Negri: Difference between revisions
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It is most naturally viewed as a [[2.3.5.7.13 subgroup|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]]. | It is most naturally viewed as a [[2.3.5.7.13 subgroup|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]]. | ||
The 7-limit version can also be viewed as joining with the [[marvel]] temperament family. See [[Semaphoresmic clan #Negri]] for technical data. For discussion on the various 11-limit extensions, see [[Negri extensions]]. | The 7-limit version can also be viewed as joining with the [[marvel]] temperament family. See [[Semaphoresmic clan #Negri]] for technical data. For discussion on the various 11-limit extensions, see [[Negri extensions]]. | ||
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<nowiki/>* In 2.3.5.7.13-subgroup [[CWE tuning]] | <nowiki/>* In 2.3.5.7.13-subgroup [[CWE tuning]] | ||
== Scales == | |||
Negri forms 9-note and 10-note [[mos scale]]s, 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]]). | |||
== History and terminology == | == History and terminology == | ||
Revision as of 10:10, 13 April 2026
| Negri |
49/48, 225/224 (2.3.5.7);
49/48, 65/64, 91/90 (2.3.5.7.13)
2.3.5.7.13 15-odd-limit: 17.8 ¢
2.3.5.7.13 15-odd-limit: 19 notes
Negri is a regular temperament generated by a generator of approximately 125 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.
The 7-limit version can also be viewed as joining with the marvel temperament family. See Semaphoresmic clan #Negri for technical data. For discussion on the various 11-limit extensions, see Negri extensions.
Interval chain
In the following table, odd harmonics and subharmonics 1–13 are in bold.
| # | 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 |
* In 2.3.5.7.13-subgroup CWE tuning
Scales
Negri forms 9-note and 10-note mos scales, 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).
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]
Tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Equilateral | CEE: ~15/14 = 124.602 ¢ | CSEE: ~15/14 = 125.284 ¢ | POEE: ~15/14 = 125.468 ¢ |
| Tenney | CTE: ~15/14 = 124.813 ¢ | CWE: ~15/14 = 125.435 ¢ | POTE: ~15/14 = 125.608 ¢ |
| Benedetti, Wilson |
CBE: ~15/14 = 124.874 ¢ | CSBE: ~15/14 = 125.429 ¢ | POBE: ~15/14 = 125.629 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Equilateral | CEE: ~14/13 = 123.471 ¢ | CSEE: ~14/13 = 124.672 ¢ | POEE: ~14/13 = 125.528 ¢ |
| Tenney | CTE: ~14/13 = 124.457 ¢ | CWE: ~14/13 = 125.354 ¢ | POTE: ~14/13 = 125.567 ¢ |
| Benedetti, Wilson |
CBE: ~14/13 = 124.756 ¢ | CSBE: ~14/13 = 125.428 ¢ | POBE: ~14/13 = 125.616 ¢ |
Tuning spectrum
| Edo generator |
Unchanged interval (eigenmonzo) |
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, 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 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
Music
- La Mulți Ani (play[dead link]) – Negri[10] in 19edo tuning
- Negri Shmegri (play[dead link]) – Negri[9] symmetric mode in 19edo
- Without a Clue (2024)
- Negri-9 Modal Fugues (YouTube playlist)
References
- ↑ Yahoo! Tuning Group | The grooviest linear temperaments for 7-limit music
- ↑ "The Nineteen-Tone System as Ten Plus Nine". Interval, Journal of Music Research and Development, pp. 11–13 of Volume 5, Number 3 (Winter 1986–1987). John Negri.
- ↑ Yahoo! Tuning Group | 25 best weighted generator steps 5-limit temperaments – "I'm calling this tertiathirds (was quadrafourths)." —Dave Keenan
- ↑ Yahoo! Tuning Group | ! middle-path 7-limit tetradic scales for kalle – "Negri [is the new name for quadrafourths]." —Gene Ward Smith
- ↑ Yahoo! Tuning Group | 98 named 7-limit temperaments – "[Negri] aka 'tertiathirds', 'negrisept' (MP)" —Herman Miller