Major third: Difference between revisions

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{{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{{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.   

3-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{{c}}.

* 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{{c}}.

* The 13-limit '''(tridecimal) ultramajor third''' is a ratio of [[13/10]], and is about 454{{c}}.

** There is also a 13-limit '''(tridecimal) submajor third''', which is a ratio of [[26/21]], and is about 370{{c}}.

* The 17-limit '''(septendecimal) submajor third''' is a ratio of [[21/17]], and is about 366{{c}}.

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]].

! EDO

! Other major thirds

| colspan="2" | 400{{c}}

| *

|  

| **

| 424{{c}}

|  

|  

|  

|  

|  

|  

| 388{{c}}

|  

| 424{{c}}

| {{nowrap|459{{c}} ≈ 13/10}}

| {{nowrap|362{{c}} ≈ 21/17|408{{c}} ≈ 81/64|452{{c}} ≈ 13/10}}

<nowiki />* These edos have an approximation to 9/7, but it's sharper than 460{{c}}, 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.

The two simplest major 3rd ratios are 5/4 and 9/7. The following notable temperaments are generated by them:

* [[Sensi]], generated by sharp supermajor thirds representing [[9/7]] and [[13/10]], such that a stack of two gives a major sixth approximating [[5/3]].