Editing Major and minor (section)

==Intonation and tuning==
Musical tuning of intervals is expressed by the ratio between the pitches' frequencies. Simple fractions can sound more harmonious than complex fractions; for instance, an [[octave]] is a simple 2:1&nbsp;ratio and a [[Perfect fifth|fifth]] is the relatively simple 3:2&nbsp;ratio. The table below gives frequency ratios that are mathematically exact for [[just intonation]], which [[meantone temperament]]s seek to approximate.
:{| class="wikitable"  style="text-align:center;vertical-align:center;"
|-
| '''Note name'''
| '''C''' || '''D''' || '''E''' || '''F''' || '''G''' || '''A''' || '''B''' || '''C'''′  
|-
| '''frequency ratio'''<br/>([[just intonation|just int.]])
| {{big|{{sfrac| 1 | 1 }} }} || {{big|{{sfrac| 9 | 8 }} }} || {{big|{{sfrac| 5 | 4 }} }} || {{big|{{sfrac| 4 | 3 }} }} || {{big|{{sfrac| 3 | 2 }} }} || {{big|{{sfrac| 5 | 3 }} }} || {{big|{{sfrac| 15 | 8 }} }} || {{big|{{sfrac| 2 | 1 }} }}
|-
| '''Interval name'''<br/>(from '''C''')
| {{small|perf&thinsp;}}{{big|1}}{{sup|&thinsp;st}} || {{sup|Maj&thinsp;}}{{big|2}}{{sup|&thinsp;nd}} || {{sup|Maj&thinsp;}}{{big|3}}{{sup|&thinsp;rd}} || {{small|perf&thinsp;}}{{big|4}}{{sup|&thinsp;th}} || {{small|perf&thinsp;}}{{big|5}}{{sup|&thinsp;th}} || {{sup|Maj&thinsp;}}{{big|6}}{{sup|&thinsp;th}} || {{sup|Maj&thinsp;}}{{big|7}}{{sup|&thinsp;th}} || {{small|perf&thinsp;}}{{big|8}}{{sup|&thinsp;th}}
|-
| '''Interval size'''<br/>(in [[musical cents|cents]])
| &nbsp; {{0}}{{0}}0{{0}}¢{{0}} &nbsp; || &nbsp; {{0}}203.9¢ &nbsp; || &nbsp; {{0}}386.3¢ &nbsp; || &nbsp; {{0}}498.0¢ &nbsp; || &nbsp; {{0}}702.0¢ &nbsp; || &nbsp; {{0}}884.4¢ &nbsp; || &nbsp; 1088.3¢ &nbsp; || &nbsp; 1200{{0}}¢ &nbsp; 
|}

In [[just intonation]], a minor chord is often (but not exclusively) tuned in the frequency ratio 10:12:15 ({{Audio|Just minor triad on C.mid|play}}). In [[12 tone equal temperament|12&nbsp;tone equal temperament]] {{nobr|(12 {{sc|TET}},}} at present the most common tuning system in the West) a minor chord has 3&nbsp;[[semitone]]s between the root and third, 4&nbsp;between the third and fifth, and 7&nbsp;between the root and fifth.

In {{nobr|12 {{sc|TET}},}} the perfect fifth (700&nbsp;[[cent (music)|cents]]) is only about two cents narrower than the justly tuned perfect fifth (3:2, or 702.0&nbsp;cents), but the minor third (300 cents) is noticeably (about 16&nbsp;cents) narrower than the just minor third (6:5, or 315.6&nbsp;cents). Moreover, the minor third (300&nbsp;cents) more closely approximates the [[19-limit]] ([[Limit (music)|Limit]]) minor third (19:16 {{audio|19th harmonic on C.mid|Play}} or, 297.5&nbsp;cents, the nineteenth [[harmonic]]) with only about a 2&nbsp;cent error.<ref name=Ellis-Helmhz-1954>A.J. Ellis, writing in {{cite book |author2-link=Alexander John Ellis |first2=A.J. |last2=Ellis |translator=[[Alexander John Ellis|Ellis, A.J.]] |author1-link=Hermann von Helmholtz |first1=H.L. |last1=von&nbsp;Helmholtz |title-link=Sensations of Tone |title=On the Sensations of Tone as a Physiological Basis for the Theory of Music |page=455 |publisher=Dover Publications |place=New York, NY |year=1954 |edition=reprint}}</ref>

[[Alexander John Ellis|A.J. Ellis]] proposed that the conflict between mathematicians and physicists on one hand and practicing musicians on the other regarding the supposed inferiority of the minor chord and scale to the major may be explained due to physicists' comparison of just minor and just major triads, in which case minor comes out the loser, versus the musicians' comparison of the equal tempered triads, in which case minor comes out the winner, since the {{nobr|12 {{sc|TET}} }} major third is about 14&nbsp;cents sharp from the just major third (5:4, or 386.3&nbsp;cents), but only about 4&nbsp;cents narrower than the 19&nbsp;limit major third (24:19, or 404.4&nbsp;cents); while the {{nobr|12 {{sc|TET}} }} minor third closely approximates the 19:16&nbsp;minor third which many find pleasing.<ref name=Ellis-Helmhz-1954/>{{rp|style=ama|p=298}}{{efn|In the 16th through 18th&nbsp;centuries, prior to 12&nbsp;TET, the minor third in [[Quarter-comma meantone#Construction of the chromatic scale|meantone temperament]] was 310&nbsp;cents {{audio|Quarter-comma meantone minor third on C.mid|Play}} and much rougher than the 300&nbsp;cent {{nobr|12 {{sc|TET}} }} minor third.<ref name=Ellis-Helmhz-1954/>{{rp|style=ama|p=298}} }}