Dyadic chord: Difference between revisions

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{{Interwiki

| de =

| en = Dyadic chord

| es = Relación diádica

| ja =

}}

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The significance of dyadic chords and of the paradigm where all interval pairs are examined in the chord has the psychoacoustic basis of timbral fusion and emergence of the [[virtual fundamental]]. In the above examples, it can be shown that the lower harmonics of each note in the first chord blends better than in the second. Meanwhile, the virtual fundamental of the first chord appears 5/1 below the bass, whereas that of the second appears much lower, at 35/1 below the bass as the denominators "fight" each other. For these reasons we tend to find the first chord more consonant than the second.

== ~~Types of~~ dyadic ~~chords~~ ==

== Essentially tempered dyadic chord ==

In [[regular temperament theory]], we may speak of a just or tempered dyadic chord. By a ''just'' dyadic chord is meant a chord in just intonation which is dyadic, so that each of its notes in relation to the lowest note is a rational number belonging to the set of consonances, and moreover each interval between the notes belongs to the set of consonances. An ''essentially just'' dyadic chord is a chord which is considered to be an approximation of a just dyadic chord, such that each of its intervals is considered to be an approximation of the corresponding interval in the just dyadic chord. So, for instance, [[4:5:6|1 – 5/4 – 3/2]] is a just dyadic chord when the consonance set is the [[5-odd-limit]] diamond with [[octave equivalence]], and 0 – 10 – 18 in 31edo with consonance set {8, 10, 13, 18, 21, 23, 31} modulo 31 is an essentially just dyadic chord approximating 1 – 5/4 – 3/2.

~~== Essentially tempered dyadic chord ==~~

[[File:Essentially tempered chord.png|400px|thumb|right|A more in-depth work-through of the starling 1-6/5-10/7 essentially tempered chord example]]

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== Innate comma chord ==

An '''innate comma chord''', proposed by [[Kite Giedraitis]], is the type of chord that cannot be mapped to just intonation in a given prime limit and odd limit. However, instead of specifying the targeted JI ratios as in an [[#~~Essentially_tempered_dyadic_chord~~|essentially tempered chord]], an innate comma chord describes only the general chord shape: the [[mapped interval]] representations or the [[interval span|sizes]] of its constituent intervals. For example, the [[augmented triad]] in 5-limit JI is an innate comma chord below the 25-odd-limit, because it is impossible to tune all three major thirds (in the four-note chord doubling the root up an octave) to [[5/4]] or any other 5-limit interval with odd limit below 25: the innate comma here is 128/125 (41¢). In practice, it might be sung or played justly but with a large odd limit (containing [[wolf interval]]s): for example, 1–5/4–8/5–2, or 1–5/4–25/16–2 (or even 1–5/4–25/16–125/64). Or it might be tempered, e.g. in 12edo as 0¢–400¢–800¢–1200¢. In 7-limit JI, one of the major thirds can be tuned to 9/7, reducing the innate comma to 225/224 (only 8¢). This comma can be distributed among the three thirds, tempering each by only a few cents, which is usually close enough to be acceptable. In 11-limit JI, the augmented chord is not an innate comma chord, because it can be tuned justly as 7:9:11:14, a low enough odd limit to "ring". (However, it is debatable whether this chord qualifies as an augmented triad, because the middle [[11/9]] interval is a neutral third rather than a major third.)

An '''innate comma chord''', proposed by [[Kite Giedraitis]], is the type of chord that cannot be mapped to just intonation in a given prime limit and odd limit. However, instead of specifying the targeted JI ratios as in an [[#Essentially tempered dyadic chord|essentially tempered chord]], an innate comma chord describes only the general chord shape: the [[mapped interval]] representations or the [[interval span|sizes]] of its constituent intervals.

