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'''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]]. Tempering it out defines the '''[[monardismic temperament]]''', which enables [[monardismic chords]].
'''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]].  
 
== Temperaments ==
[[Tempering out]] this comma in the 2.3.5.7.11.17 [[subgroup]] results in the '''monardic temperament''', or in the full 17-limit, the '''monardismic temperament''', enabling [[monardismic chords]].
 
=== 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:
* [[CTE]]: ~2 = 1200.0000, ~3/2 = 701.7805, ~5/4 = 386.6883, ~7/4 = 969.3735, ~11/8 = 550.4865, ~13/8 = 840.5277
* [[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|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]]


== See also ==
== See also ==

Revision as of 14:46, 25 February 2025

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¢
Names monardisma,
tsaharuk comma
Color name 17o1org1,
solorugu unison,
Solorugu comma
FJS name [math]\displaystyle{ \text{d1}^{11,17}_{5,7} }[/math]
Special properties superparticular,
reduced
Tenney height (log2 nd) 18.2611
Weil height (log2 max(n, d)) 18.2637
Wilson height (sopfr(nd)) 51
Comma size unnoticeable
S-expression S33 × S34
Open this interval in xen-calc

561/560, the monardisma or tsaharuk comma is a 17-limit superparticular comma of about 3.1 cents. It is the difference between 33/32 and 35/34.

Temperaments

Tempering out this comma in the 2.3.5.7.11.17 subgroup results in the monardic temperament, or in the full 17-limit, the monardismic temperament, enabling monardismic chords.

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:

  • CTE: ~2 = 1200.0000, ~3/2 = 701.7805, ~5/4 = 386.6883, ~7/4 = 969.3735, ~11/8 = 550.4865, ~13/8 = 840.5277
  • 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

See also

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