Xenharmonic Wiki

Garibaldi: Difference between revisions

← Older editNewer edit →
VisualWikitext
m FloraC moved page Schismic and garibaldi to Garibaldi over redirect: Schismic will get its own page
Relegate the discussion on the 5-limit part to a distinct article
Line 1: Line 1:
'''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). 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.
'''Garibaldi''' is a [[7-limit]] (and higher) [[regular temperament|temperament]] of the [[schismatic family #Garibaldi|schismatic family]]. It is an [[extension]] of [[helmholtz (temperament)|helmholtz]] temperament beyond the 5-limit but with the same simple [[chain of fifths|chain-of-fifths]] structure (so that [[chain-of-fifths notation|standard notation]] may be used). The garibaldi temperament tempers together the Pythagorean, syntonic, and archytas commas into a singular generalized "comma", which can be used to reach intervals of 3, 5, and 7. As in helmholtz temperament, [[5/4]] is mapped to the diminished fourth (e.g. C–F♭; a comma-flat major third), and the new mapping specific to garibaldi is that [[7/4]] is mapped to the double-diminished octave (e.g. C–C𝄫; a comma-flat minor seventh). This makes garibaldi a [[marvel temperaments|marvel temperament]] and a [[hemifamity temperaments|hemifamity temperament]]. Tuning the fifth a fraction of a cent sharp gives the best tunings.  


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


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.  
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 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.  
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.  
Retrieved from "https://en.xen.wiki/w/Garibaldi"