56/45: Difference between revisions

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**Imported revision 300765024 - Original comment: **
 
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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 = septimal narrow major third, marvel-ptolemaic major third, hemifamity diminished fourth
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-11 11:42:50 UTC</tt>.<br>
| Color name = zg4, zogu 4th
: The original revision id was <tt>300765024</tt>.<br>
| Sound = jid_56_45_pluck_adu_dr220.mp3
: 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>
'''56/45''', the '''septimal narrow major third''', or '''marvel-ptolemaic major third''', is the interval between various [[9-odd-limit]] consonances – [[5/4]] and [[14/9]], [[9/7]] and [[8/5]], [[9/8]] and [[7/5]], and [[10/7]] and [[16/9]] – with it also being [[6/5]] away from the marvelous fifth, [[112/75]]. It is flat of the [[81/64|Pythagorean major third]] by [[3645/3584]], namely a [[81/80|syntonic comma]] plus a [[225/224|marvel comma]]. [[Marvel]] tempering makes all of these intervals into classical major thirds, and turns chords like 1–5/4–8/5 or 1–5/4–14/9 into [[essentially tempered dyadic chord]]s. It is a third of a cent flatter than the 6\19 major third of [[19edo]]; however, it functionally doubles as a type of diminished fourth, flat of the [[8192/6561|Pythagorean diminished fourth]] by [[5120/5103]]. This gives rise to the name '''hemifamity diminished fourth'''.
<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">According to Ozan Yarman, the perde segah of Turkish maqam music should range from 56/45, which is flatter than 5/4 by the [[septimal kleisma]] of 225/224, and 5/4.</pre></div>
According to [[Ozan Yarman]], the perde segah of [[Arabic, Turkish, Persian music|Turkish maqam music]] should range from 56/45 to 5/4.
<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;56_45&lt;/title&gt;&lt;/head&gt;&lt;body&gt;According to Ozan Yarman, the perde segah of Turkish maqam music should range from 56/45, which is flatter than 5/4 by the &lt;a class="wiki_link" href="/septimal%20kleisma"&gt;septimal kleisma&lt;/a&gt; of 225/224, and 5/4.&lt;/body&gt;&lt;/html&gt;</pre></div>
== Approximation ==
{{Interval edo approximation|56/45}}
 
== See also ==
* [[45/28]] – its [[octave complement]]
* [[135/112]] – its [[fifth complement]]
* [[Gallery of just intervals]]
 
[[Category:Third]]
[[Category:Major third]]
[[Category:Fourth]]
[[Category:Diminished fourth]]
[[Category:Aberschismic]]

Latest revision as of 04:09, 11 June 2026

Interval information
Ratio 56/45
Factorization 23 × 3-2 × 5-1 × 7
Monzo [3 -2 -1 1⟩
Size in cents 378.6022¢
Names septimal narrow major third,
marvel-ptolemaic major third,
hemifamity diminished fourth
Color name zg4, zogu 4th
FJS name [math]\displaystyle{ \text{d4}^{7}_{5} }[/math]
Special properties reduced
Tenney norm (log2 nd) 11.2992
Weil norm (log2 max(n, d)) 11.6147
Wilson norm (sopfr(nd)) 24

[sound info]
Open this interval in xen-calc

56/45, the septimal narrow major third, or marvel-ptolemaic major third, is the interval between various 9-odd-limit consonances – 5/4 and 14/9, 9/7 and 8/5, 9/8 and 7/5, and 10/7 and 16/9 – with it also being 6/5 away from the marvelous fifth, 112/75. It is flat of the Pythagorean major third by 3645/3584, namely a syntonic comma plus a marvel comma. Marvel tempering makes all of these intervals into classical major thirds, and turns chords like 1–5/4–8/5 or 1–5/4–14/9 into essentially tempered dyadic chords. It is a third of a cent flatter than the 6\19 major third of 19edo; however, it functionally doubles as a type of diminished fourth, flat of the Pythagorean diminished fourth by 5120/5103. This gives rise to the name hemifamity diminished fourth.

According to Ozan Yarman, the perde segah of Turkish maqam music should range from 56/45 to 5/4.

Approximation

Edo approximations for 56/45 (378.60 ¢)
≤ 80edo, relative error ≤ 10%
Edo Step size Cents (¢) Absolute error (¢) Relative error (%)
3 1\3 400.00 +21.40 +5.35
16 5\16 375.00 -3.60 -4.80
19 6\19 378.95 +0.35 +0.55
22 7\22 381.82 +3.22 +5.90
32 10\32 375.00 -3.60 -9.61
35 11\35 377.14 -1.46 -4.26
38 12\38 378.95 +0.35 +1.09
41 13\41 380.49 +1.89 +6.44
51 16\51 376.47 -2.13 -9.06
54 17\54 377.78 -0.82 -3.71
57 18\57 378.95 +0.35 +1.64
60 19\60 380.00 +1.40 +6.99
70 22\70 377.14 -1.46 -8.51
73 23\73 378.08 -0.52 -3.16
76 24\76 378.95 +0.35 +2.19
79 25\79 379.75 +1.14 +7.54

See also

Retrieved from "https://en.xen.wiki/index.php?title=56/45&oldid=232147"