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19-limit: Difference between revisions

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**Imported revision 215739836 - Original comment: **
 
→Edo approximation: calibrate the list according to my research results
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Prime limit navigation|19}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
The '''19-limit''' consists of [[just intonation]] [[interval]]s whose [[ratio]]s contain no [[prime factor]]s higher than 19. It is the 8th [[prime limit]] and is a superset of the [[17-limit]] and a subset of the [[23-limit]].  
: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-03-31 03:39:50 UTC</tt>.<br>
 
: The original revision id was <tt>215739836</tt>.<br>
The 19-limit is a [[rank and codimension|rank-8]] system, and can be modeled in a 7-dimensional [[lattice]], with the primes 3, 5, 7, 11, 13, 17, and 19 represented by each dimension. The prime 2 does not appear in the typical 19-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, an eighth dimension is needed.
: The revision comment was: <tt></tt><br>
 
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
These things are contained by the 19-limit, but not the 17-limit:  
<h4>Original Wikitext content:</h4>
* The [[19-odd-limit|19-]] and [[21-odd-limit]];
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">see [[Harmonic Limit]]</pre></div>
* Mode 10 and 11 of the harmonic or subharmonic series.  
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;19-limit&lt;/title&gt;&lt;/head&gt;&lt;body&gt;see &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;Harmonic Limit&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
== Terminology and notation ==
[[Interval_region|Interval categories]] of [[harmonic class|HC19]] are relatively clear. [[19/16]] is most commonly considered a minor third, as 1–19/16–3/2 is an important {{w|tertian}} chord (the [[Functional Just System]] and [[Helmholtz–Ellis notation]] agree). However, 19/16 may act as an augmented second in certain cases. This is more complex on its own but may simplify certain combinations with other intervals, especially if [[17/16]] is considered an augmented unison and/or if [[23/16]] is considered an augmented fourth. Perhaps most interestingly, [[Sagittal notation]] provides an accidental to enharmonically spell intervals of HC19 this way.
 
== Edo approximation ==
Here is a list of [[edo]]s with progressively better tunings for 19-limit intervals ([[monotonicity limit]] ≥ 19 and decreasing [[TE error]]): {{EDOs| 34dh, 38df, 41, 50, 53, 58h, 68, 72, 94, 103h, 111, 121, 130, 140, 152fg, 159, 161, 183, 190g, 193, 212gh, 217, 243e, 270, 311, 400, 422, 460, 525, 581, 742, 935, 954h }} and so on. For a more comprehensive list, see [[Sequence of equal temperaments by error]].
 
Here is a list of edos which provides relatively good tunings for 19-limit intervals ([[TE relative error]] < 5%): {{EDOs| 72, 111, 217, 243e, 270, 282, 311, 354, 364, 373g, 400, 422, 460, 494(h), 525, 540, 581, 597, 624, 643, 653, 692, 718, 742, 764h, 814, 836f, 882, 908, 925, 935, 954h and }} so on.
 
: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "34dh" means taking the second closest approximations of harmonics 7 and 19.
 
== Intervals ==
 
Here are all the [[21-odd-limit]] intervals of 19-limit:  
 
{| class="wikitable"
! Ratio
! Cents Value
! colspan="2" | Color Name
! Interval Name
|-
| [[20/19]]
| 88.801
| 19uy1
| nuyo 1son
| small undevicesimal semitone
|-
| [[19/18]]
| 93.603
| 19o2
| ino 2nd
| large undevicesimal semitone
|-
| [[21/19]]
| 173.268
| 19uz2
| nuzo 2nd
| small undevicesimal whole tone
|-
| [[19/17]]
| 192.558
| 19o17u2
| nosu 2nd
| large undevicesimal whole tone, quasi-meantone
|-
| [[22/19]]
| 253.805
| 19u1o2
| nulo 2nd
| undevicesimal second-third
|-
| [[19/16]]
| 297.513
| 19o3
| ino 3rd
| undevicesimal minor third
|-
| [[24/19]]
| 404.442
| 19u3
| inu 3rd
| small undevicesimal major third
|-
| [[19/15]]
| 409.244
| 19og4
| nogu 4th
| large undevicesimal major third
|-
| [[19/14]]
| 528.687
| 19or4
| noru 4th
| undevicesimal acute fourth
|-
| [[26/19]]
| 543.015
| 19u3o4
| nutho 4th
| undevicesimal super fourth
|-
| [[19/13]]
| 656.985
| 19o3u5
| nothu 5th
| undevicesimal subfifth
|-
| [[28/19]]
| 671.313
| 19uz5
| nuzo 5th
| undevicesimal gravefifth
|-
| [[30/19]]
| 790.756
| 19uy5
| nuyo 5th
| small undevicesimal minor sixth
|-
| [[19/12]]
| 795.558
| 19o6
| ino 6th
| large undevicesimal minor sixth
|-
| [[32/19]]
| 902.487
| 19u6
| inu 6th
| undevicesimal major sixth
|-
| [[19/11]]
| 946.195
| 19o1u7
| nolu 7th
| undevicesimal sixth-seventh
|-
| [[34/19]]
| 1007.442
| 19u17o7
| nuso 7th
| small undevicesimal minor seventh
|-
| [[38/21]]
| 1026.732
| 19or7
| noru 7th
| large undevicesimal minor seventh
|-
| [[36/19]]
| 1106.397
| 19u7
| inu 7th
| small undevicesimal major seventh
|-
| [[19/10]]
| 1111.199
| 19og8
| nogu 8ve
| large undevicesimal major seventh
|}
 
== Music ==
; [[Domin]]
* [https://www.youtube.com/watch?v=WTo5YihoLqs ''Asuttan''] (2024)
* [https://www.youtube.com/watch?v=OPt3Y9VSliU ''Asuttan Bouta''] (2024)
 
; [[Joseph Monzo]]
* [https://www.youtube.com/watch?v=it5avwRE8PI ''Theme from Invisible Haircut''] (1990)
 
[[Category:19-limit| ]] <!-- main article -->
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