisma
|-
|-
|  
|  
| 7/6
| 7/6
| 702.209
| 702.2086
| 7-odd-limit minimax
| 7-odd-limit minimax, 1/15 septimal schisma
|-
|
| 49/48
| 702.2174
| 2/29 septimal schisma
|-
|-
|  
|  
| 7/4
| 7/4
| 702.227
| 702.2267
| 1/14 septimal schisma
|-
|  
|  
| 21/16
| 702.2476
| 1/13 septimal schisma
|-
|-
| 24\41
|  
|  
| 702.4390
| 64/63
| Lower bound of 13- and 15-odd-limit diamond monotone
| 702.2720
| 1/12 septimal schisma
|-
|
| 19/15
| 702.3111
|
|-
| '''[[41edo|24\41]]'''
|
| '''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'''
|-
|
| 19/14
| 702.6079
|
|-
|-
|  
|  
| 11/9
| 11/9
| 702.630
| 702.6296
| 11-odd-limit minimax
| 11-odd-limit minimax
|-
|-
|  
|  
| 11/6
| 11/6
| 702.665
| 702.6651
|
|-
|
| 21/19
| 702.6732
|  
|  
|-
|-
|  
|  
| 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
|
|-
|
| 19/11
| 703.0797
|
|-
|
| 21/20
| 703.1066
|
|-
|
| 19/13
| 703.1659
|  
|  
|-
|-
|  
|  
| 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, and 15-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
|
|-
|
| 21/13
| 701.7817
|
|-
|
| 19/10
| 702.2399
|
|-
|
| 21/11
| 703.8926
|  
|  
|-
|-
|  
|  
| 13/7
| 13/7
| 704.043
| 704.0426
|  
|  
|-
|-
|  
|  
| 11/7
| 11/7
| 704.377
| 704.3770
|  
|  
|}
|}


==== Helenus ====
==== Helenus ====
Gencom: [2 4/3; 99/98 176/175 275/273 847/845]
{| class="wikitable mw-collapsible mw-collapsed center-all left-4"
 
! Edo<br>generator
Gencom mapping: {{mapping| 1 2 -1 -3 -9 -10 | 0 -1 8 14 30 33 }}
! Unchanged interval<br>(eigenmonzo)*
 
{| class="wikitable center-all left-4"
|-
! Edo<br>Generator
! Eigenmonzo<br>(Unchanged-interval)
! 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'''
|-
|
| 19/16
| 700.8290
| 1/3 undevicesimal schisma
|-
|-
|  
|  
| 11/7
| 11/7
| 701.094
| 701.0942
|
|-
|
| 19/12
| 701.1105
| 1/4 undevicesimal schisma
|-
|
| 21/11
| 701.1149
|  
|  
|-
|-
|  
|  
| 13/7
| 13/7
| 701.489
| 701.4894
|
|-
|
| 21/13
| 701.5127
|  
|  
|-
|-
| 38\65
| '''[[65edo|38\65]]'''
|  
|  
| 701.5385
| '''701.5385'''
| Lower bound of 13- and and 15-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
|-
|
| 19/11
| 701.7109
|  
|  
|-
|-
|  
|  
| 5/4
| 5/4
| 701.711
| 701.7108
| 1/8 schisma
|-
|  
|  
| 25/24
| 701.7252
| 2/17 schisma
|-
|-
|  
|  
| 5/3
| 5/3
| 701.738
| 701.7379
| 5-odd-limit minimax
| 5-odd-limit minimax, 1/9 schisma
|-
|-
|  
|  
| 9/5
| 9/5
| 701.760
| 701.7596
| 1/10 schisma
|-
|  
|  
| 81/80
| 701.7922
| 1/12 schisma
|-
|-
|  
|  
| 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'''
| '''Upper bound of 11-, 13-, 15-odd-limit, <br>2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|
| 19/13
| 701.8995
|  
|  
| 701.8868
| Upper bound of 11-, 13-, and 15-odd-limit diamond monotone
|-
|-
|  
|  
| 3/2
| 3/2
| 701.955
| 701.9550
| Pythagorean tuning
|-
|
| 36/35
| 702.0321
|  
|  
|-
|-
|  
|  
| 9/7
| 9/7
| 702.193
| 702.1928
| 9-odd-limit minimax
| 9-odd-limit minimax, 1/16 septimal schisma
|-
|-
|  
|  
| 7/6
| 7/6
| 702.209
| 702.2086
| 7-odd-limit minimax
| 7-odd-limit minimax, 1/15 septimal schisma
|-
|
| 49/48
| 702.2174
| 2/29 septimal schisma
|-
|-
|  
|  
| 7/4
| 7/4
| 702.227
| 702.2267
| 1/14 septimal schisma
|-
|
| 19/10
| 702.2399
|
|-
|  
|  
| 21/16
| 702.2476
| 1/13 septimal schisma
|-
|-
| 24\41
|
| 64/63
| 702.2720
| 1/12 septimal schisma
|-
|
| 19/15
| 702.3111
|
|-
| [[41edo|24\41]]
|  
|  
| 702.4390
| 702.4390
| 41ef val
|-
|
| 19/14
| 702.6079
|
|-
|
| 21/19
| 702.6732
|  
|  
|-
|-
|  
|  
| 15/14
| 15/14
| 702.778
| 702.7775
|  
|  
|-
|-
|  
|  
| 7/5
| 7/5
| 702.915
| 702.9146
|
|-
|
| 21/20
| 703.1066
|  
|  
|-
|-
| 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/>* Besides the octave


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

Latest revision as of 10:10, 27 October 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). It 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, 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 a septischismic one, where 7/4 is mapped to the double-diminished octave (e.g. C–C𝄫; a comma-flat minor seventh). This makes garibaldi a marvel, gariboh, octagar and aberschismic 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 (not Giuseppe 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:

  • Minor: 7/6~19/16~6/5
  • Major: 5/4~19/15~9/7

In garibaldi/cassandra:

The choice of nomenclature is due to the fact that cassandra is a relabeling of the 3-limit – it does not split primes in half, so it has no truly neutral intervals. Notice also the little interval between hyperminor and hypomajor, ~243/242. This interval spans 41 generator steps. 41edo tempers it out so that it merges hyperminor and hypominor into a true 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.

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.

Cassandra nomenclature for
selected intervals
Ratio Nominal Example
3/2 Perfect fifth C–G
5/4 Submajor third C–↓E
7/4 Subminor seventh C–↓B♭
11/8 Hyperfourth C–↑↑F
13/8 Hyperminor sixth C–↑↑A♭
19/16 Minor third C–E♭
Andromeda nomenclature for selected intervals
Ratio Nominal Example
11/8 Subdiminished fifth
Hypo-augmented fourth 		C–↓G♭
C–↓↓F♯
13/8 Hypomajor sixth C–↓↓A
Helenus nomenclature for selected intervals
Ratio Nominal Example
11/8 Hypodiminished fifth
3-subaugmented fourth 		C–↓↓G♭
C–↓3F♯
13/8 3-submajor sixth C–↓3A

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, due to the harder fifths.

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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