Microtonal, just intonation, electronic music software Microtonal, just intonation, electronic music software

Encyclopedia of Microtonal Music Theory

@ 00 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Login   |  Encyclopedia Index

Kirnberger III well-temperament

[Joe Monzo]

In octave-equivalent terms, here's how Kirnberger III works:

if C = n^0,

So the next "5th" above F# would be C# 33 * 5 = ~92.17871646 cents, which is only a skhisma (~2 cents) higher than the Db 3-5 = ~90.22499567 cents which we have already derived. This ~2-cent difference is outside the tuning resolution of any human tuner of Bach's time, so that in effect the open well-tempered chain becomes a closed well-tempered cycle, and F# = ~Gb, Db = ~C#, Ab = ~G#, Eb = ~D#, Bb = ~A#, etc.

Well-temperaments have a rich variety of specific sizes for generic interval-classes. For example, Kirnberger III has :

Minor 3rds
----------

~cents  [3 5] vector  instances

  310    [ 0 -3/4]    E:G, A:C
  305    [-1 -1/2]    D:F, B:D
  300    [-2 -1/4]    F#:A, G:Bb
  296    [ 5  1  ]    Db:E, Eb:F#, Ab:B
  294    [-3  0  ]    C:Eb, F:Ab, Bb:Db
					

Because of the irrelevance of the skhisma, the 296-cent interval, [5 1], which is the JI ratio 1215:1024, can be interpreted as several different instances of the Pythagorean minor 3rd 32:27 = [-3 0] = ~294 cents, as follows:

Db:E  = ~C#:E,, Db:~Fb
Eb:F# = ~D#:F#, Eb:~Gb
Ab:B  = ~G#:B, Ab:~Cb
		

Note that the intervals F#:A and G:Bb are basically identical to the 12edo minor 3rds.

We can see that there's an even richer variety of major 3rds:

Major 3rds
----------

~cents  [3 5] vector  instances

  408    [ 4  0  ]    Db:F, Ab:C
  406    [-4 -1  ]    E:Ab, F#:Bb, B:Eb
  402    [ 3  1/4]    Eb:G
  400    [-5 -3/4]    A:Db
  397    [ 2  1/2]    D:F#, Bb:D
  392    [ 1  3/4]    F:A, G:B
  386    [ 0  1  ]    C:E
					

Because of the irrelevance of the skhisma, the 406-cent interval, [-4 -1], which is the JI ratio 512:405, can be interpreted as several different instances of the Pythagorean major 3rd 81:64 = [4 0] = ~408 cents, as follows:

E:Ab  = E:~G#, ~Fb:Ab
F#:Bb = F#:~A#, ~Gb:Bb
B:Eb  = B:~D#, ~Cb:Eb
		

Note that the interval A:Db is basically identical to the 12edo major 3rd. also note that in the case of the major 3rd, there are also intervals which are only +2 and -3 cents different from this one: Eb:G and (D:F#, Bb:D) respectively. i would say that all of this would automatically be equivalent as well, because they are also near the limit of tuning resolution.

Below is a Monzo lattice-diagram of Kirnberger III tuning. [click on the diagram for an explanation of my lattice formula]

Note that, because of the equivalence described above in connection with the skhisma, the top and bottom ends of the lattice would wrap around to become a cylinder.

Below is the interval matrix for Kirnberger III, given in cents:

Below is a table showing the cents values of the 3rds and 5ths for all major and minor triads in Kirnberger III tuning:

------------ major triads ----------------
                              ------------- minor triads ---------------

                      3rds        5th          3rds

E    386.3137139                                         E   386.3137139
                  296.0887182               386.3137139
C#    90.22499567              696.5784285               C     0
                  400.4897103               310.2647146
A    889.7352854                                         A   889.7352854


Eb   294.1349974                                         D#  294.1349974
                  294.1349974               405.8662827
C      0                       701.9550009               B  1088.268715
                  407.8200035               296.0887182
Ab   792.1799965                                         G#  792.1799965


D    193.1568569                                         D   193.1568569
                  304.8881422               397.0668587
B   1088.268715                696.5784285               Bb  996.0899983
                  391.6902863               299.5115698
G    696.5784285                                         G   696.5784285


C#    90.22499567                                        C#   90.22499567
                  294.1349974               400.4897103
A#   996.0899983               700.0012801               A   889.7352854
                  405.8662827               299.5115698
F#   590.2237156                                         F#  590.2237156


C      0                                                 C     0
                  310.2647146               407.8200035
A    889.7352854               701.9550009               Ab  792.1799965
                  391.6902863               294.1349974
F    498.0449991                                         F   498.0449991


B   1088.268715                                          B  1088.268715
                  296.0887182               391.6902863
G#   792.1799965               701.9550009               G   696.5784285
                  405.8662827               310.2647146
E    386.3137139                                         E   386.3137139


Bb   996.0899983                                         G#  996.0899983
                  299.5115698               405.8662827
G    696.5784285               701.9550009               F#  590.2237156
                  402.4434311               296.0887182
Eb   294.1349974                                         D#  294.1349974


A    889.7352854                                         A   889.7352854
                  299.5115698               391.6902863
F#   590.2237156               696.5784285               F   498.0449991
                  397.0668587               304.8881422
D    193.1568569                                         D   193.1568569


Ab   792.1799965                                         G#  792.1799965
                  294.1349974               405.8662827
F    498.0449991               701.9550009               E   386.3137139
                  407.8200035               296.0887182
Db    90.22499567                                        C#   90.22499567


G    696.5784285                                         G   696.5784285
                  310.2647146               402.4434311
E    386.3137139               696.5784285               Eb  294.1349974
                  386.3137139               294.1349974
C      0                                                 C     0


F#   590.2237156                                         F#  590.2237156
                  296.0887182               397.0668587
D#   294.1349974               701.9550009               D   193.1568569
                  405.8662827               304.8881422
B   1088.268715                                          B  1088.268715


F    498.0449991                                         F   498.0449991
                  304.8881422               407.8200035
D    193.1568569               701.9550009               Db   90.22499567
                  397.0668587               294.1349974
Bb   996.0899983                                         Bb  996.0899983
		
. . . . . . . . .

The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. Thank you.

support level