273/272: Difference between revisions

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Created page with "{{Infobox Interval | Icon = | Ratio = 273/272 | Monzo = -4 1 0 1 0 1 -1 | Cents = 6.35316 | Name = tannisma | Color name = Suthozo | FJS name = | Sound = }} '''273/272''',..."
 
 
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{{Infobox Interval
{{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]], as well as the difference between [[39/32]] and [[17/14]]. Equating these two 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]]

Latest revision as of 10:25, 22 March 2026

Interval information
Ratio 273/272
Factorization 2-4 × 3 × 7 × 13 × 17-1
Monzo [-4 1 0 1 0 1 -1⟩
Size in cents 6.35316¢
Names tannisma,
prototannisma
Color name 17u3oz1, suthozo 1sn,
Suthozo comma
FJS name [math]\displaystyle{ \text{P1}^{7,13}_{17} }[/math]
Special properties superparticular,
reduced
Tenney norm (log2 nd) 16.1802
Weil norm (log2 max(n, d)) 16.1855
Wilson norm (sopfr(nd)) 48
Comma size small
Open this interval in xen-calc

273/272, the tannisma, or the prototannisma, is a small 17-limit (also 2.3.7.13.17-subgroup) comma with a value of roughly 6.35 cents. 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.

Commatic relations

This comma is the difference between the following superparticular pairs:

It factors into the following superparticular pairs:

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 chords 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[1].

Prototannic

Subgroup: 2.3.7.13.17

Comma list: 273/272

Subgroup-val mapping: [⟨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 tunings:

  • WE: ~2 = 1200.4469 ¢, ~3/2 = 701.2262 ¢, ~7/4 = 967.0489 ¢, ~13/8 = 837.6531 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2150 ¢, ~7/4 = 967.0685 ¢, ~13/8 = 837.7810 ¢

Optimal ET sequence: 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:

[⟨ 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 tunings:

  • WE: ~2 = 1200.4469 ¢, ~3/2 = 701.2262 ¢, ~5/4 = 385.4186 ¢, ~7/4 = 967.0489 ¢, ~11/8 = 549.9753 ¢, ~13/8 = 837.6531 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2150 ¢, ~5/4 = 385.7085 ¢, ~7/4 = 967.0685 ¢, ~11/8 = 550.4163 ¢, ~13/8 = 837.7810 ¢

Optimal ET sequence: 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

References

  1. ↑ Relevant Facebook post by Scott Dakota
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