Dyadic chord: Difference between revisions

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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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| [[Keemic chords]] || [[Keemic]] || 100/99, 385/384

|-

| [[Orwell tetrad]] || [[~~Guanyin~~]] || ~~176~~/~~175~~, ~~540~~/~~539~~

| [[Orwell tetrad]] || [[Orwellian]] || 99/98, 385/384

|-

| [[Tutonic hexad|Tutonic hexads]] || [[Meantone]] || 81/80, 99/98, 126/125

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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 23: / Line 23: @@
 == 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 42: / Line 44: @@
 == 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 57: / Line 73: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Didymic chords]] || [[Meantone]] || [[81/80]]
@@ Line 69: / Line 85: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Mothwellsmic chords]] || [[Mothwellsmic]] || [[99/98]]
@@ Line 90: / Line 106: @@
 |-
 | [[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 105: / Line 123: @@
 | [[Keemic chords]] || [[Keemic]] || 100/99, 385/384
 |-
-| [[Orwell tetrad]] || [[Guanyin]] || 176/175, 540/539
+| [[Orwell tetrad]] || [[Orwellian]] || 99/98, 385/384
 |-
 | [[Tutonic hexad|Tutonic hexads]] || [[Meantone]] || 81/80, 99/98, 126/125
@@ Line 117: / Line 135: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Buzurgismic chords]] || [[Buzurgismic]] || [[169/168]]
@@ Line 126: / Line 144: @@
 |-
 | [[Marveltwin chords]] || [[Marveltwin]] || [[325/324]]
+|-
+| [[Sooty fox chords]] || [[Sooty fox]] || [[343/338]]
 |-
 | [[Ratwolfsmic chords]] || [[Ratwolfsmic]] || [[351/350]]
@@ Line 163: / Line 183: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Island chords]] || [[The Archipelago|Island]] || [[676/675]]
@@ Line 181: / Line 201: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Augustmic chords]] || [[Augustmic]] || [[154/153]]
@@ Line 219: / Line 239: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Photismic chords]] || [[Photismic]] || [[324/323]]
@@ Line 267: / Line 287: @@
 {| 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 279: / Line 299: @@
 | [[Heartlandismic chords]] || [[Heartlandismic]] || [[3971/3969]]
 |-
-| [[Schisminic chords]] || [[Schisminic]] || [[4096/4095]]
+| [[Minismic chords]] || [[Minismic]] || [[4096/4095]]
 |-
 | [[Baladismic chords]] || [[Baladismic]] || [[4914/4913]]
@@ Line 291: / Line 311: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Scanismic chords]] || [[Scanismic]] || [[460/459]]
@@ Line 351: / Line 371: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Tunbarsmic chords]] || [[Tunbarsmic]] || [[625/624]]
@@ Line 377: / Line 397: @@
 {| 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 387: / Line 407: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Vinecute chords]] || [[Vinecute]] || [[4641/4640]]
@@ Line 401: / Line 421: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Kibismic chords]] || [[Kibismic]] || [[1024/1023]]
@@ Line 413: / Line 433: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Wizardharry chords]] || [[Wizardharry]] || [[4000/3993]]
@@ Line 423: / Line 443: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Lummic chords]] || [[Lummic]] || [[1716/1715]]
@@ Line 431: / Line 451: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Bullionismic chords]] || [[Bullionismic]] || [[5292/5291]]
@@ Line 439: / Line 459: @@
 {| class="wikitable"
 |-
-! Chords !! Associated Temperament !! Associated Commas
+! Chords !! Associated temperament !! Associated commas
 |-
 | [[Harmonismic chords]] || [[Harmonismic]] || [[10648/10647]]
@@ Line 447: / Line 467: @@
 {| 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]]
 |}