Neutral temperaments: Difference between revisions
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== Rastmatic == | == Rastmatic == | ||
Rastmatic is the neutral temperament in the 2.3.11 subgroup, which equates 11-limit neutral intervals to their exact neutral (2.sqrt(3/2)) counterparts. The generator represents both 11/9 and 27/22; this is one of the most accurate prime subgroups for the temperament. Because the temperament tempers out 243/242, it is a rastmic temperament, hence "rastmic" may be used informally to refer to Rastmatic or its scales. The generator is best tuned around 350 cents, about 3 cents off from the justly tuned 11/9 and 4 cents off from just 27/22. | [[File:Mohajira7.png|thumb|Mohajira heptatonic MOS]] | ||
Rastmatic is the neutral temperament in the 2.3.11 subgroup, which equates 11-limit neutral intervals to their exact neutral (2.sqrt(3/2)) counterparts. The generator represents both 11/9 and 27/22; this is one of the most accurate prime subgroups for the temperament. Because the temperament tempers out 243/242, it is a rastmic temperament, hence "rastmic" may be used informally to refer to Rastmatic or its scales. The generator is best tuned around 350 cents, about 3 cents off from the justly tuned 11/9 and 4 cents off from just 27/22. Mohajira extends rastmatic to 2.3.5.11. | |||
=== Etymology === | === Etymology === | ||
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== Namo == | == Namo == | ||
[[File:Suhajira7.png|thumb|Suhajira heptatonic MOS]] | |||
''Namo'', ''Intertridecimal'', or ''Harmoneutral'' is the temperament of 512/507, which is 7 & 10 in the 2.3.13 subgroup. It prefers a sharp tuning of the fifth. | ''Namo'', ''Intertridecimal'', or ''Harmoneutral'' is the temperament of 512/507, which is 7 & 10 in the 2.3.13 subgroup. It prefers a sharp tuning of the fifth. | ||
It is often framed as a (somewhat inaccurate) extension to Rastmatic. Its generator is best tuned around 355c. | It is often framed as a (somewhat inaccurate) extension to Rastmatic. Its generator is best tuned around 355c. | ||
Namo extends to 7 with the temperament Suhajira, which takes the fifth as an [[archy]] fifth. | |||
== Patent vals == | == Patent vals == | ||
Latest revision as of 02:10, 2 June 2026
Neutral temperaments are any temperaments represented by the edo join 7 & 10, or any reasonable extension of such a temperament, such that the generator is a neutral third of some kind which splits 3/2 into two. They are a subset of and largely cover the dicot temperament archetype, and impose upon it the condition that the neutral third must be mapped to 2\7 and 3\10. The two most well-known neutral temperaments are the 2.3.11 (Rastmatic) and 2.3.5 (Dicot) versions.
10edo is a contorted 5edo in 2.3.7, hence 7 & 10 in that subgroup represents monocot Archy temperament.
Notation
Neutral temperaments may be notated with neutral chain-of-fifths notation.
| Note | 24edo | Notation | 2.3... | 5 (Dicot) | 11 (Rastmatic) | 13 (Namo) | 13-limit (no-fives) | 13-limit |
|---|---|---|---|---|---|---|---|---|
| A | 0 | P1 | 1/1 | |||||
| At | 50 | sA1 | 81/80 | 33/32 | ||||
| Bb | 100 | m2 | 256/243 | |||||
| Bd | 150 | n2 | 10/9, 16/15 | 12/11 | ||||
| B | 200 | M2 | 9/8 | 8/7 | 11/10 | |||
| C | 300 | m3 | 32/27 | 7/6 | ||||
| Ct | 350 | n3 | 5/4, 6/5 | 11/9 | 16/13 | |||
| C# | 400 | M3 | 81/64 | 9/7 | ||||
| Dd | 450 | sd4 | 14/11 | |||||
| D | 500 | P4 | 4/3 | |||||
| Dt | 550 | sA4 | 27/20 | 11/8 | 18/13 | 10/7 | ||
| Ed | 650 | sd5 | 40/27 | 16/11 | 13/9 | 7/5 | ||
| E | 700 | P5 | 3/2 | |||||
| Et | 750 | sA5 | 11/7 | |||||
