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[[Category:Temperaments]]
{{Infobox regtemp
[[Category:Compton family]]
| Title = Compton
'''Compton''' is a 5-limit regular temperament similar to [[12edo]], except that instead of being mapped to one of 12edo's intervals, the [[5/1|fifth harmonic]] is given its own generator. Equivalently, it is the rank-2 temperament which tempers out the Pythagorean comma [[531441/524288]]. This equates Pythagorean comma-flat or -sharp intervals with their simpler counterparts (for example, the comma-flat major third [[8192/6561]] with the standard major third [[81/64]]), and if the comma-flat third is seen as a diminished fourth, it can be seen as tempering together the two kinds of Pythagorean semitones, diatonic [[256/243]] and chromatic [[2187/2048]], 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 or 81/80. The generator does not have any explicit constraints, unlike in many other temperaments.  
| 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.  


As such, in terms of equal temperaments, compton is only supported by equal temperaments that are a multiple of 12, with 60edo, 72edo, 84edo, 96edo, 108edo, and [[240edo]] perhaps being best.
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]].  


For technical data, see [[Compton family#Compton]].
In terms of [[equal temperament]]s, compton is only [[support]]ed 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.  


== Higher-limit extensions ==
For technical data, see [[Compton family #Compton]].
If 7-limit intervals are desired, 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]]. This implies that [[72edo]] is a good tuning, as it tempers both the 5-limit and 7-limit intervals of compton close to their just counterparts. For the 2.3.7 subgroup, [[36edo]] is a good tuning.


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.
== Extensions ==
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.
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 ==
{{Todo|verify|inline=1|text=Is this 11- and 13-limit mapping correct?}}
In the following table, odd harmonics 1–21 are in '''bold'''.
{| class="wikitable"
 
{| class="wikitable center-1 right-2 right-4 right-6"
|+
|+
!Generator steps
! rowspan="2" | Period
!Cents
! colspan="2" | Generator 0
!Intervals (down from 400c)
! colspan="2" | Generator 1
!Intervals (up from 400c)
! colspan="2" | Generator 2
!Intervals (up from 0c)
|-
!Harmonics
! 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
|-
|-
|1
| 7
|0
| 700.0
|81/64
| '''3/2'''
|81/64
| 684.1
|1/1
| 40/27
|3/2, 17/16, 19/16
| 668.3
| 147/100
|-
|-
|2
| 8
|15.4
| 800.0
|5/4
| 100/63
|
| 784.1
|81/80
| 63/40
|5/4
| 768.3
| 14/9
|-
|-
|3
| 9
|30.8
| 900.0
|
| 27/16, 42/25
|9/7
| 884.1
|64/63
| 5/3
|7/4
| 868.3
| 400/243
|-
|-
|4
| 10
|46.2
| 1000.0
|11/9
| 16/9
|13/10
| 984.1
|
| 441/250
|11/8
| 968.3
| '''7/4'''
|-
|-
|5
| 11
|61.5
| 1100.0
|39/32
| 189/100
|
| 1084.1
|
| '''15/8''', 28/15
|13/8
| 1068.3
| 50/27
|-
| 12
| 1200.0
| '''2/1'''
| 1184.1
| 125/63, 160/81
| 1168.3
| 49/25, 63/32
|}
|}
<nowiki/>* In 7-limit CWE tuning, octave reduced
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Equilateral
| CEE: ~5/4 = 386.3137{{c}}
| CSEE: ~5/4 = 385.6620{{c}}
| POEE: ~5/4 = 384.2658{{c}}
|-
! Tenney
| CTE: ~5/4 = 386.3137{{c}}
| CWE: ~5/4 = 385.3590{{c}}
| POTE: ~5/4 = 384.8824{{c}}
|-
! Benedetti, <br>Wilson
| CBE: ~5/4 = 386.3137{{c}}
| CSBE: ~5/4 = 385.2276{{c}}
| POBE: ~5/4 = 384.8036{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Equilateral
| CEE: ~5/4 = 384.7931{{c}}
| CSEE: ~5/4 = 384.240{{c}}
| POEE: ~5/4 = 383.0895{{c}}
|-
! Tenney
| CTE: ~5/4 = 384.9217{{c}}
| CWE: ~5/4 = 384.1429{{c}}
| POTE: ~5/4 = 383.7752{{c}}
|-
! Benedetti, <br>Wilson
| CBE: ~5/4 = 385.0380{{c}}
| CSBE: ~5/4 = 384.0879{{c}}
| POBE: ~5/4 = 383.8295{{c}}
|}
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(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
|}
<nowiki/>* Besides the octave
[[Category:Compton| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Compton family]]
[[Category:Marvel temperaments]]
[[Category:Landscape microtemperaments]]

Latest revision as of 08:44, 11 March 2026

Compton
Subgroups 2.3.5, 2.3.5.7
Comma basis 531441/524288 (5-limit);
225/224, 250047/250000 (7-limit)
Reduced mapping ⟨12; 0 1 2]
ET join 12 & 72
Generators (CWE) ~5/4 = 384.1 ¢
MOS scales 12L 12s, 12L 24s
Ploidacot dodecaploid acot
Minimax error 5-odd-limit: 1.96 ¢;
9-odd-limit: 3.91 ¢
Target scale size 5-odd-limit: 24 notes;
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

5-limit norm-based 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 ¢
7-limit norm-based tunings
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

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