Compton: Difference between revisions
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{{Infobox regtemp | |||
| Title = Compton | |||
| Subgroups = 2.3.5, 2.3.5.7 | |||
| Comma basis = [[531441/524288]] (5-limit); <br>[[225/224]], [[250047/250000]] (7-limit) | |||
| Edo join 1 = 12 | Edo join 2 = 72 | |||
| Mapping = 12; 0 1 2 | |||
| Generators = 5/4 | |||
| Generators tuning = 384.1 | |||
| Optimization method = CWE | |||
| MOS scales = [[12L 12s]], [[12L 24s]] | |||
| Odd limit 1 = 5 | Mistuning 1 = 1.96 | Complexity 1 = 24 | |||
| Odd limit 2 = 9 | Mistuning 2 = 3.91 | Complexity 2 = 36 | |||
}} | |||
'''Compton''' is a [[regular temperament|temperament]] that takes [[12edo]]'s [[circle of fifths]] for the [[3-limit]], but the [[5/1|fifth harmonic]] is given its own generator instead of being mapped to one of 12edo's intervals. Essentially, it is the [[5-limit]] temperament which [[tempering out|tempers out]] the Pythagorean comma, [[531441/524288]]. This equates any Pythagorean interval with its [[enharmonic]] counterparts, for example, the diminished fourth [[8192/6561]] with the major third [[81/64]], and the two kinds of Pythagorean semitones, diatonic [[256/243]] and chromatic [[2187/2048]], are merged into a single interval of 1/12 octave, which serves as the [[period]]. The [[generator]] can then be seen as any ptolemaic interval (the alteration of a Pythagorean interval by a [[syntonic comma]]), but is most usefully [[5/4]], the ptolemaic major third, or 81/80, the syntonic comma itself. | '''Compton''' is a [[regular temperament|temperament]] that takes [[12edo]]'s [[circle of fifths]] for the [[3-limit]], but the [[5/1|fifth harmonic]] is given its own generator instead of being mapped to one of 12edo's intervals. Essentially, it is the [[5-limit]] temperament which [[tempering out|tempers out]] the Pythagorean comma, [[531441/524288]]. This equates any Pythagorean interval with its [[enharmonic]] counterparts, for example, the diminished fourth [[8192/6561]] with the major third [[81/64]], and the two kinds of Pythagorean semitones, diatonic [[256/243]] and chromatic [[2187/2048]], are merged into a single interval of 1/12 octave, which serves as the [[period]]. The [[generator]] can then be seen as any ptolemaic interval (the alteration of a Pythagorean interval by a [[syntonic comma]]), but is most usefully [[5/4]], the ptolemaic major third, or 81/80, the syntonic comma itself. | ||
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Assuming one has chosen to approximate the 7-limit, the comma generator will be around 15–17 cents. This means that the [[11/1|11th harmonic]] can be reached either by going up or down three steps. Going down three steps results in the canonical extension of compton. Stepping down one more comma, depending on the tuning, can lead to the [[13/1|13th harmonic]], and results in the canonical tridecimal compton. This works best with tunings of the comma around 15 cents. | Assuming one has chosen to approximate the 7-limit, the comma generator will be around 15–17 cents. This means that the [[11/1|11th harmonic]] can be reached either by going up or down three steps. Going down three steps results in the canonical extension of compton. Stepping down one more comma, depending on the tuning, can lead to the [[13/1|13th harmonic]], and results in the canonical tridecimal compton. This works best with tunings of the comma around 15 cents. | ||
Alternatively, any prime may be merged into its 12edo mapping, making the smallest available prime the generator. This is done in [[catler]] (which shares 12edo's 5-limit and is generated by 7) and [[duodecim]] (which shares 12edo's 7-limit and is generated by 11). This is a natural choice for 17 and 19, as 12edo tunes those primes especially well, so compton can be seen as a 19-limit temperament. The next step down, which ends up at around 75 cents flat of the corresponding 12edo interval, can be used for 23 and 29, though two steps up (30 | Alternatively, any prime may be merged into its 12edo mapping, making the smallest available prime the generator. This is done in [[catler]] (which shares 12edo's 5-limit and is generated by 7) and [[duodecim]] (which shares 12edo's 7-limit and is generated by 11). This is a natural choice for 17 and 19, as 12edo tunes those primes especially well, so compton can be seen as a 19-limit temperament. The next step down, which ends up at around 75 cents flat of the corresponding 12edo interval, can be used for 23 and 29, though two steps up (30 cents sharp) is also a reasonable (but less intuitive) choice. | ||