For example, the [[augmented triad]] in 5-limit JI is an innate comma chord below the 25-odd-limit, because it is impossible to tune all three major thirds (in the four-note chord doubling the root up an octave) to [[5/4]] or any other 5-limit interval with odd limit below 25: the innate comma here is 128/125 (41¢). In practice, it might be sung or played justly but with a large odd limit (containing [[wolf interval]]s): for example, 1–5/4–8/5–2, or 1–5/4–25/16–2 (or even 1–5/4–25/16–125/64). Or it might be tempered, e.g. in 12edo as 0¢–400¢–800¢–1200¢. In 7-limit JI, one of the major thirds can be tuned to 9/7, reducing the innate comma to 225/224 (only 8¢). This comma can be distributed among the three thirds, tempering each by only a few cents, which is usually close enough to be acceptable. In 11-limit JI, the augmented chord is not an innate comma chord, because it can be tuned justly as 7:9:11:14, a low enough odd limit to "ring". (However, it is debatable whether this chord qualifies as an augmented triad, because the middle [[11/9]] interval is a neutral third rather than a major third.)

== Anomalous saturated suspension ==

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== List of essentially tempered dyadic chords ==

Here are some pages on certain essentially tempered dyadic chords, sorted by the lowest odd limit in which they are available. See [[Dyadic chord/Pattern of essentially tempered chords]] for some notable abstract chord patterns.

Here are some pages on certain essentially tempered dyadic chords, sorted by the lowest odd limit in which they are available (<span style="background-color:#D4D4D4">shaded rows</span> mark [[octave-splitting comma]] chords, which are induced by octave-splitting commas). See [[Dyadic chord/Pattern of essentially tempered chords]] for some notable abstract chord patterns.

=== 5-odd-limit ===

{| class="wikitable"

|-

! Chords !! Associated Temperament !! Associated Commas

|-

| bgcolor="#D4D4D4" | [[Augmented triad]] || bgcolor="#D4D4D4" | [[Augmented (temperament)|Augmented]] || bgcolor="#D4D4D4" | [[128/125]]

|}

=== 7-odd-limit ===

{| class="wikitable"

|-

! Chords !! Associated ~~Temperament~~ !! Associated ~~Commas~~

! Chords !! Associated temperament !! Associated commas

|-

| bgcolor="#D4D4D4" | [[Jubilismic chords]] || bgcolor="#D4D4D4" | [[Jubilismic]] || bgcolor="#D4D4D4" | [[50/49]]

|-

| [[Archytas chords]] || [[~~Archytas clan|~~Archytas]] || [[64/63]]

| [[Archytas chords]] || [[Archytas]] || [[64/63]]

|-

| [[Starling chords]] || [[Starling]] || [[126/125]]

|-

| bgcolor="#D4D4D4" | [[5edo|Blackwood pentad]] || bgcolor="#D4D4D4" | [[Blackwood]] || bgcolor="#D4D4D4" | [[28/27]], [[49/48]]

|-

| bgcolor="#D4D4D4" | [[Diminished seventh chord]] || bgcolor="#D4D4D4" | [[Diminished (temperament)|Diminished]] || bgcolor="#D4D4D4" | [[36/35]], [[50/49]]

|}

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{| class="wikitable"

|-

! Chords !! Associated ~~Temperament~~ !! Associated ~~Commas~~

! Chords !! Associated temperament !! Associated commas

|-

| [[Didymic chords]] || [[Meantone]] || [[81/80]]

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{| class="wikitable"

|-

! Chords !! Associated ~~Temperament~~ !! Associated ~~Commas~~

! Chords !! Associated temperament !! Associated commas

|-

| [[Mothwellsmic chords]] || [[Mothwellsmic]] || [[99/98]]

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|-

| [[Pentacircle chords]] || [[Pentacircle]] || [[896/891]]

|-

| bgcolor="#D4D4D4" | [[Jubilismic chords|Undecimal jubilismic chords]] || bgcolor="#D4D4D4" | [[Jubilismic family|Undecimal jubilismic]] || bgcolor="#D4D4D4" | [[50/49]], [[99/98]]

|-

| [[Undecimal marvel chords]] || [[Marvel]] || 225/224, 385/384

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{| class="wikitable"

|-

! Chords !! Associated ~~Temperament~~ !! Associated ~~Commas~~

! Chords !! Associated temperament !! Associated commas

|-

| [[Buzurgismic chords]] || [[Buzurgismic]] || [[169/168]]

