In [[13-limit]] [[Just Intonation]], 13/11 is the tridecimal minor third (or [[Neo-Gothic]] minor third), measuring about 289.2¢. It is the difference between the 11th and 13th harmonics. The (octave-reduced) 11th harmonic ([[11_8|11/8]], about 551.3¢) and 13th harmonic ([[13_8|13/8]], about 840.5¢) are both quite xenharmonic and demand new interval categories, while 13/11 can be likened unto some kind of relatively complex minor third. It can even function as such in a 13-limit Neo-Gothic minor triad of 22:26:33, with a [[3_2|3/2]] perfect fifth between 33 and 22. Compare this to 22:26:32 (11:13:16), which has the much more dissonant [[16_11|16/11]] as the outside interval in place of 3/2. The latter triad sounds more like a xenharmonic version of a diminished triad, and could not be confused with simpler diminished triads such as 5:6:7.
In [[13-limit|13-limit]] [[Just_intonation|Just Intonation]], 13/11 is the tridecimal minor third (or [[Neo-Gothic|Neo-Gothic]] minor third), measuring about 289.2¢. It is the difference between the 11th and 13th harmonics. The (octave-reduced) 11th harmonic ([[11/8|11/8]], about 551.3¢) and 13th harmonic ([[13/8|13/8]], about 840.5¢) are both quite xenharmonic and demand new interval categories, while 13/11 can be likened unto some kind of relatively complex minor third. It can even function as such in a 13-limit Neo-Gothic minor triad of 22:26:33, with a [[3/2|3/2]] perfect fifth between 33 and 22. Compare this to 22:26:32 (11:13:16), which has the much more dissonant [[16/11|16/11]] as the outside interval in place of 3/2. The latter triad sounds more like a xenharmonic version of a diminished triad, and could not be confused with simpler diminished triads such as 5:6:7.
13/11 is the classic [[mediant|mediant]] between the simpler and more familiar ratios [[6/5|6/5]] and [[7/6|7/6]], as it can be given as (6+7)/(5+6). This puts in between the latter ratios, slightly closer to 7/6. More complex minor thirds can be generated by taking the mediant between 13/11 and 7/6 (which yields (13+7)/(11+6) = [[20/17|20/17]], the septendecimal subminor third, about 281.4¢) and between 13/11 and 6/5 (which yields (13+6)/(11+5) = [[19/16|19/16]], the overtone minor third of [[19-limit|19-limit]] JI, about 297.5¢). (See the diagram below.)
{| class="wikitable"
|-
! | subminor and minor third
| style="text-align:center;" | 7/6
266.9¢
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" | 6/5
315.6¢
|-
! | interval in between
| style="text-align:center;" |
| style="text-align:center;" | <<
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" | [[36/35|36:35]]
48.7¢
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" | >>
| |
|-
! | add mediant (13/11)
| style="text-align:center;" | 7/6
266.9¢
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" | 13/11
289.2¢
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" | 6/5
315.6¢
|-
! | intervals in between
| style="text-align:center;" |
| style="text-align:center;" | <<
| style="text-align:center;" | [[78/77|78:77]]
22.3¢
| style="text-align:center;" | >>
| |
| style="text-align:center;" | <<
| style="text-align:center;" | [[66/65|66:65]]
26.4¢
| style="text-align:center;" | >>
| |
|-
! | add mediants (20/17 and 19/16)
| style="text-align:center;" | 7/6
266.9¢
| style="text-align:center;" |
| style="text-align:center;" | 20/17
281.4¢
| style="text-align:center;" |
| style="text-align:center;" | '''13/11'''
'''289.2¢'''
| style="text-align:center;" |
| style="text-align:center;" | 19/16
13/11 is the classic [[mediant]] between the simpler and more familiar ratios [[6_5|6/5]] and [[7_6|7/6]], as it can be given as (6+7)/(5+6). This puts in between the latter ratios, slightly closer to 7/6. More complex minor thirds can be generated by taking the mediant between 13/11 and 7/6 (which yields (13+7)/(11+6) = [[20_17|20/17]], the septendecimal subminor third, about 281.4¢) and between 13/11 and 6/5 (which yields (13+6)/(11+5) = [[19_16|19/16]], the overtone minor third of [[19-limit]] JI, about 297.5¢). (See the diagram below.)