== Interval chain == | == Interval chain == | ||
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== Tunings == | == Tunings == | ||
{| class="wikitable mw-collapsible mw-collapsed" | {| class="wikitable mw-collapsible mw-collapsed" | ||
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit | |+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings | ||
|- | |- | ||
! rowspan="2" | | ! rowspan="2" | | ||
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{| class="wikitable mw-collapsible mw-collapsed" | {| class="wikitable mw-collapsible mw-collapsed" | ||
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit | |+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings | ||
|- | |- | ||
! rowspan="2" | | ! rowspan="2" | | ||
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|- | |- | ||
! Edo<br>generator | ! Edo<br>generator | ||
! [[Eigenmonzo| | ! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]* | ||
! Generator (¢) | ! Generator (¢) | ||
! Comments | ! Comments | ||
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|- | |- | ||
| | | | ||
| 5 | | 7/5 | ||
| 382.512 | | 382.512 | ||
| | | | ||
| Line 244: | Line 257: | ||
| 49/48 | | 49/48 | ||
| 383.924 | | 383.924 | ||
| | |||
|- | |||
| 73\228 | |||
| | |||
| 384.211 | |||
| | | | ||
|- | |- | ||
| Line 259: | Line 277: | ||
| 21/20 | | 21/20 | ||
| 384.467 | | 384.467 | ||
| | |||
|- | |||
| 50\156 | |||
| | |||
| 384.615 | |||
| | |||
|- | |||
| 77\240 | |||
| | |||
| 385.000 | |||
| | | | ||
|- | |- | ||
Latest revision as of 08:44, 11 March 2026
| Compton |
225/224, 250047/250000 (7-limit)
9-odd-limit: 3.91 ¢
9-odd-limit: 36 notes
Compton is a temperament that takes 12edo's circle of fifths for the 3-limit, but the fifth harmonic is given its own generator instead of being mapped to one of 12edo's intervals. Essentially, it is the 5-limit temperament which tempers out the Pythagorean comma, 531441/524288. This equates any Pythagorean interval with its enharmonic counterparts, for example, the diminished fourth 8192/6561 with the major third 81/64, and the two kinds of Pythagorean semitones, diatonic 256/243 and chromatic 2187/2048, are merged into a single interval of 1/12 octave, which serves as the period. The generator can then be seen as any ptolemaic interval (the alteration of a Pythagorean interval by a syntonic comma), but is most usefully 5/4, the ptolemaic major third, or 81/80, the syntonic comma itself.
One may observe that simple septimal intervals are usually about twice as far away from 12edo intervals as simple classical intervals are. As such, the septimal comma 64/63 can be mapped to two syntonic commas. This tempers out 413343/409600, and can also be seen as tempering out 225/224.
In terms of equal temperaments, compton is only supported by those that are a multiple of 12, such as 60edo, 72edo, 84edo, 96edo, 108edo. 240edo is a recommendable tuning for both the 5- and 7-limit.
For technical data, see Compton family #Compton.
Extensions
Assuming one has chosen to approximate the 7-limit, the comma generator will be around 15–17 cents. This means that the 11th harmonic can be reached either by going up or down three steps. Going down three steps results in the canonical extension of compton. Stepping down one more comma, depending on the tuning, can lead to the 13th harmonic, and results in the canonical tridecimal compton. This works best with tunings of the comma around 15 cents.
Alternatively, any prime may be merged into its 12edo mapping, making the smallest available prime the generator. This is done in catler (which shares 12edo's 5-limit and is generated by 7) and duodecim (which shares 12edo's 7-limit and is generated by 11). This is a natural choice for 17 and 19, as 12edo tunes those primes especially well, so compton can be seen as a 19-limit temperament. The next step down, which ends up at around 75 cents flat of the corresponding 12edo interval, can be used for 23 and 29, though two steps up (30 cents sharp) is also a reasonable (but less intuitive) choice.