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|-

| [[Marveltwin chords]] || [[Marveltwin]] || [[325/324]]

|-

| [[Sooty fox chords]] || [[Sooty fox]] || [[343/338]]

|-

| [[Ratwolfsmic chords]] || [[Ratwolfsmic]] || [[351/350]]

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{| class="wikitable"

|-

! Chords !! Associated ~~Temperament~~ !! Associated ~~Commas~~

! Chords !! Associated temperament !! Associated commas

|-

| [[Island chords]] || [[The Archipelago|Island]] || [[676/675]]

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{| class="wikitable"

|-

! Chords !! Associated ~~Temperament~~ !! Associated ~~Commas~~

! Chords !! Associated temperament !! Associated commas

|-

| [[Augustmic chords]] || [[Augustmic]] || [[154/153]]

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{| class="wikitable"

|-

! Chords !! Associated ~~Temperament~~ !! Associated ~~Commas~~

! Chords !! Associated temperament !! Associated commas

|-

| [[Photismic chords]] || [[Photismic]] || [[324/323]]

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{| class="wikitable"

|-

! Chords !! Associated ~~Temperament~~ !! Associated ~~Commas~~

! Chords !! Associated temperament !! Associated commas

|-

| [[Slendric pentad]] || [[~~Gamelismic clan|~~Gamelismic]] || [[1029/1024]]

| [[Slendric pentad]] || [[Gamelismic]] || [[1029/1024]]

|-

| [[Palingenetic chords]] || [[Palingenetic]] || [[1701/1700]]

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| [[Heartlandismic chords]] || [[Heartlandismic]] || [[3971/3969]]

|-

| [[~~Schisminic~~ chords]] || [[~~Schisminic~~]] || [[4096/4095]]

| [[Minismic chords]] || [[Minismic]] || [[4096/4095]]

|-

| [[Baladismic chords]] || [[Baladismic]] || [[4914/4913]]

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{| class="wikitable"

|-

! Chords !! Associated ~~Temperament~~ !! Associated ~~Commas~~

! Chords !! Associated temperament !! Associated commas

|-

| [[Scanismic chords]] || [[Scanismic]] || [[460/459]]

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{| class="wikitable"

|-

! Chords !! Associated ~~Temperament~~ !! Associated ~~Commas~~

! Chords !! Associated temperament !! Associated commas

|-

| [[Tunbarsmic chords]] || [[Tunbarsmic]] || [[625/624]]

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{| class="wikitable"

|-

! Chords !! Associated ~~Temperament~~ !! Associated ~~Commas~~

! Chords !! Associated temperament !! Associated commas

|-

| [[Ragismic chords]] || [[~~Ragismic microtemperaments|~~Ragismic]] || [[4375/4374]]

| [[Ragismic chords]] || [[Ragismic]] || [[4375/4374]]

|-

| [[Hagendorfismic chords]] || [[Hagendorfismic]] || [[8075/8073]]

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{| class="wikitable"

|-

! Chords !! Associated ~~Temperament~~ !! Associated ~~Commas~~

! Chords !! Associated temperament !! Associated commas

|-

| [[Vinecute chords]] || [[Vinecute]] || [[4641/4640]]

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{| class="wikitable"

|-

! Chords !! Associated ~~Temperament~~ !! Associated ~~Commas~~

! Chords !! Associated temperament !! Associated commas

|-

| [[Kibismic chords]] || [[Kibismic]] || [[1024/1023]]

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{| class="wikitable"

|-

! Chords !! Associated ~~Temperament~~ !! Associated ~~Commas~~

! Chords !! Associated temperament !! Associated commas

|-

| [[Wizardharry chords]] || [[Wizardharry]] || [[4000/3993]]

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{| class="wikitable"

|-

! Chords !! Associated ~~Temperament~~ !! Associated ~~Commas~~

! Chords !! Associated temperament !! Associated commas

|-

| [[Lummic chords]] || [[Lummic]] || [[1716/1715]]

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{| class="wikitable"

|-

! Chords !! Associated ~~Temperament~~ !! Associated ~~Commas~~

! Chords !! Associated temperament !! Associated commas

|-

| [[Bullionismic chords]] || [[Bullionismic]] || [[5292/5291]]

