Nicolic chords: Difference between revisions

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This is of pattern 1b
 
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'''Nicolic chords''' are [[Dyadic chord|essentially tempered chords]] tempered by the nicola, [[1575/1573]].
'''Nicolic chords''' are [[dyadic chord|essentially tempered chords]] tempered by the nicola, [[1575/1573]].


There are three triads, six tetrads and two pentads as [[15-odd-limit]] essentially tempered chords.
Nicolic chords are of [[dyadic chord/Pattern of essentially tempered chords|pattern 1b]] in the [[15-odd-limit]]. There are 3 triads, 6 tetrads and 2 pentads for a total of 11 distinct chord structures.  


The nicolic triads include the following palindromic form,
The nicolic triads include the following palindromic form since it identifies [[13/7]] by a stack of two [[15/11]]'s:
* 1-[[15/11]]-[[13/7]] with steps of 15/11-15/11-[[14/13]],
* 1–15/11–13/7 with steps of 15/11, 15/11, [[14/13]].


and a pair in inverse relationship,
And a pair in inverse relationship:
* 1-15/11-[[11/7]] with steps of 15/11-[[15/13]]-[[14/11]], and
* 1–15/11–[[11/7]] with steps of 15/11, [[15/13]], [[14/11]], and its inverse
* 1-15/11-[[26/15]] with steps of 15/11-14/11-15/13.
* 1–15/11–[[26/15]] with steps of 15/11, 14/11, 15/13.


For tetrads, there are two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are  
For tetrads, there are two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are  
* 1-15/13-15/11-11/7 with steps of 15/13-[[13/11]]-15/13-14/11;
* 1–15/13–15/11–11/7 with steps of 15/13, [[13/11]], 15/13, 14/11;
* 1-15/13-[[22/15]]-11/7 with steps of 15/13-14/11-[[15/14]]-14/11.
* 1–15/13–[[22/15]]–11/7 with steps of 15/13, 14/11, [[15/14]], 14/11.


The inversely related pairs of chords are  
The inversely related pairs of chords are  
* 1-15/11-11/7-13/7 with steps of 15/11-15/13-13/11-14/13, and its inverse  
* 1–15/11–11/7–13/7 with steps of 15/11, 15/13, 13/11, 14/13, and its inverse  
* 1-15/11-22/15-26/15 with steps of 15/11-14/13-13/11-15/13;
* 1–15/11–22/15–26/15 with steps of 15/11, 14/13, 13/11, 15/13;
* 1-15/11-26/15-13/7 with steps of 15/11-14/11-15/14-14/13, and its inverse  
* 1–15/11–26/15–13/7 with steps of 15/11, 14/11, 15/14, 14/13, and its inverse  
* 1-15/11-22/15-11/7 with steps of 15/11-14/13-15/14-14/11.
* 1–15/11–22/15–11/7 with steps of 15/11, 14/13, 15/14, 14/11.


They can be extended to pentads:  
They can be extended to pentads:  
* 1-15/14-15/13-15/11-11/7 with steps of 15/14-14/13-13/11-15/13-14/11, and its inverse  
* 1–15/14–15/13–15/11–11/7 with steps of 15/14, 14/13, 13/11, 15/13, 14/11, and its inverse  
* 1-15/13-15/11-22/15-11/7 with steps of 15/13-13/11-14/13-15/14-14/11.
* 1–15/13–15/11–22/15–11/7 with steps of 15/13, 13/11, 14/13, 15/14, 14/11.


[[Equal temperament]]s with nicolic chords include {{Optimal ET sequence| 29, 31, 43, 58, 72, 87, 94, 103, 121, 130, 159, 190, 217, 224, 311, 414, 441, 665 and 855 }}.
[[Equal temperament]]s with nicolic chords include {{Optimal ET sequence| 29, 31, 43, 58, 72, 87, 94, 103, 121, 130, 159, 190, 217, 224, 311, 414, 441, 665 and 855 }}.
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[[Category:15-odd-limit]]
[[Category:15-odd-limit chords]]
[[Category:Essentially tempered chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Triads]]

Latest revision as of 14:29, 20 March 2025

Nicolic chords are essentially tempered chords tempered by the nicola, 1575/1573.

Nicolic chords are of pattern 1b in the 15-odd-limit. There are 3 triads, 6 tetrads and 2 pentads for a total of 11 distinct chord structures.

The nicolic triads include the following palindromic form since it identifies 13/7 by a stack of two 15/11's:

  • 1–15/11–13/7 with steps of 15/11, 15/11, 14/13.

And a pair in inverse relationship:

  • 1–15/11–11/7 with steps of 15/11, 15/13, 14/11, and its inverse
  • 1–15/11–26/15 with steps of 15/11, 14/11, 15/13.

For tetrads, there are two palindromic chords and two pairs of chords in inverse relationship. The palindromic chords are

  • 1–15/13–15/11–11/7 with steps of 15/13, 13/11, 15/13, 14/11;
  • 1–15/13–22/15–11/7 with steps of 15/13, 14/11, 15/14, 14/11.

The inversely related pairs of chords are

  • 1–15/11–11/7–13/7 with steps of 15/11, 15/13, 13/11, 14/13, and its inverse
  • 1–15/11–22/15–26/15 with steps of 15/11, 14/13, 13/11, 15/13;
  • 1–15/11–26/15–13/7 with steps of 15/11, 14/11, 15/14, 14/13, and its inverse
  • 1–15/11–22/15–11/7 with steps of 15/11, 14/13, 15/14, 14/11.

They can be extended to pentads:

  • 1–15/14–15/13–15/11–11/7 with steps of 15/14, 14/13, 13/11, 15/13, 14/11, and its inverse
  • 1–15/13–15/11–22/15–11/7 with steps of 15/13, 13/11, 14/13, 15/14, 14/11.

Equal temperaments with nicolic chords include 29, 31, 43, 58, 72, 87, 94, 103, 121, 130, 159, 190, 217, 224, 311, 414, 441, 665 and 855.

Here is a listening sample, tempered to TE tuning of nicolic.

Ratio 1/1 15/13 15/11 11/7
Cents 0.000 247.099 536.086 783.185
Sound
[info]
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