# Kirnberger III well-temperament

[Joe Monzo]

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

if C = n^0,

• the 4 "5ths" between C and E are tuned in 1/4-comma meantone: G = 5^(1/4), D = 5^(1/2), A = 5^(3/4), E = 5^1;
• all the rest of the "5ths" are tuned Pythagorean down from C and up from E: Db...C = 3^(-5...0) and E...F# = 3^(0...2) * 5 .

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
```
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