13/11: Difference between revisions

| Name = tridecimal minor third, major minthmic minor third

In [[13-limit]] [[just intonation]], '''13/11''' is a '''tridecimal minor third''', specifically the '''major minthmic minor third''', measuring about 289.2 [[cent]]s. It is the difference between the [[11/1|11th]] and [[13/1|13th]] [[harmonic]]s. The octave-reduced 11th harmonic ([[11/8]], about 551.3{{c}}) and 13th harmonic ([[13/8]], about 840.5{{c}}) are both quite xenharmonic and demand new interval categories, while 13/11 can be likened unto some kind of relatively complex minor third – it is a [[352/351|major minthma (352/351)]] narrower than the [[32/27|Pythagorean minor third (32/27)]]. It is the simplest [[neogothic major and minor|neogothic minor third]], and can function as such in a 13-limit neogothic minor triad of [[22:26:33]], with a [[3/2]] perfect fifth between 33 and 22<ref group="note">This is a [[minor minthmic chords|minor minthmic chord]] where 13/11 and [[14/11]] sum to a perfect fifth. Shown here is the simplest JI representation. </ref>. 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 it 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{{c}}) 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{{c}}). See the diagram below.

! Subminor and minor third

| 7/6 <br> 266.9{{c}}

| 6/5 <br> 315.6{{c}}

! Interval in between

| [[36/35|36:35]] <br> 48.7{{c}}

! Add mediant (13/11)

| 7/6 <br> 266.9{{c}}

| 13/11 <br> 289.2{{c}}

| 6/5 <br> 315.6{{c}}

! Intervals in between

| [[78/77|78:77]] <br> 22.3{{c}}

| [[66/65|66:65]] <br> 26.4{{c}}

! Add mediants (20/17 and 19/16)

| 7/6 <br> 266.9{{c}}

| [[20/17]] <br> 281.4{{c}}

| '''13/11''' <br> '''289.2{{c}}'''

| [[19/16]] <br> 297.5{{c}}

| 6/5 <br> 315.6{{c}}

! Intervals in between

| &lt;&lt; [[120/119|120:119]] &gt;&gt; <br> 14.5{{c}}

| &lt;&lt; [[221/220|221:220]] &gt;&gt; <br> 7.9{{c}}

| &lt;&lt; [[209/208|209:208]] &gt;&gt; <br> 8.3{{c}}

| &lt;&lt; [[96/95|96:95]] &gt;&gt; <br> 18.1{{c}}