| F | 800 | m6 | 128/81 | 14/9 | ||||
| Ft | 850 | n6 | 5/3, 8/5 | 18/11 | 13/8 | |||
| F# | 900 | M6 | 27/16 | 12/7 | ||||
| G | 1000 | m7 | 16/9 | 7/4 | 20/11 | |||
| Gt | 1050 | n7 | 15/8, 9/5 | 11/6 | ||||
| G# | 1100 | M7 | 243/128 | |||||
| Ad | 1150 | sd8 | 160/81 | 64/33 | ||||
| A | 1200 | P8 | 2/1 |
These intervals may additionally be arranged on a chart which explains their mappings to 7edo and 10edo:
| 0\7 | 1\7 | 2\7 | 3\7 | 4\7 | 5\7 | 6\7 | 7\7 | |
|---|---|---|---|---|---|---|---|---|
| 0\10 | P1 | m2 | d3 | |||||
| 1\10 | sA1 | n2 | sd3 | |||||
| 2\10 | A1 | M2 | m3 | d4 | ||||
| 3\10 | sA2 | n3 | sd4 | |||||
| 4\10 | A2 | M3 | P4 | d5 | d6 | |||
| 5\10 | sA3 | sA4 | sd5 | sd6 | ||||
| 6\10 | A3 | A4 | P5 | m6 | d7 | |||
| 7\10 | sA5 | n6 | sd7 | |||||
| 8\10 | A5 | M6 | m7 | d8 | ||||
| 9\10 | sA6 | n7 | sd8 | |||||
| 10\10 | A6 | M7 | P8 |
Rastmatic
Rastmatic is the neutral temperament in the 2.3.11 subgroup, which equates 11-limit neutral intervals to their exact neutral (2.sqrt(3/2)) counterparts. The generator represents both 11/9 and 27/22; this is one of the most accurate prime subgroups for the temperament. Because the temperament tempers out 243/242, it is a rastmic temperament, hence "rastmic" may be used informally to refer to Rastmatic or its scales. The generator is best tuned around 350 cents, about 3 cents off from the justly tuned 11/9 and 4 cents off from just 27/22. Mohajira extends rastmatic to 2.3.5.11.
Etymology
Rastmatic is named after the rastma, the comma it tempers out, which is in turn named after the maqam Rast which utilizes a scale with several neutral intervals.
Dicot
Dicot[note 1], not to be confused with the dicot archetype as a whole, is the neutral temperament in the 2.3.5 subgroup. an exotemperament that can be defined to temper out 25/24, the Dicot comma. The provided edo join also tempers out 45/44 and 64/63 in the 11-limit, representing the extension Dichotic and also tempering out 55/54. Alternative extensions include 4 & 7 (which conflates 9/7~7/6~6/5~5/4). 7 & 10 and 10 & 17 are both reasonable edo joins, suggesting Dicot as a 3-, 7-, or 10-form temperament.
Dicot makes 4:5:6 equidistant, suggesting the simplified structure of tertian harmony, the same way Semaphore does for chthonic harmony. As a result, the temperament archetype dicot is named after it.
Etymology
Dicot originates from the term "dicot" in botany, referring to plants with two embryonic leaves, perhaps by analogy with 3/2 being split into two generators. The name Dicot would also inspire Tetracot, Alphatricot, and by extension the ploidacot temperament archetype naming system as a whole.
Tuning considerations
A perfect ~351c tuning of the generator, while useful for understanding tertian harmony and suggested by some temperament tuning optimization systems, does not reasonably approximate either 5/4 or 6/5. The optimal tunings of Dicot are roughly bimodal, with ~360c (around 10edo) and ~343c (around 7edo) both being better tunings.
History
According to Unque, dicot temperament was discussed by Vicentino as a descriptive model of vocal music; the reason it is considered an exotemperament today is, if not members of the xenharmonic community tending to be sensitive to detuning, likely at least partially the fact that the tonality system motivates a distinction between 4:5:6 and 10:12:15. (Note that the tuning error on 5/4 in 10edo is not much greater than the error on 3/2 in the same tuning.)
Namo
Namo, Intertridecimal, or Harmoneutral is the temperament of 512/507, which is 7 & 10 in the 2.3.13 subgroup. It prefers a sharp tuning of the fifth.
