Syntonic–chromatic equivalence continuum: Difference between revisions

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{{Technical data page}}<br><br>
{{Technical data page}}
The '''syntonic–chromatic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equate a number of [[81/80|syntonic commas (81/80)]] with the [[2187/2048|apotome (2187/2048)]]. This continuum is theoretically interesting in that these are all 5-limit temperaments [[support]]ed by [[7edo]].  
The '''syntonic–chromatic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equate a number of [[81/80|syntonic commas (81/80)]] with the [[2187/2048|apotome (2187/2048)]]. This continuum is theoretically interesting in that these are all 5-limit temperaments [[support]]ed by [[7edo]].  


All temperaments in the continuum satisfy {{nowrap|(81/80)<sup>''n''</sup> ~ 2187/2048}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 7edo due to it being the unique equal temperament that [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is 5.2861…, and temperaments near this tend to be the most accurate ones.  
All temperaments in the continuum satisfy {{nowrap| (81/80)<sup>''n''</sup> ~ 2187/2048 }}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 7edo due to it being the unique equal temperament that [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is 5.2861…, and temperaments near this tend to be the most accurate ones.  


2187/2048 is the characteristic [[3-limit]] comma tempered out in [[7edo]], and has many advantages as a target. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1|harmonic 3]] in the generator chain. For example:  
2187/2048 is the characteristic [[3-limit]] comma tempered out in [[7edo]], and has many advantages as a target. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1|harmonic 3]] in the generator chain. For example:  
* [[Mavila]] {{nowrap|(''n'' {{=}} 1)}} is generated by a fifth;
* [[Mavila]] ({{nowrap| ''n'' {{=}} 1 }}) is generated by a fifth;
* [[Dicot]] {{nowrap|(''n'' {{=}} 2)}} splits its fifth in two;  
* [[Dicot]] ({{nowrap| ''n'' {{=}} 2 }}) splits its fifth in two;  
* [[Porcupine]] {{nowrap|(''n'' {{=}} 3)}} splits its fourth in three;  
* [[Porcupine]] ({{nowrap| ''n'' {{=}} 3 }}) splits its fourth in three;  
* Etc.
* Etc.


At {{nowrap|''n'' {{=}} 7}}, the corresponding temperament splits the ''octave'' into seven instead, as after a stack of seven syntonic commas, both the orders of 3 and 5 are multiples of 7 again.  
At {{nowrap|''n'' {{=}} 7}}, the corresponding temperament splits the ''octave'' into seven instead, as after a stack of seven syntonic commas, both the orders of 3 and 5 are multiples of 7 again.  


If we let {{nowrap|''k'' {{=}} ''n'' &minus; 2}} so that {{nowrap|''k'' {{=}} 0}} means {{nowrap|''n'' {{=}} 2}}, {{nowrap|''k'' {{=}} &minus;1}} means {{nowrap|''n'' {{=}} 1}}, etc. then the continuum corresponds to {{nowrap|(81/80)<sup>''k''</sup> {{=}} 25/24}}. Some prefer this way of conceptualising it because:
If we let {{nowrap| ''k'' {{=}} ''n'' − 2 }} so that {{nowrap| ''k'' {{=}} 0 }} means {{nowrap| ''n'' {{=}} 2 }}, {{nowrap| ''k'' {{=}} −1 }} means {{nowrap| ''n'' {{=}} 1 }}, etc. then the continuum corresponds to {{nowrap| (81/80)<sup>''k''</sup> {{=}} 25/24 }}. Some prefer this way of conceptualising it because:
* 25/24 is the classic chromatic semitone, notable in the 5-limit as the difference between 5/4 and 6/5, so this shifted continuum could also logically be termed the "syntonic–chromatic equivalence continuum". This means that at {{nowrap|''k'' {{=}} 0}}, 5/4 and 6/5 are mapped to the same interval while 81/80 becomes independent of 25/24 (meaning 81/80 may or may not be tempered) because the relation becomes {{nowrap|(81/80)<sup>0</sup> ~ 1/1 ~ 25/24}}.
* 25/24 is the classic chromatic semitone, notable in the 5-limit as the difference between 5/4 and 6/5, so this shifted continuum could also logically be termed the "syntonic–chromatic equivalence continuum". This means that at {{nowrap| ''k'' {{=}} 0 }}, 5/4 and 6/5 are mapped to the same interval while 81/80 becomes independent of 25/24 (meaning 81/80 may or may not be tempered) because the relation becomes {{nowrap| (81/80)<sup>0</sup> ~ 1/1 ~ 25/24 }}.
* {{nowrap|''k'' {{=}} 1}} and upwards up to a point represent temperaments with the potential for reasonably good accuracy as equating at least one 81/80 with 25/24 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be gravity {{nowrap|(''k'' {{=}} 4)}}, with the only exception being meantone {{nowrap|(''n'' {{=}} ''k'' {{=}} &infin;)}}. Temperaments corresponding to {{nowrap|''k'' {{=}} 0, &minus;1, &minus;2, …}} are comparatively low-accuracy to the point of developing various intriguing structures and consequences.
* {{nowrap| ''k'' {{=}} 1 }} and upwards up to a point represent temperaments with the potential for reasonably good accuracy as equating at least one 81/80 with 25/24 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be gravity ({{nowrap| ''k'' {{=}} 4 }}), with the only exception being meantone ({{nowrap| ''n'' {{=}} ''k'' {{=}} ∞ }}). Temperaments corresponding to {{nowrap| ''k'' {{=}} 0, −1, −2, … }} are comparatively low-accuracy to the point of developing various intriguing structures and consequences.
* 25/24 is the simplest ratio to be tempered in the continuum, the second simplest being 81/80 at unsigned infinity, which together are the two smallest 5-limit [[List of superparticular intervals|superparticular intervals]] and the only superparticular intervals in the continuum.
* 25/24 is the simplest ratio to be tempered in the continuum, the second simplest being 81/80 at unsigned infinity, which together are the two smallest 5-limit [[List of superparticular intervals|superparticular intervals]] and the only superparticular intervals in the continuum.


