273/272: Difference between revisions

← Older edit

VisualWikitext

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Icon =
+| Name = tannisma, prototannisma
-| Ratio = 273/272
+| Color name = 17u3oz1, suthozo 1sn,<br>Suthozo comma
-| Monzo = -4 1 0 1 0 1 -1
+| Comma = yes
-| Cents = 6.35316
-| Name = tannisma
-| Color name = Suthozo
-| FJS name =
-| Sound =
 }}
+'''273/272''', the '''tannisma''', or the '''prototannisma''', is a [[small comma|small]] [[17-limit]] (also 2.3.7.13.17-[[subgroup]]) [[comma]] with a value of roughly 6.35 [[cent]]s. It forms the difference between [[21/17]] and [[16/13]], and the difference between [[39/32]] and [[17/14]], as well as the difference between [[17/13]] and [[21/16]].
-'''273/272''', the '''tannisma''', is a [[small comma|small]] [[17-limit]] comma with a value of roughly 6.35 [[cent]]s.  It forms the difference between [[21/17]] and [[16/13]], and the difference between [[39/32]] and [[17/14]], as well as the difference between [[17/13]] and [[21/16]].  Equating these three sets of intervals is characteristic of '''tannismic temperaments'''.
+== Commatic relations ==
+This comma is the difference between the following superparticular pairs:
+* [[35/34]] and [[40/39]]
+* [[52/51]] and [[64/63]]
+* [[65/64]] and [[85/84]]
+* [[91/90]] and [[136/135]]
+* [[154/153]] and [[352/351]]
+* [[169/168]] and [[442/441]]
+* [[221/220]] and [[1156/1155]]
+* [[225/224]] and [[1275/1274]]
+* [[256/255]] and [[4096/4095]]
-[[Category:17-limit]]
+It factors into the following superparticular pairs:
-[[Category:Small comma]]
+* [[441/440]] and [[715/714]]
-[[Category:Superparticular]]
+* [[385/384]] and [[936/935]]
+* [[375/374]] and [[1001/1000]]
+* [[364/363]] and [[1089/1088]]
+* [[351/350]] and [[1225/1224]]
+* [[325/324]] and [[1701/1700]]
+* [[289/288]] and [[4914/4913]]
+It also factors neatly into three superparticular commas: [[715/714]], [[833/832]], and [[936/935]].
+== Temperaments ==
+[[Tempering out]] this comma in the 17-limit leads to the rank-6 '''prototannismic''' temperament, or in the 2.3.7.13.17 subgroup, the rank-4 '''prototannic''' temperament, both characterized by the equivalences introduced above, and lead to a type of [[essentially tempered chord]]s called [[prototannismic chords]]. The prototannic temperament has a notable [[weak extension]] to the full 17-limit called [[tannic]]. According to [[Scott Dakota]], tannic is of similar overall utility to [[marvel]]<ref>[https://www.facebook.com/groups/497105067092502/posts/1158638974272438/ Relevant Facebook post by Scott Dakota]</ref>.
+=== Prototannic ===
+[[Subgroup]]: 2.3.7.13.17
+[[Comma list]]: 273/272
+{{Mapping|legend=2| 1 0 0 0 -4 | 0 1 0 0 1 | 0 0 1 0 1 | 0 0 0 1 1 }}
+: mapping generators: ~2, ~3, ~7, ~13
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.4469{{c}}, ~3/2 = 701.2262{{c}}, ~7/4 = 967.0489{{c}}, ~13/8 = 837.6531{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2150{{c}}, ~7/4 = 967.0685{{c}}, ~13/8 = 837.7810{{c}}
+{{Optimal ET sequence|legend=1| 10, 17g, 24, 26, 31, 36, 113, 149 }}
+[[Badness]] (Sintel): 0.113
+=== Prototannismic ===
+[[Subgroup]]: 2.3.5.7.11.13.17
+[[Comma list]]: 273/272
+[[Mapping]]: <br>
+{| class="right-all"
+|-
+| [⟨ || 1 || 0 || 0 || 0 || 0 || 0 || -4 || ],
+|-
+| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || 1 || ],
+|-
+| ⟨ || 0 || 0 || 1 || 0 || 0 || 0 || 0 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || 1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || 0 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 1 || ]]
+|}
+: mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.4469{{c}}, ~3/2 = 701.2262{{c}}, ~5/4 = 385.4186{{c}}, ~7/4 = 967.0489{{c}}, ~11/8 = 549.9753{{c}}, ~13/8 = 837.6531{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2150{{c}}, ~5/4 = 385.7085{{c}}, ~7/4 = 967.0685{{c}}, ~11/8 = 550.4163{{c}}, ~13/8 = 837.7810{{c}}
+{{Optimal ET sequence|legend=1| 17cg, 19eg, 22, 26, 27eg, 29g, 31, 41, 46, 72, 103, 130g, 149, 159, 221ef, 262df, 308def, 334cdf }}
+[[Badness]] (Sintel): 0.678
+== Etymology ==
+This comma was named by Scott Dakota, who also devised tannic temperament, as the ''tannisma'' no later than 2017. The name ''tannisma'' does not adhere to the wiki comma naming standard as it would induce ''tannic'' as a temperament that tempers out the comma alone in its minimal prime subgroup, 2.3.7.13.17 {273/272}, for which Scott's 17-limit tannic is a weak extension and cannot share the same time. ''Prototannisma'' has thus been proposed as an alternative name.
+== See also ==
+* [[List of superparticular intervals]]
+== References ==
+<references/>
+[[Category:Prototannismic]]
+[[Category:Commas with unknown etymology]]