6/5: Difference between revisions

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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.

== Approximation ~~by edos~~ ==

== Approximation ==

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

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 (↑) or flat (↓).

{~~| 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¢]])~~

~~! ↕~~

~~! 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 || ↑ ||~~

|-

~~| [[19edo|19]] || 5\19 || 0.1482 || 0.2346 || ↑ ||~~

~~[[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 || ↓ ||~~

|-

~~| [[34edo|34]] || 9\34 || 2.0058 || 5.683 || ↑ ||~~

|-

~~| [[42edo|42]] || 11\42 || 1.3556 || 4.7445 || ↓ ||~~

|-

~~| [[53edo|53]] || 14\53 || 1.3398 || 5.9176 || ↑ ||~~

|-

~~| [[61edo|61]] || 16\61 || 0.8872 || 4.5099 || ↓ ||~~

|-

~~| [[72edo|72]] || 19\72 || 1.0254 || 6.1523 || ↑ ||~~

|-

~~| [[80edo|80]] || 21\80 || 0.6413 || 4.2752 || ↓ ||~~

|-

~~| [[91edo|91]] || 24\91 || 0.8422 || 6.3869 || ↑ ||~~

|-

~~| [[99edo|99]] || 26\99 || 0.4898 || 4.0406 || ↓ ||~~

|-

~~| [[110edo|110]] || 29\110 || 0.7223 || 6.6215 || ↑ ||~~

|-

~~| [[118edo|118]] || 31\118 || 0.387 || 3.806 || ↓ ||~~

|-

~~| [[129edo|129]] || 34\129 || 0.6378 || 6.8562 || ↑ ||~~

|-

~~| [[137edo|137]] || 36\137 || 0.3128 || 3.5714 || ↓ ||~~

|-

~~| [[156edo|156]] || 41\156 || 0.2567 || 3.3367 || ↓ ||~~

|-

~~| [[175edo|175]] || 46\175 || 0.2127 || 3.1021 || ↓ ||~~

|-

~~| [[194edo|194]] || 51\194 || 0.1774 || 2.8675 || ↓ ||~~

|}

== See also ==

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[[Category:Third]]

[[Category:Minor third]]

[[Category:Over-5]]

[[Category:Over-5 intervals]]

@@ Line 5: / Line 5: @@
 }}
 {{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.
-== Approximation by edos ==
+== Approximation ==
 /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 ==
@@ Line 91: / Line 26: @@
 [[Category:Third]]
 [[Category:Minor third]]
-[[Category:Over-5]]
+[[Category:Over-5 intervals]]