Line 32: Line 32:
| [[Nadir]]
| [[Nadir]]
| [[1162261467/1048576000]]
| [[1162261467/1048576000]]
| {{monzo| -23 19 -3 }}
| {{Monzo| -23 19 -3 }}
|-
|-
| −4
| −4
| −2
| −2
| Nethertone
| [[Nethertone]]
| [[14348907/13107200]]
| [[14348907/13107200]]
| {{monzo| -19 15 -2 }}
| {{Monzo| -19 15 -2 }}
|-
|-
| −3
| −3
| −1
| −1
| Deeptone a.k.a. tragicomical
| [[Deeptone]] a.k.a. tragicomical
| [[177147/163840]]
| [[177147/163840]]
| {{monzo| -15 11 -1 }}
| {{Monzo| -15 11 -1 }}
|-
|-
| −2
| −2
Line 50: Line 50:
| [[Whitewood]]
| [[Whitewood]]
| [[2187/2048]]
| [[2187/2048]]
| {{monzo| -11 7 }}
| {{Monzo| -11 7 }}
|-
|-
| −1
| −1
Line 56: Line 56:
| [[Mavila]]
| [[Mavila]]
| [[135/128]]
| [[135/128]]
| {{monzo| -7 3 1 }}
| {{Monzo| -7 3 1 }}
|-
|-
| 0
| 0
Line 62: Line 62:
| [[Dicot]]
| [[Dicot]]
| [[25/24]]
| [[25/24]]
| {{monzo| -3 -1 2 }}
| {{Monzo| -3 -1 2 }}
|-
|-
| 1
| 1
Line 68: Line 68:
| [[Porcupine]]
| [[Porcupine]]
| [[250/243]]
| [[250/243]]
| {{monzo| 1 -5 3 }}
| {{Monzo| 1 -5 3 }}
|-
|-
| 2
| 2
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| [[Tetracot]]
| [[Tetracot]]
| [[20000/19683]]
| [[20000/19683]]
| {{monzo| 5 -9 4 }}
| {{Monzo| 5 -9 4 }}
|-
|-
| 3
| 3
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| [[Amity]]
| [[Amity]]
| [[1600000/1594323]]
| [[1600000/1594323]]
| {{monzo| 9 -13 5 }}
| {{Monzo| 9 -13 5 }}
|-
|-
| 4
| 4
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| [[Gravity]]
| [[Gravity]]
| [[129140163/128000000]]
| [[129140163/128000000]]
| {{monzo| -13 17 -6 }}
| {{Monzo| -13 17 -6 }}
|-
|-
| 5
| 5
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| [[Absurdity]]
| [[Absurdity]]
| 10460353203/10240000000
| 10460353203/10240000000
| {{monzo| -17 21 -7 }}
| {{Monzo| -17 21 -7 }}
|-
|-
| …
| …
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| [[Meantone]]
| [[Meantone]]
| [[81/80]]
| [[81/80]]
| {{monzo| -4 4 -1 }}
| {{Monzo| -4 4 -1 }}
|}
|}


We may invert the continuum by setting ''m'' such that {{nowrap|{{frac|1|''m''}} + {{frac|1|''n''}} {{=}} 1}}. This may be called the ''mavila/pelogic-chromatic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.2333… The [[135/128|mavila comma]] is both larger and more complex than the syntonic comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.  
We may invert the continuum by setting ''m'' such that {{nowrap| 1/''m'' + 1/''n'' {{=}} 1}}. This may be called the ''mavila–chromatic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.2333… The [[135/128|mavila comma]] is both larger and more complex than the syntonic comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.  