Interval chain
In the following table, odd harmonics 1–21 are in bold.
| Period | Generator 0 | Generator 1 | Generator 2 | |||
|---|---|---|---|---|---|---|
| Cents* | Approx. ratios | Cents* | Approx. ratios | Cents* | Approx. ratios | |
| 0 | 0.0 | 1/1 | ||||
| 1 | 100.0 | 200/189 | 84.1 | 21/20 | 68.3 | 25/24, 28/27 |
| 2 | 200.0 | 9/8 | 184.1 | 10/9 | 168.3 | 441/400 |
| 3 | 300.0 | 25/21, 32/27 | 284.1 | 147/125 | 268.3 | 7/6 |
| 4 | 400.0 | 63/50 | 384.1 | 5/4 | 368.3 | 100/81 |
| 5 | 500.0 | 4/3 | 484.1 | 250/189 | 468.3 | 21/16 |
| 6 | 600.0 | 567/400 | 584.1 | 7/5 | 568.3 | 25/18 |
| 7 | 700.0 | 3/2 | 684.1 | 40/27 | 668.3 | 147/100 |
| 8 | 800.0 | 100/63 | 784.1 | 63/40 | 768.3 | 14/9 |
| 9 | 900.0 | 27/16, 42/25 | 884.1 | 5/3 | 868.3 | 400/243 |
| 10 | 1000.0 | 16/9 | 984.1 | 441/250 | 968.3 | 7/4 |
| 11 | 1100.0 | 189/100 | 1084.1 | 15/8, 28/15 | 1068.3 | 50/27 |
| 12 | 1200.0 | 2/1 | 1184.1 | 125/63, 160/81 | 1168.3 | 49/25, 63/32 |
* In 7-limit CWE tuning, octave reduced
Tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Equilateral | CEE: ~5/4 = 386.3137 ¢ | CSEE: ~5/4 = 385.6620 ¢ | POEE: ~5/4 = 384.2658 ¢ |
| Tenney | CTE: ~5/4 = 386.3137 ¢ | CWE: ~5/4 = 385.3590 ¢ | POTE: ~5/4 = 384.8824 ¢ |
| Benedetti, Wilson |
CBE: ~5/4 = 386.3137 ¢ | CSBE: ~5/4 = 385.2276 ¢ | POBE: ~5/4 = 384.8036 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Equilateral | CEE: ~5/4 = 384.7931 ¢ | CSEE: ~5/4 = 384.240 ¢ | POEE: ~5/4 = 383.0895 ¢ |
| Tenney | CTE: ~5/4 = 384.9217 ¢ | CWE: ~5/4 = 384.1429 ¢ | POTE: ~5/4 = 383.7752 ¢ |
| Benedetti, Wilson |
CBE: ~5/4 = 385.0380 ¢ | CSBE: ~5/4 = 384.0879 ¢ | POBE: ~5/4 = 383.8295 ¢ |
Tuning spectrum
| Edo generator |
Unchanged interval (eigenmonzo)* |
Generator (¢) | Comments |
|---|---|---|---|
| 15\48 | 375.000 | 48d val, lower bound of 7- and 9-odd-limit diamond monotone | |
| 19\60 | 380.000 | ||
| 15/14 | 380.557 | ||
| 9/5 | 382.404 | ||
| 9/7 | 382.458 | ||
| 7/5 | 382.512 | ||
| 23\72 | 383.333 | ||
| 7/6 | 383.435 | ||
| 36/35 | 383.743 | ||
| 49/48 | 383.924 | ||
| 73\228 | 384.211 | ||
| 5/3 | 384.359 | ||
| 7/4 | 384.413 | ||
| 21/20 | 384.467 | ||
| 50\156 | 384.615 | ||
| 77\240 | 385.000 | ||
| 25/24 | 385.336 | ||
| 21/16 | 385.390 | ||
| 27\84 | 385.714 | ||
| 5/4 | 386.314 | ||
| 31\96 | 387.500 | ||
| 15/8 | 388.269 | ||
| 4\12 | 400.000 | Upper bound of 7- and 9-odd-limit diamond monotone |
* Besides the octave