561/560: Difference between revisions

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Created page with "{{Infobox Interval | Name = monardisma, <br>tsaharuk comma | Color name = 17o1org1, <br>solorugu unison, <br>Solorugu comma | Comma = yes }} '''561/560''', the '''monardisma'..."
 
 
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{{Infobox Interval
{{Infobox Interval
| Name = monardisma, <br>tsaharuk comma
| Name = monardisma
| Color name = 17o1org1, <br>solorugu unison, <br>Solorugu comma
| Color name = 17o1org1, <br>solorugu unison, <br>Solorugu comma
| Comma = yes
| Comma = yes
}}
}}
'''561/560''', the '''monardisma''' is an [[unnoticeable comma|unnoticeable]] [[17-limit]] [[superparticular]] [[comma]] of about 3.09 [[cent]]s. It is the difference between the pentacircle minor third, [[33/28]], and the diatismic minor third, [[20/17]], on account of it being the difference between the diatisma, [[136/135]], and the pentacircle comma, [[896/891]]. It also represents the difference between the supraminor thirds of [[17/14]] and [[40/33]], and between the quartertones of [[33/32]] and [[35/34]].


'''561/560''', the '''monardisma''' or '''tsaharuk comma''' is a [[17-limit]] [[superparticular]] [[comma]] of about 3.1 [[cent]]s. It is the difference between [[33/32]] and [[35/34]]. By tempering it out is defined the '''monardismic temperament''', which enables the [[monardismic chords]].
== Temperaments ==
[[Tempering out]] this comma in the full 17-limit results in the rank-6 '''monardismic''' temperament, or in the 2.3.5.7.11.17 [[subgroup]], the rank-5 '''monardic''' temperament, enabling [[monardismic chords]]. This temperament has a number of notable properties:
* [[14/11]] and [[20/17]] become [[fifth complement]]s;
* [[11/10]] and [[17/14]] become [[fourth complement]]s.
 
=== Monardic ===
[[Subgroup]]: 2.3.5.7.11.17
 
[[Subgroup val|Subgroup-val]] [[mapping]]: <br>
{| class="right-all"
|-
| [⟨ || 1 || 0 || 0 || 0 || 0 || 4 || ],
|-
| ⟨ || 0 || 1 || 0 || 0 || 0 || -1 || ],
|-
| ⟨ || 0 || 0 || 1 || 0 || 0 || 1 || ],
|-
| ⟨ || 0 || 0 || 0 || 1 || 0 || 1 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || 1 || -1 || ]]
|}
: mapping generators: ~2, ~3, ~5, ~7, ~11
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2042{{c}}, ~3/2 = 701.6222{{c}}, ~5/4 = 386.1804{{c}}, ~7/4 = 968.8197{{c}}, ~11/8 = 550.0934{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6204{{c}}, ~5/4 = 386.3575{{c}}, ~7/4 = 968.9591{{c}}, ~11/8 = 550.2059{{c}}
 
{{Optimal ET sequence|legend=1| 12, 14c, 15, 17cg, 19g, 22, 31, 41g, 43, 46, 58, 68, 72, 118, 183, 255, 301, 373g, 674eg, 768egg, 1290beeeggg }}
 
[[Badness]] (Sintel): 0.204
 
=== Monardismic ===
[[Subgroup]]: 2.3.5.7.11.13.17
 
[[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 || 1 || ],
|-
| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || 1 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || -1 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
|}
: mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2042{{c}}, ~3/2 = 701.6222{{c}}, ~5/4 = 386.1804{{c}}, ~7/4 = 968.8197{{c}}, ~11/8 = 550.0934{{c}}, ~13/8 = 839.9145{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6204{{c}}, ~5/4 = 386.3575{{c}}, ~7/4 = 968.9591{{c}}, ~11/8 = 550.2059{{c}}, ~13/8 = 840.0917{{c}}
 
{{Optimal ET sequence|legend=1| 12f, 14cf, 15, 17cg, 19g, 22, 26, 29g, 31, 38df, 41g, 43, 46, 58, 72, 103, 111, 130, 140, 183, 243e, 301, 373g, 886eefgg, 1362bdeeefgggg }} *
 
<nowiki/>* [[optimal patent val]]: [[484edo|484]]
 
[[Badness]] (Sintel): 0.398
 
== Etymology ==
This comma was named by [[Scott Dakota]] in 2023 after the [[Monarda]] scale, a 12-tone scale of [[tannic]], where this is one of the commas being tempered out.  