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{| class="wikitable"

|-

! Chords !! Associated ~~Temperament~~ !! Associated ~~Commas~~

! Chords !! Associated temperament !! Associated commas

|-

| [[Harmonismic chords]] || [[Harmonismic]] || [[10648/10647]]

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{| class="wikitable"

|-

! Chords !! Associated ~~Temperament~~ !! Associated ~~Commas~~

! Chords !! Associated temperament !! Associated commas

|-

| [[Jacobin chords]] || [[The Jacobins|Jacobin]] || [[6656/6655]]

|}

~~== List of innate comma chords ==~~

~~{| class="wikitable"~~

|-

~~! Chords !! Associated Temperament !! Associated Commas~~

|-

~~| [[Diminished seventh chord]] || [[Diminished (temperament)|Diminished]] || [[36/35]], [[50/49]]~~

|-

~~| [[Augmented triad]] || [[Augmented (temperament)|Augmented]] || [[128/125]]~~

|}

@@ Line 1: / Line 1: @@
+{{Interwiki
+| de =
+| en = Dyadic chord
+| es = Relación diádica
+| ja =
+}}
 {{Distinguish| Dyad }}
@@ Line 13: / Line 19: @@
 The significance of dyadic chords and of the paradigm where all interval pairs are examined in the chord has the psychoacoustic basis of timbral fusion and emergence of the [[virtual fundamental]]. In the above examples, it can be shown that the lower harmonics of each note in the first chord blends better than in the second. Meanwhile, the virtual fundamental of the first chord appears 5/1 below the bass, whereas that of the second appears much lower, at 35/1 below the bass as the denominators "fight" each other. For these reasons we tend to find the first chord more consonant than the second.
-== Types of dyadic chords ==
+== Essentially tempered dyadic chord ==
 In [[regular temperament theory]], we may speak of a just or tempered dyadic chord. By a ''just'' dyadic chord is meant a chord in just intonation which is dyadic, so that each of its notes in relation to the lowest note is a rational number belonging to the set of consonances, and moreover each interval between the notes belongs to the set of consonances. An ''essentially just'' dyadic chord is a chord which is considered to be an approximation of a just dyadic chord, such that each of its intervals is considered to be an approximation of the corresponding interval in the just dyadic chord. So, for instance, [[4:5:6|1&thinsp;–&thinsp;5/4&thinsp;–&thinsp;3/2]] is a just dyadic chord when the consonance set is the [[5-odd-limit]] diamond with [[octave equivalence]], and 0&thinsp;–&thinsp;10&thinsp;–&thinsp;18 in 31edo with consonance set {8, 10, 13, 18, 21, 23, 31} modulo 31 is an essentially just dyadic chord approximating 1&thinsp;–&thinsp;5/4&thinsp;–&thinsp;3/2.
-== Essentially tempered dyadic chord ==
 [[File:Essentially tempered chord.png|400px|thumb|right|A more in-depth work-through of the starling 1-6/5-10/7 essentially tempered chord example]]
@@ Line 25: / Line 29: @@
 == Innate comma chord ==
-An '''innate comma chord''', proposed by [[Kite Giedraitis]], is the type of chord that cannot be mapped to just intonation in a given prime limit and odd limit. However, instead of specifying the targeted JI ratios as in an [[#Essentially_tempered_dyadic_chord|essentially tempered chord]], an innate comma chord describes only the general chord shape: the [[mapped interval]] representations or the [[interval span|sizes]] of its constituent intervals. For example, the [[augmented triad]] in 5-limit JI is an innate comma chord below the 25-odd-limit, because it is impossible to tune all three major thirds (in the four-note chord doubling the root up an octave) to [[5/4]] or any other 5-limit interval with odd limit below 25: the innate comma here is 128/125 (41¢). In practice, it might be sung or played justly but with a large odd limit (containing [[wolf