Formal comma: Difference between revisions

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A '''~~mapping~~ comma''' (also called a '''~~formal~~ comma''') for a prime number P > 3 is a comma that maps every [[~~Prime~~ subgroup|2.3.P]] interval to a nearby conventional 3-limit interval. For example, [[81/80]] maps every 5-limit interval to the 3-limit.

A '''formal comma''' (also called a '''mapping comma'''{{idiosyncratic}}) for a prime number {{nowrap| ''p'' > 3 }} is a comma in some [[musical notation]] that maps every [[prime subgroup|2.3.''p''-subgroup]] interval to a nearby conventional [[3-limit]] interval. For example, [[81/80]] maps every 5-limit interval to the 3-limit.

A ~~mapping~~ comma can be identified by the prime P and the 3-limit interval that P/1 (octave-reduced) maps to. The 3-limit interval is named conventionally. Thus both "prime 5 = major ~~3rd~~" and "5/4 = M3" unambiguously indicate 81/80.

A formal comma can be identified by the prime ''p'' and the 3-limit interval that ''p''/1 (octave-reduced) maps to. The 3-limit interval is named conventionally. Thus both "prime 5 = major third" and "5/4 = M3" unambiguously indicate 81/80.

A mapping comma's [[monzo]] or prime-count vector always has a P-count of ±1. The 2-count and 3-count are almost always non-zero, and all other counts are always zero. For example, the usual mapping comma for prime 19 is [[513/512]] ~~= [~~-9 3 0 0 0 0 0 1>.

A mapping comma's [[monzo]] or prime-count vector always has a ''p''-count of ±1. The 2-count and 3-count are almost always non-zero, and all other counts are always zero. For example, the usual mapping comma for prime 19 is [[513/512]] ({{monzo| -9 3 0 0 0 0 0 1 }}).

=== Usage in JI notations ===

== Usage in JI notations ==

A JI notation will typically have for each prime P a pair of [[~~Inflections~~ and alterations|inflections]] that raise/lower a note by P's ~~mapping~~ comma. Thus 5/4 from C is notated as an inflected E, 7/4 as an inflected ~~B flat~~, etc. Ratios like [[35/32]] and [[49/48]] are inflected twice, and the commas accumulate, so complex ratios may be quite distant from the uninflected 3-limit note.

A JI notation will typically have for each prime ''p'' a pair of [[inflections and alterations|inflections]] that raise/lower a note by ''p'''s formal comma. Thus 5/4 from C is notated as an inflected E, 7/4 as an inflected B♭, etc. Ratios like [[35/32]] and [[49/48]] are inflected twice, and the commas accumulate, so complex ratios may be quite distant from the uninflected 3-limit note.

Each JI notation assumes certain ~~mapping~~ commas. The notations largely agree but do diverge for certain primes, because the exact method for choosing the best ~~mapping~~ commas is disputed. Ideally, both the 3-count and the size in cents is minimized, and there are other considerations as well. The choice for neutral-sounding primes like 11 (P4 vs. A4) and 13 (m6 vs. M6) is particularly tricky. Certain choices map the ratio [[13/11]] = ~~289¢~~ to either M2 or M3~~! Color~~ notation ~~allows~~ one to replace the ~~mapping~~ comma for such a prime with its [[~~apotome~~]] ~~complement~~.

Each JI notation assumes certain formal commas. The notations largely agree but do diverge for certain primes, because the exact method for choosing the best formal commas is disputed. Ideally, both the 3-count and the size in cents is minimized, and there are other considerations as well. The choice for neutral-sounding primes like 11 (P4 vs. A4) and 13 (m6 vs. M6) is particularly tricky. Certain choices map the ratio [[13/11]] = 289{{c}} to either M2 or M3. Some notation systems allow one to replace the formal comma for such a prime with its [[chromatic]] and/or [[enharmonic]] counterparts.

~~{{Todo|inline=1|text=add ratios}}~~

{| class="wikitable"

{| class="wikitable center-all"

|+~~Mapping~~ commas used by various JI notations

|+Formal commas used by various JI notations

!

