# kleisma

(Greek: "lock", "lid"; plural: kleismata)

[John Chalmers, Divisions of the Tetrachord]

The kleisma is a small interval composed of 6 major thirds up and 5 fifths down [= 3-556].

Its ratio is 15625/15552 [= 0.08 Semitones = ~8.10728 cents]. The term was introduced by Shohé Tanaka in 1890.

. . . . . . . . .
[Joe Monzo]

Tanaka described the kleisma thus:

```Fb:A#= 4 * (5/6)^6

E#:A# = 3:4

E#:Fb = [ 4 * (5/6)^6 ] * 3/4  =  15625/15552 = 214/213 (approximately)

[after Tanaka 1890, p 9-10]
```

```  2,3,5-monzo            ratio        ~cents

[ 2  0,  0>             4           1200         * 2
+ [-1 -1,  1> * 6      * (5/6)^6    + -315.641287  * 6
+ [-2  1,  0>          *  3/4       + -498.0449991
----------------      ----------    -------------------
[-6 -5,  6>         15625/15552        8.107278862
```

and is shown below on a 3,5-prime-space lattice:

The kleisma's importance in Tanaka's tuning system lay not in any explicit use as a musical interval (i.e., harmonic or melodic), but rather, in the fact that it is treated as a unison-vector: together with the skhisma, it is one of the promos which defines a 53-tone periodicity-block ("periodic parallelogram" was Tanaka's term, possibly the earliest recognition of the concept) which is audibly indistinguishable from 5-limit just intonation, and which is nearly identical to pythagorean tuning.

Below is one lattice of a 53-tone periodicity-block derived from these two promos ... it differs slightly from Tanaka's actual diagram, in that some of this pitches were a kleisma higher.

Below is Tanaka's actual diagram. The central periodicity-block is clearly marked. Other replications of the "periodic parallelogram" are marked with various combinations of S and K to represent the transposition of the "periodic parallelogram" by skhismata and kleismata.

See the 53et bingo-card for more.

The approximate ratio of 214/213 has a small error of ~1/641 cent from the actual kleisma:

```		  2,3,5,71,107-monzo          ratio            ~cents

[ 1 -1  0 -1  1>           214/213          8.108839410
- [-6 -5  6  0  0>      ÷  15625/15552      - 8.107278862
------------------      ----------------    -------------------
[ 7  4 -6 -1  1>       1109376/1109375      0.001560548

```
. . . . . . . . .
[Joe Monzo]

# septimal kleisma

Another interval that has become known as a kleisma (and in fact is now the one most commonly referred as such) is the one which forms the difference between the 5-limit "augmented 6th" and the harmonic "7th":

```       interval                ratio    2,3,5,7-monzo    ~cents

5-limit "augmented-6th"     225:128    [-7 2, 2 0>    976.5374295
- "harmonic-7th"                7:4      [-2 0, 0 1>    968.8259065
--------------------------    -------    -----------    ------------
septimal-kleisma            225:224    [-5 2, 2 -1>     7.711522991
```

Fokker often invoked the septimal-kleisma in precisely the way i describe here -- to make singers become familiar with the 5-limit JI "augmented-6th", then have them use it as a "target pitch" when he notates a harmonic-7th in his compositions. (This was at least as early as his 1949 English book on singing JI.)

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###### Reference

"Studien im Gebiete der reinen Stimmung"
Vierteljahrsschrift fÃ¼r Musikwissenschaft vol. 6 no. 1,
Friedrich Chrysander, Philipp Spitta, Guido Adler (eds.),
Breitkopf und HÃ¤rtel, Leipzig, pp. 1-90.

English translation of pages 8 to 18 by Daniel J. Wolf,
"Studies in the Realm of Just Intonation",
XenharmonikÃ´n vol. 16, autumn 1995, pp. 118-125.

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