interval]]s): for example, 1–5/4–8/5–2, or 1–5/4–25/16–2 (or even 1–5/4–25/16–125/64). Or it might be tempered, e.g. in 12edo as 0¢–400¢–800¢–1200¢. In 7-limit JI, one of the major thirds can be tuned to 9/7, reducing the innate comma to 225/224 (only 8¢). This comma can be distributed among the three thirds, tempering each by only a few cents, which is usually close enough to be acceptable. In 11-limit JI, the augmented chord is not an innate comma chord, because it can be tuned justly as 7:9:11:14, a low enough odd limit to "ring". (However, it is debatable whether this chord qualifies as an augmented triad, because the middle [[11/9]] interval is a neutral third rather than a major third.)
+An '''innate comma chord''', proposed by [[Kite Giedraitis]], is the type of chord that cannot be mapped to just intonation in a given prime limit and odd limit. However, instead of specifying the targeted JI ratios as in an [[#Essentially tempered dyadic chord|essentially tempered chord]], an innate comma chord describes only the general chord shape: the [[mapped interval]] representations or the [[interval span|sizes]] of its constituent intervals.
+For example, the [[augmented triad]] in 5-limit JI is an innate comma chord below the 25-odd-limit, because it is impossible to tune all three major thirds (in the four-note chord doubling the root up an octave) to [[5/4]] or any other 5-limit interval with odd limit below 25: the innate comma here is 128/125 (41¢). In practice, it might be sung or played justly but with a large odd limit (containing [[wolf interval]]s): for example, 1–5/4–8/5–2, or 1–5/4–25/16–2 (or even 1–5/4–25/16–125/64). Or it might be tempered, e.g. in 12edo as 0¢–400¢–800¢–1200¢. In 7-limit JI, one of the major thirds can be tuned to 9/7, reducing the innate comma to 225/224 (only 8¢). This comma can be distributed among the three thirds, tempering each by only a few cents, which is usually close enough to be acceptable. In 11-limit JI, the augmented chord is not an innate comma chord, because it can be tuned justly as 7:9:11:14, a low enough odd limit to "ring". (However, it is debatable whether this chord qualifies as an augmented triad, because the middle [[11/9]] interval is a neutral third rather than a major third.)
 == Anomalous saturated suspension ==
@@ Line 44: / Line 50: @@
 == List of essentially tempered dyadic chords ==
-Here are some pages on certain essentially tempered dyadic chords, sorted by the lowest odd limit in which they are available. See [[Dyadic chord/Pattern of essentially tempered chords]] for some notable abstract chord patterns.
+Here are some pages on certain essentially tempered dyadic chords, sorted by the lowest odd limit in which they are available (<span style="background-color:#D4D4D4">shaded rows</span> mark [[octave-splitting comma]] chords, which are induced by octave-splitting commas). See [[Dyadic chord/Pattern of essentially tempered chords]] for some notable abstract chord patterns.
+=== 5-odd-limit ===
+{| class="wikitable"
+|-
+! Chords !! Associated Temperament !! Associated Commas
+|-
+| bgcolor="#D4D4D4" | [[Augmented triad]] || bgcolor="#D4D4D4" | [[Augmented (temperament)|Augmented]] || bgcolor="#D4D4D4" | [[128/125]]
+|}
 === 7-odd-limit ===
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
+|-
+| bgcolor="#D4D4D4" | [[Jubilismic chords]] || bgcolor="#D4D4D4" | [[Jubilismic]] || bgcolor="#D4D4D4" | [[50/49]]
 |-
-| [[Archytas chords]] || [[Archytas clan|Archytas]] || [[64/63]]
+| [[Archytas chords]] || [[Archytas]] || [[64/63]]
 |-
 | [[Starling chords]] || [[Starling]] || [[126/125]]
+|-
+| bgcolor="#D4D4D4" | [[5edo|Blackwood pentad]] || bgcolor="#D4D4D4" | [[Blackwood]] || bgcolor="#D4D4D4" | [[28/27]], [[49/48]]
+|-
