56/45: Difference between revisions

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Rework to reflect the progress thanks to the establishment of the aberschisma
 
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{{Infobox Interval
{{Infobox Interval
| Icon =
| Name = septimal narrow major third, marvel-ptolemaic major third, aberschismic diminished fourth
| Ratio = 56/45
| Color name = zg4, zogu 4th
| Monzo = 3 -2 -1 1
| Cents = 378.60219
| Name = septimal narrow major third, <br> marvelous major third
| Color name =  
| Sound = jid_56_45_pluck_adu_dr220.mp3
| Sound = jid_56_45_pluck_adu_dr220.mp3
}}
}}
'''56/45''', conventionally known as the '''septimal narrow major third''' or '''marvel-ptolemaic major third''', is flat of the [[81/64|Pythagorean major third]] by [[3645/3584]], namely a [[81/80|syntonic comma]] plus a [[225/224|marvel comma]]. However, it functions more commonly as a diminished fourth, flat of the [[8192/6561|Pythagorean diminished fourth]] by an [[aberschisma]]. This gives rise to the more precise name '''aberschismic diminished fourth''', and in fact it arises as the interval between various [[9-odd-limit]] consonances – between [[5/4]] and [[14/9]], between [[9/7]] and [[8/5]], between [[9/8]] and [[7/5]], and between [[10/7]] and [[16/9]] – with it also being [[6/5]] away from the marvelous fifth, [[112/75]].


'''56/45''', the '''septimal narrow major third''', or '''marvelous major third''', is the interval between various 9-limit consonances: 5/4 and 14/9, 9/7 and 8/5, 9/8 and 7/5, and 10/7 and 16/9. [[Marvel]] tempering makes all of these intervals into major thirds like any other, and turns chords like 1-5/4-8/5 or 1-5/4-14/9 into [[Dyadic_chord|essentially tempered dyadic chords]]. The 56/45 third is [[6/5]] away from the marvelous fifth, [[112/75]], and may also be called the marvelous major third. It is a third of a cent flatter than the 6\19 major third of [[19et]]. According to [[Ozan Yarman]], the perde segah of Turkish [[maqam music]] should range from 56/45.
According to [[Ozan Yarman]], the perde segah of [[Arabic, Turkish, Persian music|Turkish maqam music]] should range from 56/45 to 5/4.
 
== Temperaments ==
[[Marvel]] tempering equates this interval with the [[5/4|classical major third]] and turns chords like 1–5/4–8/5 or 1–5/4–14/9 into [[essentially tempered chord]]s.
 
== Approximation ==
This interval is a third of a cent flatter than 6\19, the major third of [[19edo]].  
 
{{Interval edo approximation|56/45}}


== See also ==
== See also ==
 
* [[45/28]] – its [[octave complement]]
* [[45/28]] -- its octave complement
* [[135/112]] – its [[fifth complement]]
* [[135/112]] -- its fifth complement
* [[Gallery of just intervals]]
* [[Gallery of just intervals]]


[[Category:7-limit]]
[[Category:Just interval]]
[[Category:Third]]
[[Category:Third]]
[[Category:Major third]]
[[Category:Major third]]
[[Category:19edo]]
[[Category:Fourth]]
[[Category:Diminished fourth]]
[[Category:Aberschismic]]

Latest revision as of 19:32, 30 July 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,
aberschismic 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, conventionally known as the septimal narrow major third or marvel-ptolemaic major third, is flat of the Pythagorean major third by 3645/3584, namely a syntonic comma plus a marvel comma. However, it functions more commonly as a diminished fourth, flat of the Pythagorean diminished fourth by an aberschisma. This gives rise to the more precise name aberschismic diminished fourth, and in fact it arises as the interval between various 9-odd-limit consonances – between 5/4 and 14/9, between 9/7 and 8/5, between 9/8 and 7/5, and between 10/7 and 16/9 – with it also being 6/5 away from the marvelous fifth, 112/75.

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

Temperaments

Marvel tempering equates this interval with the classical major third and turns chords like 1–5/4–8/5 or 1–5/4–14/9 into essentially tempered chords.

Approximation

This interval is a third of a cent flatter than 6\19, the major third of 19edo.


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

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