It is often framed as a (somewhat inaccurate) extension to Rastmatic. Its generator is best tuned around 355c.
Namo extends to 7 with the temperament Suhajira, which takes the fifth as an archy fifth.
Patent vals
List of patent vals
| EDO | Mappings supported | Generator tuning | 3/2 tuning |
|---|---|---|---|
| 10 | 5, 13, 11 | 360.0c | 720.0c |
| 37 | 13 | 356.8c | 713.5c |
| 27 | 13 | 355.6c | 711.1c |
| 71 | 13 | 354.9c | 709.9c |
| 44 | 13 | 354.5c | 709.1c |
| 61 | 13 | 354.1c | 708.2c |
| 78 | 13 | 353.8c | 707.7c |
| 95 | 13 | 353.7c | 707.4c |
| 17 | 5, 13, 11 | 352.9c | 705.9c |
| 75 | 13 | 352.0c | 704.0c |
| 58 | 13, 11 | 351.7c | 703.4c |
| 41 | 13, 11 | 351.2c | 702.4c |
| 147 | 11 | 351.0c | 702.0c |
| 106 | 11 | 350.9c | 701.9c |
| 171 | 11 | 350.9c | 701.8c |
| 65 | 13, 11 | 350.8c | 701.5c |
| 219 | 11 | 350.7c | 701.4c |
| 154 | 11 | 350.6c | 701.3c |
| 243 | 11 | 350.62c | 701.23c |
| 332 | 11 | 350.60c | 701.20c |
| 89 | 11 | 350.56c | 701.12c |
| 380 | 11 | 350.53c | 701.05c |
| 291 | 11 | 350.52c | 701.03c |
| 202 | 11 | 350.50c | 700.99c |
| 517 | 11 | 350.48c | 700.97c |
| 315 | 11 | 350.48c | 700.95c |
| 428 | 11 | 350.47c | 700.93c |
| 541 | 11 | 350.46c | 700.92c |
| 113 | 11 | 350.44c | 700.88c |
| 476 | 11 | 350.42c | 700.84c |
| 363 | 11 | 350.41c | 700.83c |
| 250 | 11 | 350.40c | 700.80c |
| 387 | 11 | 350.39c | 700.78c |
| 137 | 11 | 350.36c | 700.73c |
| 435 | 11 | 350.34c | 700.69c |
| 298 | 11 | 350.34c | 700.67c |
| 459 | 11 | 350.33c | 700.65c |
| 161 | 11 | 350.31c | 700.62c |
| 346 | 11 | 350.29c | 700.58c |
| 185 | 11 | 350.27c | 700.54c |
| 394 | 11 | 350.25c | 700.51c |
| 209 | 11 | 350.24c | 700.48c |
| 233 | 11 | 350.21c | 700.43c |
| 257 | 11 | 350.19c | 700.39c |
| 281 | 11 | 350.18c | 700.36c |
| 305 | 11 | 350.16c | 700.33c |
| 329 | 11 | 350.15c | 700.30c |
| 353 | 11 | 350.14c | 700.28c |
| 24 | 13, 11 | 350.00c | 700.00c |
| 247 | 11 | 349.80c | 699.60c |
| 223 | 11 | 349.78c | 699.55c |
| 199 | 11 | 349.7c | 699.5c |
| 175 | 11 | 349.7c | 699.4c |
| 151 | 11 | 349.7c | 699.3c |
| 127 | 11 | 349.6c | 699.2c |
| 103 | 11 | 349.5c | 699.0c |
| 79 | 11 | 349.4c | 698.7c |
| 55 | 13, 11 | 349.1c | 698.2c |
| 31 | 13, 11 | 348.4c | 696.8c |
| 38 | 13, 11 | 347.4c | 694.7c |
| 45 | 13 | 346.7c | 693.3c |
| 7 | 5, 13, 11 | 342.9c | 685.7c |
Other temperaments with neutral third generators
Hemififths
Hemififths, 41 & 58, is a temperament in the 2.3.5.7 subgroup which tempers out 2401/2400 = S49 (aka the breedsma) equating 49/40 to its 3/2-complement and additionally tempers out 5120/5103 making it an aberschismic temperament. Note that 7 & 10 in 2.3.5.7 is dichotic.