!5

! 5

!7

! 7

!11

! 11

!13

! 13

!17

! 17

!19

! 19

! 23

|-

![[Color notation]]

! [[Color notation]]

|M3

| [[81/80|80/81]] (M3)

|m7

| [[64/63|63/64]] (m7)

|P4/A4

| [[33/32]] (P4)<br>[[729/704|704/729]] (A4)

|m6/M6

| [[1053/1024]] (m6)<br> [[27/26|26/27]] (M6)

|m2

| [[4131/4096]] (m2)

|m3

| [[513/512]] (m3)

| [[16767/16384]] (d5)

|-

![[~~FJS|FJS notation~~]]

! [[Functional Just System]]

|M3

| 80/81 (M3)

|m7

| 63/64 (m7)

|P4

| 33/32 (P4)

|m6

| 1053/1024 (m6)

|m2

| 4131/4096 (m2)

|m3

| 513/512 (m3)

| [[736/729]] (A4)

|-

![[~~HEJI|HEJI~~ notation]]

! [[Helmholtz–Ellis notation]]

|M3

| 80/81 (M3)

|m7

| 63/64 (m7)

|P4

| 33/32 (P4)

|M6

| 26/27 (M6)

|A1

| [[2187/2176]] (A1)

|m3

| 513/512 (m3)

| 736/729 (A4)

|-

!Prime-factor [[Sagittal notation|Sagittal]]

! Prime-factor [[Sagittal notation|Sagittal]]

|M3

| 80/81 (M3)

|m7

| 63/64 (m7)

|P4

| 33/32 (P4)

|M6

| 26/27 (M6)

|m2

| 4131/4096 (m2)

|m3

| 513/512 (m3)

| 736/729 (A4)

|}

== See also ==

~~https://forum~~.~~sagittal.org/viewtopic.php?t=99~~

* [[Solfege string]], a concise way to indicate a series of formal commas.

[~~[Solfege string~~]~~], a concise way to indicate a series of mapping commas.~~

== External links ==

* [https://forum.sagittal.org/viewtopic.php?t=99 The Sagittal Forum | ''Prime-factor Sagittal JI notation (one symbol per prime)'']

