56/45: Difference between revisions
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'''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'''. | '''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'''. | ||
According to [[Ozan Yarman]], the perde segah of [[Arabic, Turkish, Persian music|Turkish maqam music]] should range from 56/45 to 5/4. | According to [[Ozan Yarman]], the perde segah of [[Arabic, Turkish, Persian music|Turkish maqam music]] should range from 56/45 to 5/4. | ||
Latest revision as of 04:09, 11 June 2026
| Interval information |
marvel-ptolemaic major third,
hemifamity diminished fourth
[sound info]
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 | 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 |