Negri

Negri is a regular temperament generated by a generator of approximately 125.7549 cents, which can be identified with a tempered 16/15, such that:

Two of them make a tempered 7/6~8/7~15/13
Three of them make a tempered 5/4~16/13
Four of them make a tempered 4/3.

It is most naturally viewed as a 2.3.5.7.13 subgroup temperament, tempering out 49/48, 65/64 and 91/90. This is sometimes called negra, and it is realized consistently in 19edo and 29edo. Other edos which may be usable as a negri or negra tuning include 9edo, 10edo, 28edo, 47edo, and 48edo, all of which are consistent through (at least) the 5-odd-limit, since in the broadest sense, negri is defined as tempering out the negri comma in the 5-limit.

Negri is generated by a large "semitone" of around 125 cents and forms 9-note and 10-note mosses, negri[9] and negri[10], at 1L 8s and 9L 1s respectively. In 19edo, the negri generator is the "diatonic half step" of 2\19, which allows these mosses to be written fairly simply in conventional notation. For example, the ssssLssss mode of 19edo could be written as E F Gb G# A B C Db D# E. This mode is particularly useful as it has identical ssss pentachords (analogous to the tetrachords of classical Greek music theory) on the 1/1 and 3/2. It is also notable in that a subset of these notes form the E Double Harmonic Major scale, E F G# A B C D# E, which features in a wide variety of world musical traditions. In fact, all modes of negri[9] and negri[10] contain at least one mode of the double harmonic scale as a subset.

Another useful mode of negri[9] is Lssssssss, which in 19edo would be A B C Db D# E F Gb G# A. This has a minor triad (A-C-E) for a tonic chord, which can be extended to a 7-limit utonal tetrad (A-C-E-D#), as well as 7-limit otonal tetrads on E and F that can function as, respectively, a dominant seventh chord and a German Augmented Sixth chord. This scale also contains the popular Hungarian Minor mode of the Double Harmonic scale, A B C D# E F G# A.

4 of the 9 modes of negri[9] are like the Locrian mode of the diatonic major scale in that they do not have a note a perfect 5th above the tonic. These are more difficult to apply conventional music theory to. However even in these modes there are a number of chords built on the tonic that can provide a measure of consonance and stability, such as 13:16:20:24 and 6:7:8.

Negri[10] also has a number of useful features. One of these features is the fact that it makes 4:5:6 and 10:12:15 share the same "shape" of generic intervals in the scale (as in other rank-2 decatonic scales such as pajara and blackwood scales; this is because 5/4 and 6/5 get tempered to the same thing in 10edo).

The 7-limit version can also be viewed as joining with the marvel temperament family. See Slendro clan #Negri for more technical data. For the various 11-limit extensions, see negri extensions.

Interval chain

#	Cents*	Approximate Ratios
0	0.0	1/1
1	125.6	16/15, 15/14, 14/13, 13/12
2	251.1	8/7, 7/6, 15/13
3	376.7	5/4, 16/13
4	502.3	4/3
5	627.8	10/7, 13/9
6	753.4	14/9, 32/21, 20/13
7	879.0	5/3
8	1004.5	16/9
9	1130.1	35/18, 40/21, 52/27
10	55.7	25/24, 28/27, 50/49, 64/63

* In 2.3.5.7.13 POTE tuning

History and terminology

Negri was named by Paul Erlich in 2001^[1] after John Negri's 10-out-of-19 maximally even scale^[2]. It used to be known by distinct names in the 5- and 7-limit as negripent and negrisept, respectively (for more information on this, see Temperament names#Diminished and dimipent). It was also earlier known as "quadrafourths" and "tertiathirds". ^[3] ^[4] ^[5]