Garibaldi: Difference between revisions

← Older edit Newer edit →

VisualWikitext

@@ Line 315: / Line 315: @@
 |}
 <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]]. 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 a [[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.
+{| class="wikitable center-all mw-collapsible mw-collapsed"
+|-
+! rowspan="2" | #
+! rowspan="2" | Interval<br>category
+! colspan="3" style="border-left: double;" | "Double-Sub"
+! colspan="3" style="border-left: double;" | "Sub"
+! colspan="3" style="border-left: double;" | "Plain"
+! colspan="3" style="border-left: double;" | "Super"
+! colspan="3" style="border-left: double;" | "Double-super"
+|-
+! style="border-left: double;" | Gen. || Cents* || Ratios
+! style="border-left: double;" | Gen. || Cents* || Ratios
+! style="border-left: double;" | Gen. || Cents* || Ratios
+! style="border-left: double;" | Gen. || Cents* || Ratios
+! style="border-left: double;" | Gen. || Cents* || Ratios
+|-
+| 0
+| P1
+| style="border-left: double;" |  ||  ||
+| style="border-left: double;" |  ||  ||
+| style="border-left: double;" | 0 || 0.0 || 1/1
+| style="border-left: double;" | 12 || 25.2 || 64/63~81/80
+| style="border-left: double;" | 24 || 50.5 || 33/32~36/35
+|-
+| 1
+| m2
+| style="border-left: double;" |  ||  ||
+| style="border-left: double;" | -17 || 64.2 || 27/26~28/27
+| style="border-left: double;" | -5 || 89.5 || 20/19~21/20
+| style="border-left: double;" | 7 || 114.7 || 15/14~16/15
+| style="border-left: double;" | 19 || 140.0 || 13/12
+|-
+| 2
+| M2
+| style="border-left: double;" | -22 || 153.7 || 12/11
+| style="border-left: double;" | -10 || 178.9 || 10/9
+| style="border-left: double;" | 2 || 204.2 || 9/8
+| style="border-left: double;" | 14 || 229.5 || 8/7
+| style="border-left: double;" | 26 || 254.7 || 22/19
+|-
+| 3
+| m3
+| style="border-left: double;" |  ||  ||
+| style="border-left: double;" | -15 || 268.4 || 7/6
+| style="border-left: double;" | -3 || 293.6 || 13/11~19/16
+| style="border-left: double;" | 9 || 318.9 || 6/5
+| style="border-left: double;" | 21 || 344.2 || 11/9
+|-
+| 4
+| M3
+| style="border-left: double;" | -20 || 357.9 || 16/13
+| style="border-left: double;" | -8 || 383.2 || 5/4
+| style="border-left: double;" | 4 || 408.4 || 19/15~24/19
+| style="border-left: double;" | 16 || 433.7 || 9/7
+| style="border-left: double;" |  ||  ||
+|-
+| 5
+| P4
+| style="border-left: double;" | -25 || 447.4 || 35/27
+| style="border-left: double;" | -13 || 472.6 || 21/16
+| style="border-left: double;" | -1 || 497.9 || 4/3
+| style="border-left: double;" | 11 || 523.2 || 19/14
+| style="border-left: double;" | 23 || 548.4 || 11/8
+|-
+| 6
+| A4, d5
+| style="border-left: double;" | -18 || 562.1 || 18/13
+| style="border-left: double;" | -6 || 587.4 || 7/5
+| style="border-left: double;" | 6 || 612.6 || 10/7
+| style="border-left: double;" | 18 || 637.9 || 13/9
+| style="border-left: double;" |  ||  ||
+|-
+| 7
+| P5
+| style="border-left: double;" | -23 || 651.6 || 16/11
+| style="border-left: double;" | -11 || 676.8 || 28/19
+| style="border-left: double;" | 1 || 702.1 || 3/2
+| style="border-left: double;" | 13 || 727.4 || 32/21
+| style="border-left: double;" | 25 || 752.6 || 54/35
+|-
+| 8
+| m6
+| style="border-left: double;" |  ||  ||
+| style="border-left: double;" | -16 || 766.3 || 14/9
+| style="border-left: double;" | -4 || 791.6 || 19/12~30/19
+| style="border-left: double;" | 8 || 816.8 || 8/5
+| style="border-left: double;" | 20 || 842.1 || 13/8
+|-
+| 9
+| M6
+| style="border-left: double;" | -21 || 855.8 || 18/11
+| style="border-left: double;" | -9 || 881.1 || 5/3
+| style="border-left: double;" | 3 || 906.3 || 22/13~27/16
+| style="border-left: double;" | 15 || 931.6 || 12/7
+| style="border-left: double;" |  ||  ||
+|-
+| 10
+| m7
+| style="border-left: double;" | -26 || 945.3 || 19/11
+| style="border-left: double;" | -14 || 970.5 || 7/4
+| style="border-left: double;" | -2 || 995.8 || 16/9
+| style="border-left: double;" | 10 || 1021.1 || 9/5
+| style="border-left: double;" | 22 || 1046.3 || 11/6
+|-
+| 11
+| M7
+| style="border-left: double;" | -19 || 1060.0 || 24/13
+| style="border-left: double;" | -7 || 1085.3 || 15/8~28/15
+| style="border-left: double;" | 5 || 1110.5 || 19/10~40/21
+| style="border-left: double;" | 17 || 1135.8 || 27/14~52/27
+| style="border-left: double;" |  ||  ||
+|-
+| 12
+| P8
+| style="border-left: double;" | -24 || 1149.5 || 35/18~64/33
+| style="border-left: double;" | -12 || 1174.7 || 63/32~65/33
+| style="border-left: double;" | 0 || 1200.0 || 2/1
+| style="border-left: double;" |  ||  ||
+| style="border-left: double;" |  ||  ||
+|}
+See the diagrams on the right for isomorphic versions.
 == Notation ==