6/5: Difference between revisions

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{{Wikipedia|Minor third}}
{{Wikipedia|Minor third}}
In [[5-limit]] [[just intonation]], '''6/5''' is the '''just minor third''', '''classic(al) minor third''', or '''ptolemaic minor third'''<ref>For reference, see [[5/4]]. </ref>, measuring about 315.6[[cent|¢]]. It is sharp of the [[Pythagorean]] minor third of [[32/27]] (about 294.1¢) as well as the 300¢ minor third of [[4edo]], [[12edo]] and all other 4n-[[EDO|edos]]. It arises in the [[harmonic series]] between the 5th and 6th harmonics and appears in the [[5-limit]] otonal triad of 4:5:6. A 5-limit minor triad in just intonation can be written 10:12:15, with 6/5 falling between 10 and 12, [[5/4]] falling between 12 and 15, and [[3/2]] falling between 10 and 15.
In [[5-limit]] [[just intonation]], '''6/5''' is the '''just minor third''', '''classic(al) minor third''', or '''ptolemaic minor third'''<ref>For reference, see [[5-limit]]. </ref>, measuring about 315.6[[cent|¢]]. It is sharp of the [[Pythagorean]] minor third of [[32/27]] (about 294.1¢) as well as the 300¢ minor third of [[4edo]], [[12edo]] and all other 4n-[[EDO|edos]]. It arises in the [[harmonic series]] between the 5th and 6th harmonics and appears in the [[5-limit]] otonal triad of 4:5:6. A 5-limit minor triad in just intonation can be written 10:12:15, with 6/5 falling between 10 and 12, [[5/4]] falling between 12 and 15, and [[3/2]] falling between 10 and 15.


In higher-limit JI, 6/5 is only one of many minor thirds. A popular one in the [[7-limit]] is [[7/6]] (about 266.9¢), the septimal subminor third, which is [[36/35]] (about 48.8¢) flat of 6/5. Another in the [[13-limit]] is [[13/11]] (about 289.2¢), which is [[66/65]] (about 26.4¢) flat of 6/5. Both of these are more complex intervals than 6/5 and have their own character to them.
In higher-limit JI, 6/5 is only one of many minor thirds. A popular one in the [[7-limit]] is [[7/6]] (about 266.9¢), the septimal subminor third, which is [[36/35]] (about 48.8¢) flat of 6/5. Another in the [[13-limit]] is [[13/11]] (about 289.2¢), which is [[66/65]] (about 26.4¢) flat of 6/5. Both of these are more complex intervals than 6/5 and have their own character to them.

Revision as of 10:32, 15 August 2023

Interval information
Ratio 6/5
Factorization 2 × 3 × 5-1
Monzo [1 1 -1⟩
Size in cents 315.6413¢
Names just minor third,
classic(al) minor third,
ptolemaic minor third
Color name g3, gu 3rd
FJS name [math]\displaystyle{ \text{m3}_{5} }[/math]
Special properties superparticular,
reduced
Tenney norm (log2 nd) 4.90689
Weil norm (log2 max(n, d)) 5.16993
Wilson norm (sopfr(nd)) 10

[sound info]
Open this interval in xen-calc
English Wikipedia has an article on:

In 5-limit just intonation, 6/5 is the just minor third, classic(al) minor third, or ptolemaic minor third[1], measuring about 315.6¢. It is sharp of the Pythagorean minor third of 32/27 (about 294.1¢) as well as the 300¢ minor third of 4edo, 12edo and all other 4n-edos. It arises in the harmonic series between the 5th and 6th harmonics and appears in the 5-limit otonal triad of 4:5:6. A 5-limit minor triad in just intonation can be written 10:12:15, with 6/5 falling between 10 and 12, 5/4 falling between 12 and 15, and 3/2 falling between 10 and 15.

In higher-limit JI, 6/5 is only one of many minor thirds. A popular one in the 7-limit is 7/6 (about 266.9¢), the septimal subminor third, which is 36/35 (about 48.8¢) flat of 6/5. Another in the 13-limit is 13/11 (about 289.2¢), which is 66/65 (about 26.4¢) flat of 6/5. Both of these are more complex intervals than 6/5 and have their own character to them.

Approximation by edos

6/5 is very accurately approximated by 19edo (5\19), and hence the enneadecal temperament.

The following edos (up to 200) contain good approximations[2] of the interval 6/5. Errors are given by magnitude, the arrows in the table show if the edo representation is sharp (↑) or flat (↓).

Edo deg\edo Absolute
error (¢) 		Relative
error (r¢) 		↕ Equally acceptable multiples [3]
15 4\15 4.3587 5.4484 ↑
19 5\19 0.1482 0.2346 ↑

10\38, 15\57, 20\76, 25\95, 30\114, 35\133, 40\152, 45\171, 50\190

23 6\23 2.5978 4.9791 ↓
34 9\34 2.0058 5.683 ↑
42 11\42 1.3556 4.7445 ↓
53 14\53 1.3398 5.9176 ↑
61 16\61 0.8872 4.5099 ↓
72 19\72 1.0254 6.1523 ↑
80 21\80 0.6413 4.2752 ↓
91 24\91 0.8422 6.3869 ↑
99 26\99 0.4898 4.0406 ↓
110 29\110 0.7223 6.6215 ↑
118 31\118 0.387 3.806 ↓
129 34\129 0.6378 6.8562 ↑
137 36\137 0.3128 3.5714 ↓
156 41\156 0.2567 3.3367 ↓
175 46\175 0.2127 3.1021 ↓
194 51\194 0.1774 2.8675 ↓

See also

Notes

  1. ↑ For reference, see 5-limit.
  2. ↑ error magnitude below 7, both, absolute (in ¢) and relative (in r¢)
  3. ↑ Super-edos up to 200 within the same error tolerance
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