@@ Line 1: / Line 1: @@
-A '''mapping comma''' (also called a '''formal comma''') for a prime number P > 3 is a comma that maps every [[Prime subgroup|2.3.P]] interval to a nearby conventional 3-limit interval. For example, [[81/80]] maps every 5-limit interval to the 3-limit.
+A '''formal comma''' (also called a '''mapping comma'''{{idiosyncratic}}) for a prime number {{nowrap| ''p'' > 3 }} is a comma in some [[musical notation]] that maps every [[prime subgroup|2.3.''p''-subgroup]] interval to a nearby conventional [[3-limit]] interval. For example, [[81/80]] maps every 5-limit interval to the 3-limit.
-A mapping comma can be identified by the prime P and the 3-limit interval that P/1 (octave-reduced) maps to. The 3-limit interval is named conventionally. Thus both "prime 5 = major 3rd" and "5/4 = M3" unambiguously indicate 81/80.
+A formal comma can be identified by the prime ''p'' and the 3-limit interval that ''p''/1 (octave-reduced) maps to. The 3-limit interval is named conventionally. Thus both "prime 5 = major third" and "5/4 = M3" unambiguously indicate 81/80.
-A mapping comma's [[monzo]] or prime-count vector always has a P-count of ±1. The 2-count and 3-count are almost always non-zero, and all other counts are always zero. For example, the usual mapping comma for prime 19 is [[513/512]] = [-9 3 0 0 0 0 0 1>.
+A mapping comma's [[monzo]] or prime-count vector always has a ''p''-count of ±1. The 2-count and 3-count are almost always non-zero, and all other counts are always zero. For example, the usual mapping comma for prime 19 is [[513/512]] ({{monzo| -9 3 0 0 0 0 0 1 }}).
-=== Usage in JI notations ===
+== Usage in JI notations ==
-A JI notation will typically have for each prime P a pair of [[Inflections and alterations|inflections]] that raise/lower a note by P's mapping comma. Thus 5/4 from C is notated as an inflected E, 7/4 as an inflected B flat, etc. Ratios like [[35/32]] and [[49/48]] are inflected twice, and the commas accumulate, so complex ratios may be quite distant from the uninflected 3-limit note.
+A JI notation will typically have for each prime ''p'' a pair of [[inflections and alterations|inflections]] that raise/lower a note by ''p'''s formal comma. Thus 5/4 from C is notated as an inflected E, 7/4 as an inflected B♭, etc. Ratios like [[35/32]] and [[49/48]] are inflected twice, and the commas accumulate, so complex ratios may be quite distant from the uninflected 3-limit note.
-Each JI notation assumes certain mapping commas. The notations largely agree but do diverge for certain primes, because the exact method for choosing the best mapping commas is disputed. Ideally, both the 3-count and the size in cents is minimized, and there are other considerations as well. The choice for neutral-sounding primes like 11 (P4 vs. A4) and 13 (m6 vs. M6) is particularly tricky. Certain choices map the ratio [[13/11]] = 289¢ to either M2 or M3! Color notation allows one to replace the mapping comma for such a prime with its [[apotome]] complement.
+Each JI notation assumes certain formal commas. The notations largely agree but do diverge for certain primes, because the exact method for choosing the best formal commas is disputed. Ideally, both the 3-count and the size in cents is minimized, and there are other considerations as well. The choice for neutral-sounding primes like 11 (P4 vs. A4) and 13 (m6 vs. M6) is particularly tricky. Certain choices map the ratio [[13/11]] = 289{{c}} to either M2 or M3. Some notation systems allow one to replace the formal comma for such a prime with its [[chromatic]] and/or [[enharmonic]] counterparts.
-{{Todo|inline=1|text=add ratios}}
-{| class="wikitable"
+{| class="wikitable center-all"
-|+Mapping commas used by various JI notations
+|+Formal commas used by various JI notations
 !
-!5
+! 5
-!7
+! 7
-!11
+! 11
-!13
+! 13
-!17
+! 17
-!19
+! 19
+! 23
 |-
-![[Color notation]]
+! [[Color notation]]
-|M3
+| [[81/80|80/81]] (M3)
-|m7
+| [[64/63|63/64]] (m7)
-|P4/A4
+| [[33/32]] (P4)<br>[[729/704|704/729]] (A4)
-|m6/M6
+| [[1053/1024]] (m6)<br> [[27/26|26/27]] (M6)
-|m2
+| [[4131/4096]] (m2)
-|m3
+| [[513/512]] (m3)
+| [[16767/16384]] (d5)
 |-
-![[FJS|FJS notation]]
+! [[Functional Just System]]
-|M3
+| 80/81 (M3)
-|m7
+| 63/64 (m7)
-|P4
+| 33/32 (P4)
-|m6
+| 1053/1024 (m6)
-|m2
+| 4131/4096 (m2)
-|m3
+| 513/512 (m3)
+| [[736/729]] (A4)
 |-
-![[HEJI|HEJI notation]]
+! [[Helmholtz–Ellis notation]]
-|M3
+| 80/81 (M3)
-|m7
+| 63/64 (m7)
-|P4
+| 33/32 (P4)
-|M6
+| 26/27 (M6)
-|A1
+| [[2187/2176]] (A1)
-|m3
+| 513/512 (m3)
+| 736/729 (A4)
 |-
-!Prime-factor [[Sagittal notation|Sagittal]]
+! Prime-factor [[Sagittal notation|Sagittal]]
-|M3
+| 80/81 (M3)
-|m7
+| 63/64 (m7)
-|P4
+| 33/32 (P4)
-|M6
+| 26/27 (M6)
-|m2
+| 4131/4096 (m2)
-|m3
+| 513/512 (m3)
+| 736/729 (A4)
 |}
 == See also ==
-https://forum.sagittal.org/viewtopic.php?t=99
+* [[Solfege string]], a concise way to indicate a series of formal commas.
-[[Solfege string]], a concise way to indicate a series of mapping commas.
+== External links ==
+* [https://forum.sagittal.org/viewtopic.php?t=99 The Sagittal Forum | ''Prime-factor Sagittal JI notation (one symbol per prime)'']