== See also ==
== See also ==
* [[Unnoticeable comma]]
* [[List of superparticular intervals]]
* [[List of superparticular intervals]]


[[Category:Monardismic]]
[[Category:Monardismic]]
[[Category:Commas with unknown etymology]]

Latest revision as of 10:25, 22 March 2026

Interval information
Ratio 561/560
Factorization 2-4 × 3 × 5-1 × 7-1 × 11 × 17
Monzo [-4 1 -1 -1 1 0 1⟩
Size in cents 3.088732¢
Name monardisma
Color name 17o1org1,
solorugu unison,
Solorugu comma
FJS name [math]\displaystyle{ \text{d1}^{11,17}_{5,7} }[/math]
Special properties superparticular,
reduced
Tenney norm (log2 nd) 18.2611
Weil norm (log2 max(n, d)) 18.2637
Wilson norm (sopfr(nd)) 51
Comma size unnoticeable
S-expression S33⋅S34
Open this interval in xen-calc

561/560, the monardisma is an unnoticeable 17-limit superparticular comma of about 3.09 cents. It is the difference between the pentacircle minor third, 33/28, and the diatismic minor third, 20/17, on account of it being the difference between the diatisma, 136/135, and the pentacircle comma, 896/891. It also represents the difference between the supraminor thirds of 17/14 and 40/33, and between the quartertones of 33/32 and 35/34.

Temperaments

Tempering out this comma in the full 17-limit results in the rank-6 monardismic temperament, or in the 2.3.5.7.11.17 subgroup, the rank-5 monardic temperament, enabling monardismic chords. This temperament has a number of notable properties:

Monardic

Subgroup: 2.3.5.7.11.17

Subgroup-val mapping:

[⟨ 1 0 0 0 0 4 ],
⟨ 0 1 0 0 0 -1 ],
⟨ 0 0 1 0 0 1 ],
⟨ 0 0 0 1 0 1 ],
⟨ 0 0 0 0 1 -1 ]]
mapping generators: ~2, ~3, ~5, ~7, ~11

Optimal tunings:

  • WE: ~2 = 1200.2042 ¢, ~3/2 = 701.6222 ¢, ~5/4 = 386.1804 ¢, ~7/4 = 968.8197 ¢, ~11/8 = 550.0934 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6204 ¢, ~5/4 = 386.3575 ¢, ~7/4 = 968.9591 ¢, ~11/8 = 550.2059 ¢

Optimal ET sequence: 12, 14c, 15, 17cg, 19g, 22, 31, 41g, 43, 46, 58, 68, 72, 118, 183, 255, 301, 373g, 674eg, 768egg, 1290beeeggg

Badness (Sintel): 0.204

Monardismic

Subgroup: 2.3.5.7.11.13.17

Mapping:

[⟨ 1 0 0 0 0 0 4 ],
⟨ 0 1 0 0 0 0 -1 ],
⟨ 0 0 1 0 0 0 1 ],
⟨ 0 0 0 1 0 0 1 ],
⟨ 0 0 0 0 1 0 -1 ],
⟨ 0 0 0 0 0 1 0 ]]
mapping generators: ~2, ~3, ~5, ~7, ~11, ~13

Optimal tunings:

  • WE: ~2 = 1200.2042 ¢, ~3/2 = 701.6222 ¢, ~5/4 = 386.1804 ¢, ~7/4 = 968.8197 ¢, ~11/8 = 550.0934 ¢, ~13/8 = 839.9145 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6204 ¢, ~5/4 = 386.3575 ¢, ~7/4 = 968.9591 ¢, ~11/8 = 550.2059 ¢, ~13/8 = 840.0917 ¢

Optimal ET sequence: 12f, 14cf, 15, 17cg, 19g, 22, 26, 29g, 31, 38df, 41g, 43, 46, 58, 72, 103, 111, 130, 140, 183, 243e, 301, 373g, 886eefgg, 1362bdeeefgggg *

* optimal patent val: 484

Badness (Sintel): 0.398

Etymology

This comma was named by Scott Dakota in 2023 after the Monarda scale, a 12-tone scale of tannic, where this is one of the commas being tempered out.

See also

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