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''', '''classic(al)''', 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.


== Approximation by edos ==
== Approximation ==
6/5 is very accurately approximated by [[19edo]] (5\19), and hence the [[enneadecal]] temperament.
6/5 is very accurately approximated by [[19edo]] (5\19), and hence the [[enneadecal]] temperament.
 
{{Interval edo approximation}}
The following [[edo]]s (up to 200) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 6/5. Errors are given by magnitude, the arrows in the table show if the edo representation is sharp (&uarr;) or flat (&darr;).
 
{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[Edo]]
! class="unsortable" | deg\edo
! Absolute <br> error ([[Cent|¢]])
! Relative <br> error ([[Relative cent|r¢]])
! &#8597;
! class="unsortable" | Equally acceptable multiples <ref>Super-edos up to 200 within the same error tolerance</ref>
|-
| [[15edo|15]] || 4\15 || 4.3587 || 5.4484 || &uarr; ||
 
|-
| [[19edo|19]] || 5\19 || 0.1482 || 0.2346 || &uarr; ||
[[38edo|10\38]], [[57edo|15\57]], [[76edo|20\76]], [[95edo|25\95]], [[114edo|30\114]], [[133edo|35\133]], [[152edo|40\152]], [[171edo|45\171]], [[190edo|50\190]]
 
|-
| [[23edo|23]] || 6\23 || 2.5978 || 4.9791 || &darr; ||
 
|-
| [[34edo|34]] || 9\34 || 2.0058 || 5.683 || &uarr; ||
 
|-
| [[42edo|42]] || 11\42 || 1.3556 || 4.7445 || &darr; ||
 
|-
| [[53edo|53]] || 14\53 || 1.3398 || 5.9176 || &uarr; ||
 
|-
| [[61edo|61]] || 16\61 || 0.8872 || 4.5099 || &darr; ||
 
|-
| [[72edo|72]] || 19\72 || 1.0254 || 6.1523 || &uarr; ||
 
|-
| [[80edo|80]] || 21\80 || 0.6413 || 4.2752 || &darr; ||
 
|-
| [[91edo|91]] || 24\91 || 0.8422 || 6.3869 || &uarr; ||
 
|-
| [[99edo|99]] || 26\99 || 0.4898 || 4.0406 || &darr; ||
 
|-
| [[110edo|110]] || 29\110 || 0.7223 || 6.6215 || &uarr; ||
 
|-
| [[118edo|118]] || 31\118 || 0.387 || 3.806 || &darr; ||
 
|-
| [[129edo|129]] || 34\129 || 0.6378 || 6.8562 || &uarr; ||
 
|-
| [[137edo|137]] || 36\137 || 0.3128 || 3.5714 || &darr; ||
 
|-
| [[156edo|156]] || 41\156 || 0.2567 || 3.3367 || &darr; ||
 
|-
| [[175edo|175]] || 46\175 || 0.2127 || 3.1021 || &darr; ||
 
|-
| [[194edo|194]] || 51\194 || 0.1774 || 2.8675 || &darr; ||
|}


== See also ==  
== See also ==  
Line 91: Line 26:
[[Category:Third]]
[[Category:Third]]
[[Category:Minor third]]
[[Category:Minor third]]
[[Category:Over-5]]
[[Category:Over-5 intervals]]

Latest revision as of 17:29, 6 November 2025

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

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

Edo approximations for 6/5 (315.64 ¢)
≤ 80edo, relative error ≤ 10%
Edo Step size Cents (¢) Absolute error (¢) Relative error (%)
4 1\4 300.00 -15.64 -5.21
15 4\15 320.00 +4.36 +5.45
19 5\19 315.79 +0.15 +0.23
23 6\23 313.04 -2.60 -4.98
34 9\34 317.65 +2.01 +5.68
38 10\38 315.79 +0.15 +0.47
42 11\42 314.29 -1.36 -4.74
46 12\46 313.04 -2.60 -9.96
53 14\53 316.98 +1.34 +5.92
57 15\57 315.79 +0.15 +0.70
61 16\61 314.75 -0.89 -4.51
65 17\65 313.85 -1.80 -9.72
72 19\72 316.67 +1.03 +6.15
76 20\76 315.79 +0.15 +0.94
80 21\80 315.00 -0.64 -4.28

See also

Notes

  1. ↑ For reference, see 5-limit.
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