View source for Negri

{{interwiki
| de = Negri
| en = Negri
| es = 
| ja = 
}}
'''Negri''' is a [[regular temperament]] generated by a [[generator]] of approximately 125 [[cent]]s, 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 [[edo]]s 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 forms 9-note and 10-note [[mos]]ses, 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 [[tetrachord]]s 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 [[Semaphoresmic clan #Negri]] for more technical data. For the various 11-limit extensions, see [[negri extensions]]. 

== Interval chain ==
In the following table, odd harmonics and subharmonics 1–13 are in '''bold'''. 

{| class="wikitable center-1 right-2"
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
|-
| 1
| 125.4
| 13/12, 14/13, 15/14, 16/15
|-
| 2
| 250.7
| 7/6, '''8/7''', 15/13
|-
| 3
| 376.1
| '''5/4''', '''16/13'''
|-
| 4
| 501.4
| '''4/3'''
|-
| 5
| 626.8
| 10/7, 13/9
|-
| 6
| 752.1
| 14/9, 20/13, 32/21
|-
| 7
| 877.5
| 5/3
|-
| 8
| 1002.8
| '''16/9'''
|-
| 9
| 1128.2
| 35/18, 40/21, 52/27
|-
| 10
| 53.5
| 25/24, 28/27, 50/49, 64/63
|}
<nowiki/>* In 2.3.5.7.13-subgroup [[CWE tuning]]

== History and terminology ==
Negri was named by [[Paul Erlich]] in 2001<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_31054.html#31065 Yahoo! Tuning Group | ''The grooviest linear temperaments for 7-limit music'']</ref> after John Negri's 10-out-of-19 maximally even scale<ref>"The Nineteen-Tone System as Ten Plus Nine". [https://interval.xentonic.org/tables-of-contents.html  ''Interval, Journal of Music Research and Development''], pp. 11–13 of Volume 5, Number 3 (Winter 1986–1987). John Negri. </ref>. 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".<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3774#3780 Yahoo! Tuning Group | ''25 best weighted generator steps 5-limit temperaments''] – "I'm calling this tertiathirds (was quadrafourths)." —Dave Keenan</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_41392#41396 Yahoo! Tuning Group | ''! middle-path 7-limit tetradic scales for kalle''] – "Negri [is the new name for quadrafourths]." —Gene Ward Smith</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12957.html#12970 Yahoo! Tuning Group | ''98 named 7-limit temperaments''] – "[Negri] aka 'tertiathirds', 'negrisept' (MP)" —Herman Miller</ref>

== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit prime-optimized tunings
|-
! rowspan="2" | 
! colspan="2" | Euclidean
|-
! Unskewed
! Skewed
|-
! Equilateral
| CEE: ~15/14 = 124.602¢
| CSEE: ~15/14 = 125.284¢
|-
! Tenney
| CTE: ~15/14 = 124.813¢
| CWE: ~15/14 = 125.435¢
|-
! Benedetti, <br>Wilson
| CBE: ~15/14 = 124.874¢
| CSBE: ~15/14 = 125.429¢
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.7.13-subgroup prime-optimized tunings
|-
! rowspan="2" | 
! colspan="2" | Euclidean
|-
! Unskewed
! Skewed
|-
! Equilateral
| CEE: ~14/13 = 123.471¢
| CSEE: ~14/13 = 124.672¢
|-
! Tenney
| CTE: ~14/13 = 124.457¢
| CWE: ~14/13 = 125.354¢
|-
! Benedetti, <br>Wilson
| CBE: ~14/13 = 124.756¢
| CSBE: ~14/13 = 125.428¢
|}

=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! Generator (¢)
! Comments
|-
| 
| 15/8
| 111.731
| 
|-
| 
| 7/4
| 115.587
| 
|-
| 
| 15/14
| 119.443
| 
|-
| 
| 13/8
| 119.824
| 
|-
| 1\10
| 
| 120.000
| Lower bound of 7-, 9-odd-limit, <br>and 2.3.5.7.13-subgroup 13-odd-limit diamond monotone
|-
| 
| 7/5
| 123.498
| 
|-
| 
| 15/13
| 123.871
| 
|-
| 3\29
| 
| 124.138
| 
|-
| 
| 13/10
| 124.298
| 
|-
| 
| 3/2
| 124.511
| 7- and 9-odd-limit minimax
|-
| 5\48
| 
| 125.000
| 48df val
|-
| 
| 10/9
| 125.673
| 
|-
| 2\19
| 
| 126.316
| Upper bound of 9-odd-limit<br>and 2.3.5.7.13-subgroup 13-odd-limit diamond monotone
|-
| 
| 5/3
| 126.337
| 5-odd-limit minimax
|-
| 
| 13/9
| 127.324
| 
|-
| 
| 9/7
| 127.486
| 
|-
| 5\47
| 
| 127.660
| 47df val
|-
| 
| 13/7
| 128.298
| 
|-
| 3\28
| 
| 128.571
| 28df val
|-
| 
| 5/4
| 128.771
| 
|-
| 1\9
| 
| 133.333
| Upper bound of 7-odd-limit diamond monotone
|-
| 
| 7/6
| 133.435
| 
|-
| 
| 13/12
| 138.573
| 
|}

== See also ==
* [[Intervals of Negri-9]]
* [[Modes of Negri-9]]

== Music ==
; [[Mike Battaglia]]
* [https://youtu.be/bCLZChG6U6c ''Negri Comma Pump'']

; [[Sebastian Dumitrescu]]
* [https://soundcloud.com/sedumitr/la-multi-ani-19_edo ''La Mulți Ani''] ([http://micro.soonlabel.com/gene_ward_smith/Others/Dumitrescu/__La_Mul_i_Ani__negri_10___19edo__by_Sebastian_Dumitrescu.mp3 play]{{dead link}}) – Negri[10] in 19edo tuning

; [[Lillian Hearne]]
* [https://soundcloud.com/lillianhearne/negri-shmegri ''Negri Shmegri''] ([http://micro.soonlabel.com/gene_ward_smith/Others/Hearne/Negri%20Shmegri.mp3 play]{{dead link}}) – Negri[9] symmetric mode in 19edo

; [[Herman Miller]]
* [https://soundcloud.com/morphosyntax-1/without-a-clue ''Without a Clue''] (2024)

; [[Ray Perlner]]
* [https://www.youtube.com/playlist?list=PLkW9S8bpltfy3qYhWKO2vyloaMGiH4JtN ''Negri-9 Modal Fugues''] (YouTube playlist)

== Notes ==
<references/>

[[Category:Temperaments]]
[[Category:Negri| ]] <!-- main article -->
[[Category:Semaphoresmic clan]]
[[Category:Marvel temperaments]]