@@ Line 1: / Line 1: @@
-A '''major third (M3)''' in the [[5L 2s|diatonic scale]] is an interval that spans two scale steps with the major (wider) quality. It is generated by stacking 4 fifths [[octave reduction|octave reduced]], and depending on the specific tuning, it ranges from 343 to 480 [[cent]]s ([[7edo|2\7]] to [[5edo|2\5]]).
+A '''major third (M3)''' in the [[5L 2s|diatonic scale]] is an interval that spans two scale steps with the major (wider) quality. It is generated by stacking 4 fifths [[octave reduction|octave reduced]], and depending on the specific tuning, it ranges from 343 to 480{{cent}} ([[7edo|2\7]] to [[5edo|2\5]]).
 In [[just intonation]], an interval may be classified as a major third if it is reasonably mapped to 2\7 and [[24edo|8\24]] (precisely two steps of the diatonic scale and four steps of the chromatic scale). The use of 24edo's 8\24 as the mapping criteria here rather than [[12edo]]'s 4\12 better captures the characteristics of many intervals in the [[11-limit|11-]] and [[13-limit]].
-As a concrete [[interval region]], it is typically near 400 [[cents]] in size, distinct from the [[minor third]] of roughly 300 cents and the [[neutral third]] of roughly 350 cents. A rough tuning range for the major third is about 370 to 440 cents according to [[Margo Schulter]]'s theory of interval regions. ''Major third'' in this sense refers both to the ~350-450 cent range as a whole, and to a specific subdivision within it (~370–415 cents) as opposed to supermajor thirds; major thirds sharp of this are often called "supermajor thirds".
+As a concrete [[interval region]], it is typically near 400{{c}} in size, distinct from the [[minor third]] of roughly 300{{c}} and the [[neutral third]] of roughly 350{{c}}. A rough tuning range for the major third is about 370 to 440{{c}} according to [[Margo Schulter]]'s theory of interval regions. ''Major third'' in this sense refers both to the ~350–450{{c}} range as a whole, and to a specific subdivision within it (~370–415{{c}}) as opposed to supermajor thirds; major thirds sharp of this are often called "supermajor thirds".
-This article covers intervals between 360 and 460 cents. The outer range of this might be too extreme to call "major thirds", but this is done so that one can find what they're looking for easily.
+This article covers intervals between 360 and 460{{c}}. The outer range of this might be too extreme to call "major thirds", but this is done so that one can find what they're looking for easily.
 == In just intonation ==
 === By prime limit ===
--limit intervals in the range of major thirds include the '''Pythagorean major third''' of [[81/64]], about 408 cents in size, which corresponds to the mos-based interval category of the diatonic major third and is generated by [[stacking]] four just perfect fifths of [[3/2]], and the '''Pythagorean diminished fourth''' of [[8192/6561]], which is flat of 81/64 by one Pythagorean comma, and is about 384 cents in size.
+-limit intervals in the range of major thirds include the '''Pythagorean major third''' of [[81/64]], 407.8{{c}} in size, which corresponds to the mos-based interval category of the diatonic major third and is generated by [[stacking]] four just perfect fifths of [[3/2]], and the '''Pythagorean diminished fourth''' of [[8192/6561]], which is flat of 81/64 by one Pythagorean comma, and is about 384{{c}} in size.
 Much [[odd limit|simpler]] major thirds exist in higher [[prime limit|limits]], however, for example:
-* The 5-limit '''classical major third''' is a ratio of [[5/4]], and is about 386 cents.
+* The 5-limit '''classical major third''' is a ratio of [[5/4]], and is about 386{{c}}.
-* The 7-limit '''supermajor third''' is a ratio of [[9/7]], and is about 435 cents.
+* The 7-limit '''(septimal) supermajor third''' is a ratio of [[9/7]], and is about 435{{c}}.
-* The 11-limit '''neogothic major third''' is a ratio of [[14/11]], and is about 418 cents.
+* The 11-limit '''neogothic major third''' is a ratio of [[14/11]], and is about 418{{c}}.
-* The 13-limit '''ultramajor third''' is a ratio of [[13/10]], and is about 454 cents.
+* The 13-limit '''(tridecimal) ultramajor third''' is a ratio of [[13/10]], and is about 454{{c}}.
-** There is also a 13-limit '''submajor third''', which is a ratio of [[26/21]], and is about 370 cents.
+** There is also a 13-limit '''(tridecimal) submajor third''', which is a ratio of [[26/21]], and is about 370{{c}}.
-* The 17-limit '''submajor third''' is a ratio of [[21/17]], and is about 366 cents.
+* The 17-limit '''(septendecimal) submajor third''' is a ratio of [[21/17]], and is about 366{{c}}.
