56/45: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 300765024 - Original comment: ** |
Rework to reflect the progress thanks to the establishment of the aberschisma |
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{{Infobox Interval | |||
| Name = septimal narrow major third, marvel-ptolemaic major third, aberschismic diminished fourth | |||
| Color name = zg4, zogu 4th | |||
| 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]]. | |||
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 == | |||
* [[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 19:32, 30 July 2026
| Interval information |
marvel-ptolemaic major third,
aberschismic diminished fourth
[sound info]
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 | 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 |