40/39: Difference between revisions

In 13-limit just intonation, 40/39 is the difference between the third octave of the third 5/4 (5 × 2³) and the fifth of the thirteenth partial of the same root (13 × 3). Within an octave, it is the difference between 39/32 and 5/4 and thus between 13/8 and 5/3. It is also the difference between the perfect fourth (4/3) and the tridecimal naiadic (13/10), and between the Pythagorean whole tone (9/8) and the tridecimal semifourth (15/13).

Temperaments

If treated as a comma to be tempered out in the 2.3.5.13 subgroup, it leads to unintendo temperament.

Unintendo

This temperament was accidentally discovered when Vector described fendo as tempering out 40/39 in the 2.3.5.13 subgroup. The mistake has since been corrected, and the 2.3.5.13 temperament was renamed to "unintendo" to reflect its unintended discovery. Its generators are a sharp perfect fifth and a flat major third. It can be described as 7 & 15 & 10. It equates 39/32 with 5/4 and equates 13/8 with 5/3, so it does not assosciate major with greater neutral and minor with lesser neutral as one would expect (see 65/64), but the other way around.

Subgroup: 2.3.5.13

Comma list: 40/39

Sval mapping: [⟨1 0 0 3], ⟨0 1 0 -1], ⟨0 0 1 1]]

sval mapping generators: ~2, ~3, ~5

Optimal tunings:

CWE: ~2 = 1200 ¢, ~3/2 = 710.153 ¢, ~5/4 = 383.023 ¢

Badness (Sintel): 0.227

Notation

Sagittal notation

In the Sagittal system, the downward version of this comma (possibly tempered) is represented (in a secondary role) by the sagittal ⁠ ⁠ and is called the 13/5 small diesis, or 13/5S for short, because the simplest interval it notates is 13/5 (equiv. 13/10), as for example in C-F⁠ ⁠⁠ ⁠. The primary role of ⁠ ⁠ is 6400/6561 (25S). The upward version is called 5/13S or 13/5S up and is represented (in a secondary role) by ⁠ ⁠.

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-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{Infobox Interval
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+| Name = tridecimal 1/5-tone, tridecimal minor diesis, unintendo comma
-: This revision was by author [[User:Musikleeranstalt|Musikleeranstalt]] and made on <tt>2016-02-02 17:44:24 UTC</tt>.<br>
+| Color name = 3uy1, thuyo unison
-: The original revision id was <tt>573736931</tt>.<br>
+}}
-: 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>
+In [[13-limit]] [[just intonation]], '''40/39''' is the difference between the third octave of the third [[5/4]] ({{nowrap| 40 = 5 × 2<sup>3</sup> }}) and the fifth of the thirteenth partial of the same root ({{nowrap| 39 = 13 × 3 }}). Within an octave, it is the difference between [[39/32]] and [[5/4]] and thus between [[13/8]] and [[5/3]]. It is also the difference between the [[4/3|perfect fourth (4/3)]] and the [[13/10|tridecimal naiadic (13/10)]], and between the [[9/8|Pythagorean whole tone (9/8)]] and the [[15/13|tridecimal semifourth (15/13)]].
-<h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">difference between the third octave of the third 5:4 (40=5x2x2x2) and the fith of the thirteenth partial of the same root (39=13x3). possibility to change from subsystem 4 to subsystem 3 (or 5 to 13).</pre></div>
+== Temperaments ==
-<h4>Original HTML content:</h4>
+If treated as a comma to be tempered out in the 2.3.5.13 subgroup, it leads to '''unintendo''' temperament.
-<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;40_39&lt;/title&gt;&lt;/head&gt;&lt;body&gt;difference between the third octave of the third 5:4 (40=5x2x2x2) and the fith of the thirteenth partial of the same root (39=13x3). possibility to change from subsystem 4 to subsystem 3 (or 5 to 13).&lt;/body&gt;&lt;/html&gt;</pre></div>
+=== Unintendo ===
+This temperament was accidentally discovered when [[User:VectorGraphics|Vector]] described [[fendo]] as tempering out 40/39 in the 2.3.5.13 subgroup. The mistake has since been corrected, and the 2.3.5.13 temperament was renamed to "unintendo" to reflect its unintended discovery. Its generators are a sharp perfect fifth and a flat major third. It can be described as {{nowrap| 7 & 15 & 10 }}. It equates 39/32 with 5/4 and equates 13/8 with 5/3, so it does not assosciate major with greater neutral and minor with lesser neutral as one would expect (see [[65/64]]), but the other way around.
+[[Subgroup]]: 2.3.5.13
+[[Comma list]]: 40/39
+{{Mapping|legend=2| 1 0 0 3 | 0 1 0 -1 | 0 0 1 1 }}
+: sval mapping generators: ~2, ~3, ~5
+[[Optimal tuning]]s:
+* [[CWE]]: ~2 = 1200{{c}}, ~3/2 = 710.153{{c}}, ~5/4 = 383.023{{c}}
+[[Badness]] (Sintel): 0.227
+== Notation ==
+=== Sagittal notation ===
+In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented (in a secondary role) by the sagittal {{sagittal | \\! }} and is called the '''13/5 small diesis''', or '''13/5S''' for short, because the simplest interval it notates is 13/5 (equiv. 13/10), as for example in C-F{{nbhsp}}{{sagittal | \\! }}. The primary role of {{ sagittal | \\! }} is [[6561/6400 #Sagittal notation|6400/6561]] (25S). The upward version is called '''5/13S''' or '''13/5S up''' and is represented (in a secondary role) by {{sagittal| //| }}.

40/39: Difference between revisions

Latest revision as of 16:39, 6 June 2025

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Temperaments

Unintendo

Notation

Sagittal notation

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40/39: Difference between revisions

Latest revision as of 16:39, 6 June 2025

Temperaments

Unintendo

Notation

Sagittal notation

Navigation menu

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