Editing Major third (section)

== Harmonic and non-harmonic thirds ==
The major third may be derived from the [[harmonic series (music)|harmonic series]] as the interval between the fourth and fifth harmonics. 
The [[major scale]] is so named because of the presence of this interval between its [[tonic (music)|tonic]] and [[mediant]] (1st and 3rd) [[scale degrees]]. 
The [[major chord]] also takes its name from the presence of this interval built on the chord's [[root (chord)|root]] (provided that the interval of a [[perfect fifth]] from the root is also present).

A major third is slightly different in different [[musical tuning]]s: In [[just intonation]] it corresponds to a pitch ratio of 5:4, or {{small|{{math|{{sfrac| 5 | 4 }}}}}} ({{Audio|Just major third on C.mid|play}}) (fifth harmonic in relation to the fourth) or 386.31&nbsp;[[musical cents|cents]]; in [[12 equal temperament|12&nbsp;tone equal temperament]], a major third is equal to four [[semitone]]s, a ratio of 2<sup>1/3</sup>:1 (about 1.2599) or 400&nbsp;cents, 13.69&nbsp;[[cent (music)|cent]]s wider than the 5:4 ratio. The older concept of a "[[ditone]]" (two 9:8 major seconds) made a dissonant, wide major third with the ratio 81:64 (about 1.2656) or 408&nbsp;cents ({{Audio|Pythagorean major third on C.mid|play}}), about [[syntonic comma|22&nbsp;cents]] sharp from the harmonic ratio of 5:4&nbsp;. The [[septimal major third]] is 9:7 (435&nbsp;cents), the '''undecimal major third''' is 14:11 (418&nbsp;cents), and the '''tridecimal major third''' is 13:10 (452&nbsp;cents).

In 12&nbsp;tone equal temperament {{nobr|([[12 equal temperament|12&thinsp;{{sc|TET}}]])}} three major thirds in a row are equal to an octave. For example, A{{sup|{{music|flat}}}} to C, C to E, and E to G{{sup|{{music|sharp}}}} (in {{nobr|[[12 equal temperament|12&thinsp;{{sc|TET}}]],}} the differently written notes G{{sup|{{music|sharp}}}} and A{{sup|{{music|flat}}}} both represent the same pitch, but ''not'' in most other [[tuning (music)|tuning systems]]). This is sometimes called the "[[circle of thirds]]". In just intonation, however, three 5:4 major third, the 125th [[subharmonic]], is less than an octave. For example, three 5:4 major thirds from C is B{{sup|{{music|sharp}}}} (C to E, to G{{sup|{{music|sharp}}}}, to B{{sup|{{music|sharp}}}}) ({{sfrac|  B{{sup|{{music|#}}}} | C }} <math> = \tfrac{\; 5^3 \ }{\; 2^6\ } = \tfrac{\ 125\ }{ 64 }\ </math>). The difference between this just-tuned B{{sup|{{music|sharp}}}} and C, like the interval between G{{sup|{{music|sharp}}}} and A{{sup|{{music|flat}}}}, is called the "enharmonic [[diesis]]", about 41&nbsp;cents, or about two [[syntonic comma|commas]] (the [[inversion (interval)|inversion]] of the interval {{small|{{math|{{sfrac| 125 | 64 }}}} }}: <math>\ \frac{\ 128\ }{ 125 } = \frac{\; 2^7\ }{\; 5^3 }\ </math> ({{audio|5-limit limma on C.mid|play}})).