 === By delta ===
 {| class="wikitable"
-|+
-!Delta 1
-!Cents
-!Delta 2
-!Cents
-!Delta 3
-!Cents
-!Delta 4
-!Cents
-!Delta 5
-!Cents
 |-
-|5/4
+! Delta 1
-|386c
+! Cents
-|9/7
+! Delta 2
-|435c
+! Cents
-|13/10
+! Delta 3
-|454c
+! Cents
-|19/15
+! Delta 4
-|409c
+! Cents
-|22/17
+! Delta 5
-|446c
+! Cents
 |-
-|
+| 5/4
-|
+| 386{{c}}
-|
+| 9/7
-|
+| 435{{c}}
-|14/11
+| 13/10
-|418c
+| 454{{c}}
-|21/17
+| 19/15
-|366c
+| 409{{c}}
-|23/18
+| 22/17
-|424c
+| 446{{c}}
 |-
 |
 |
 |
 |
-|
+| 14/11
-|
+| 418{{c}}
-|
+| 21/17
-|
+| 366{{c}}
-|24/19
+| 23/18
-|404c
+| 424{{c}}
 |-
 |
 |
 |
 |
 |
 |
 |
 |
-|26/21
+| 24/19
-|370c
+| 404{{c}}
+|-
+|
+|
+|
+|
+|
+|
+|
+|
+| 26/21
+| 370{{c}}
 |}
 == In edos ==
 The following table lists the best tuning of 5/4 and 9/7, as well as other major thirds if present, in various significant [[edos]].
 {| class="wikitable"
-|+
-!EDO
-!5/4
-!9/7
-!Other major thirds
 |-
-|12
+! EDO
-| colspan="2" |400c
+! 5/4
-|
+! 9/7
+! Other major thirds
 |-
-|15
+| 12
-|400c
+| colspan="2" | 400{{c}}
-|**
+|
-|
 |-
-|16
+| 15
-|375c
+| 400{{c}}
-|450c
+| *
 |
 |-
-|17
+| 16
-|***
+| 375{{c}}
-|424c
+| 450{{c}}
 |
 |-
-|19
+| 17
-|379c
+| **
-|442c
+| 424{{c}}
 |
 |-
-|22
+| 19
-|382c
+| 379{{c}}
-|436c
+| 442{{c}}
 |
 |-
-|24
+| 22
-|400c
+| 382{{c}}
-|450c
+| 436{{c}}
 |
 |-
-|25
+| 24
-|384c
+| 400{{c}}
-|432c
+| 450{{c}}
 |
 |-
-|26
+| 25
-|369c
+| 384{{c}}
-|415c
+| 432{{c}}
 |
 |-
-|27
+| 26
-|400c
+| 369{{c}}
-|444c
+| 415{{c}}
 |
 |-
-|29
+| 27
-|372c
+| 400{{c}}
-|455c
+| 444{{c}}
-|414c '''≈''' 81/64, 14/11
+|
 |-
-|31
+| 29
-|388c
+| 372{{c}}
-|426c
+| 455{{c}}
-|
+| 414{{c}} '''≈''' 81/64, 14/11
 |-
-|34
+| 31
-|388c
+| 388{{c}}
-|424c
+| 426{{c}}
-|459c '''≈''' 13/10
+|
 |-
-|41
+| 34
-|381c
+| 388{{c}}
-|439c
+| 424{{c}}
-|410c '''≈''' 81/64
+| 459{{c}} '''≈''' 13/10
 |-
-|53
+| 41
-|385c
+| 381{{c}}
-|430c
+| 439{{c}}
-|362c '''≈''' 21/17, 408c '''≈''' 81/64, 452c '''≈''' 13/10
+| 410{{c}} '''≈''' 81/64
+|-
+| 53
+| 385{{c}}
+| 430{{c}}
+| 362{{c}} '''≈''' 21/17, 408{{c}} '''≈''' 81/64, 452{{c}} '''≈''' 13/10
 |}
-<nowiki>**</nowiki> These edos have an approximation to 9/7, but it's sharper than 460 cents, not really a major third.
+<nowiki />* These edos have an approximation to 9/7, but it's sharper than 460{{c}}, not really a major third.
-<nowiki>***</nowiki> These edos have an approximation to 5/4, but it's flatter than 360 cents, not really a major third.
+<nowiki />** These edos have an approximation to 5/4, but it's flatter than 360{{c}}, not really a major third.
 == In regular temperaments ==
@@ Line 170: / Line 171: @@
 === Temperaments that use 5/4 as a generator ===
 * [[Magic]], which generates 3/2 by stacking five 5/4s (octave-reduced).
 * [[Augmented (temperament)|Augmented]], which splits the octave into three equal parts, each representing [[5/4]].
@@ Line 177: / Line 177: @@
 === Temperaments that use 9/7 as a generator ===
+* [[Sensi]], generated by sharp major thirds representing [[9/7]] and [[13/10]], such that a stack of two gives a major sixth approximating [[5/3]].
-* [[Sensi]], which equates [[5/3]] with a stack of two sharp major thirds representing [[9/7]].
+* [[Squares]], generated by sharp major thirds representing [[9/7]] and [[14/11]].
 {{Navbox intervals}}

Major third: Difference between revisions