297.5¢
| style="text-align:center;" |
| style="text-align:center;" | 6/5
||~ subminor and minor third ||= 7/6
315.6¢
266.9¢ ||= ||= ||= ||= ||= ||= ||= ||= 6/5
|-
315.6¢ ||
! | intervals in between
||~ interval in between ||= ||= << ||= ||= ||= [[36_35|36:35]]
In <a class="wiki_link" href="/13-limit">13-limit</a> <a class="wiki_link" href="/Just%20Intonation">Just Intonation</a>, 13/11 is the tridecimal minor third (or <a class="wiki_link" href="/Neo-Gothic">Neo-Gothic</a> minor third), measuring about 289.2¢. It is the difference between the 11th and 13th harmonics. The (octave-reduced) 11th harmonic (<a class="wiki_link" href="/11_8">11/8</a>, about 551.3¢) and 13th harmonic (<a class="wiki_link" href="/13_8">13/8</a>, about 840.5¢) are both quite xenharmonic and demand new interval categories, while 13/11 can be likened unto some kind of relatively complex minor third. It can even function as such in a 13-limit Neo-Gothic minor triad of 22:26:33, with a <a class="wiki_link" href="/3_2">3/2</a> perfect fifth between 33 and 22. Compare this to 22:26:32 (11:13:16), which has the much more dissonant <a class="wiki_link" href="/16_11">16/11</a> as the outside interval in place of 3/2. The latter triad sounds more like a xenharmonic version of a diminished triad, and could not be confused with simpler diminished triads such as 5:6:7.<br />
<br />
13/11 is the classic <a class="wiki_link" href="/mediant">mediant</a> between the simpler and more familiar ratios <a class="wiki_link" href="/6_5">6/5</a> and <a class="wiki_link" href="/7_6">7/6</a>, as it can be given as (6+7)/(5+6). This puts in between the latter ratios, slightly closer to 7/6. More complex minor thirds can be generated by taking the mediant between 13/11 and 7/6 (which yields (13+7)/(11+6) = <a class="wiki_link" href="/20_17">20/17</a>, the septendecimal subminor third, about 281.4¢) and between 13/11 and 6/5 (which yields (13+6)/(11+5) = <a class="wiki_link" href="/19_16">19/16</a>, the overtone minor third of <a class="wiki_link" href="/19-limit">19-limit</a> JI, about 297.5¢). (See the diagram below.)<br />
<br />
13/11 is also [[352/351|352/351]] (about 4.9¢) narrower than [[32/27|32/27]], the minor third in Pythagorean ([[3-limit|3-limit]]) tuning.
<table class="wiki_table">
See: [[Gallery_of_Just_Intervals|Gallery of Just Intonation Intervals]], [[gentle_chords|gentle chords]], [[List_of_root-3rd-P5_triads_in_JI|List of root-3rd-P5 triads in JI]]
[http://dkeenan.com/Music/NobleMediant.txt The Noble Mediant] (earliest description of 13:11 as the "Neo-Gothic" minor third)
<br />
[[Category:minor_third]]
13/11 is also <a class="wiki_link" href="/352_351">352/351</a> (about 4.9¢) narrower than <a class="wiki_link" href="/32_27">32/27</a>, the minor third in Pythagorean (<a class="wiki_link" href="/3-limit">3-limit</a>) tuning.<br />
[[Category:tredecimal]]
<br />
See: <a class="wiki_link" href="/Gallery%20of%20Just%20Intervals">Gallery of Just Intonation Intervals</a>, <a class="wiki_link" href="/gentle%20chords">gentle chords</a>, <a class="wiki_link" href="/List%20of%20root-3rd-P5%20triads%20in%20JI">List of root-3rd-P5 triads in JI</a><br />
<br />
<a class="wiki_link_ext" href="http://dkeenan.com/Music/NobleMediant.txt" rel="nofollow">The Noble Mediant</a> (earliest description of 13:11 as the &quot;Neo-Gothic&quot; minor third)</body></html></pre></div>
In 13-limitJust Intonation, 13/11 is the tridecimal minor third (or Neo-Gothic minor third), measuring about 289.2¢. It is the difference between the 11th and 13th harmonics. The (octave-reduced) 11th harmonic (11/8, about 551.3¢) and 13th harmonic (13/8, about 840.5¢) are both quite xenharmonic and demand new interval categories, while 13/11 can be likened unto some kind of relatively complex minor third. It can even function as such in a 13-limit Neo-Gothic minor triad of 22:26:33, with a 3/2 perfect fifth between 33 and 22. Compare this to 22:26:32 (11:13:16), which has the much more dissonant 16/11 as the outside interval in place of 3/2. The latter triad sounds more like a xenharmonic version of a diminished triad, and could not be confused with simpler diminished triads such as 5:6:7.
13/11 is the classic mediant between the simpler and more familiar ratios 6/5 and 7/6, as it can be given as (6+7)/(5+6). This puts in between the latter ratios, slightly closer to 7/6. More complex minor thirds can be generated by taking the mediant between 13/11 and 7/6 (which yields (13+7)/(11+6) = 20/17, the septendecimal subminor third, about 281.4¢) and between 13/11 and 6/5 (which yields (13+6)/(11+5) = 19/16, the overtone minor third of 19-limit JI, about 297.5¢). (See the diagram below.)