+| bgcolor="#D4D4D4" | [[Diminished seventh chord]] || bgcolor="#D4D4D4" | [[Diminished (temperament)|Diminished]] || bgcolor="#D4D4D4" | [[36/35]], [[50/49]]
 |}
@@ Line 59: / Line 79: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Didymic chords]] || [[Meantone]] || [[81/80]]
@@ Line 71: / Line 91: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Mothwellsmic chords]] || [[Mothwellsmic]] || [[99/98]]
@@ Line 92: / Line 112: @@
 |-
 | [[Pentacircle chords]] || [[Pentacircle]] || [[896/891]]
+|-
+| bgcolor="#D4D4D4" | [[Jubilismic chords|Undecimal jubilismic chords]] || bgcolor="#D4D4D4" | [[Jubilismic family|Undecimal jubilismic]] || bgcolor="#D4D4D4" | [[50/49]], [[99/98]]
 |-
 | [[Undecimal marvel chords]] || [[Marvel]] || 225/224, 385/384
@@ Line 119: / Line 141: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Buzurgismic chords]] || [[Buzurgismic]] || [[169/168]]
@@ Line 128: / Line 150: @@
 |-
 | [[Marveltwin chords]] || [[Marveltwin]] || [[325/324]]
+|-
+| [[Sooty fox chords]] || [[Sooty fox]] || [[343/338]]
 |-
 | [[Ratwolfsmic chords]] || [[Ratwolfsmic]] || [[351/350]]
@@ Line 165: / Line 189: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Island chords]] || [[The Archipelago|Island]] || [[676/675]]
@@ Line 183: / Line 207: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Augustmic chords]] || [[Augustmic]] || [[154/153]]
@@ Line 221: / Line 245: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Photismic chords]] || [[Photismic]] || [[324/323]]
@@ Line 269: / Line 293: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
-| [[Slendric pentad]] || [[Gamelismic clan|Gamelismic]] || [[1029/1024]]
+| [[Slendric pentad]] || [[Gamelismic]] || [[1029/1024]]
 |-
 | [[Palingenetic chords]] || [[Palingenetic]] || [[1701/1700]]
@@ Line 281: / Line 305: @@
 | [[Heartlandismic chords]] || [[Heartlandismic]] || [[3971/3969]]
 |-
-| [[Schisminic chords]] || [[Schisminic]] || [[4096/4095]]
+| [[Minismic chords]] || [[Minismic]] || [[4096/4095]]
 |-
 | [[Baladismic chords]] || [[Baladismic]] || [[4914/4913]]
@@ Line 293: / Line 317: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Scanismic chords]] || [[Scanismic]] || [[460/459]]
@@ Line 353: / Line 377: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Tunbarsmic chords]] || [[Tunbarsmic]] || [[625/624]]
@@ Line 379: / Line 403: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
-| [[Ragismic chords]] || [[Ragismic microtemperaments|Ragismic]] || [[4375/4374]]
+| [[Ragismic chords]] || [[Ragismic]] || [[4375/4374]]
 |-
 | [[Hagendorfismic chords]] || [[Hagendorfismic]] || [[8075/8073]]
@@ Line 389: / Line 413: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Vinecute chords]] || [[Vinecute]] || [[4641/4640]]
@@ Line 403: / Line 427: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Kibismic chords]] || [[Kibismic]] || [[1024/1023]]
@@ Line 415: / Line 439: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Wizardharry chords]] || [[Wizardharry]] || [[4000/3993]]
@@ Line 425: / Line 449: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Lummic chords]] || [[Lummic]] || [[1716/1715]]
@@ Line 433: / Line 457: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Bullionismic chords]] || [[Bullionismic]] || [[5292/5291]]
@@ Line 441: / Line 465: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Harmonismic chords]] || [[Harmonismic]] || [[10648/10647]]
@@ Line 449: / Line 473: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Jacobin chords]] || [[The Jacobins|Jacobin]] || [[6656/6655]]
-|}
-== List of innate comma chords ==
-{| class="wikitable"
-|-
-! Chords !! Associated Temperament !! Associated Commas
-|-
-| [[Diminished seventh chord]] || [[Diminished (temperament)|Diminished]] || [[36/35]], [[50/49]]
-|-
-| [[Augmented triad]] || [[Augmented (temperament)|Augmented]] || [[128/125]]
 |}