{| class="wikitable center-1"
{| class="wikitable center-1"
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| [[Shallowtone]]
| [[Shallowtone]]
| [[295245/262144]]
| [[295245/262144]]
| {{monzo| -18 10 1 }}
| {{Monzo| -18 10 1 }}
|-
|-
| 0
| 0
| [[Whitewood]]
| [[Whitewood]]
| [[2187/2048]]
| [[2187/2048]]
| {{monzo| -11 7 }}
| {{Monzo| -11 7 }}
|-
|-
| 1
| 1
| [[Meantone]]
| [[Meantone]]
| [[81/80]]
| [[81/80]]
| {{monzo| -4 4 -1 }}
| {{Monzo| -4 4 -1 }}
|-
|-
| 2
| 2
| [[Dicot]]
| [[Dicot]]
| [[25/24]]
| [[25/24]]
| {{monzo| -3 -1 2 }}
| {{Monzo| -3 -1 2 }}
|-
|-
| 3
| 3
| [[Enipucrop]]
| [[Enipucrop]]
| [[1125/1024]]
| [[1125/1024]]
| {{monzo| -10 2 3 }}
| {{Monzo| -10 2 3 }}
|-
|-
| …
| …
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| [[Mavila]]
| [[Mavila]]
| [[135/128]]
| [[135/128]]
| {{monzo| -7 3 1 }}
| {{Monzo| -7 3 1 }}
|}
|}


Line 190: Line 190:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2/1 = 1200.000, ~3/2 = 689.879
* [[WE]]: ~2 = 1203.7664{{c}}, ~3/2 = 691.1525{{c}}
* [[CWE]]: ~2/1 = 1200.000, ~3/2 = 689.312
: [[error map]]: {{val| +3.766 -7.036 +1.298 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 689.3122{{c}}
: error map: {{val| 0.000 -12.643 -3.879 }}


{{Optimal ET sequence|legend=1| 7, 33, 40, 47, 87b }}
{{Optimal ET sequence|legend=1| 7, 33, 40, 47, 54b }}


[[Badness]] (Smith): 0.403
[[Badness]] (Sintel): 9.44


== Shallowtone (5-limit) ==
== Shallowtone (5-limit) ==
: ''For extensions, see [[Mint temperaments #Shallowtone]].''
: ''For extensions, see [[Mint temperaments #Shallowtone]].''


Shallowtone is generated by a fifth, which is typically sharper than in [[mavila]] but flatter than in [[7edo]]. The ~5/4 is reached by minus ten fifths octave-reduced, which is an augmented third (C-E𝄪) in melodic [[2L 5s|antidiatonic]] notation and a diminished third (C-E𝄫) in harmonic antidiatonic notation.
Shallowtone is generated by a fifth, which is typically sharper than in [[mavila]] but flatter than in [[7edo]]. The ~5/4 is reached by minus ten fifths octave-reduced, which is an augmented third (C–E𝄪) in melodic [[2L 5s|antidiatonic]] notation and a diminished third (C–E𝄫) in harmonic antidiatonic notation.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 211: Line 213:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~3/2 = 681.801
* [[WE]]: ~2 = 1206.3211{{c}}, ~3/2 = 686.6308{{c}}
* [[CWE]]: ~2 = 1200.000, ~3/2 = 682.662
: [[error map]]: {{val| +6.321 -9.003 -2.053 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 682.6617{{c}}
: error map: {{val| 0.000 -19.293 -12.931 }}


{{Optimal ET sequence|legend=1| 7, 30b, 37b, 44b, 51b, 58bc, 65bbc }}
{{Optimal ET sequence|legend=1| 7, 30b, 37b, 44b, 51b, 58bc, 65bbc }}


[[Badness]] (Smith): 0.666
[[Badness]] (Sintel): 15.6


== Nethertone ==
== Nethertone ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~2560/2187 = 345.946
* [[WE]]: ~2 = 1203.1196{{c}}, ~2560/2187 = 346.2857{{c}}
* [[CWE]]: ~2 = 1200.000, ~2560/2187 = 345.599
: [[error map]]: {{val| +3.120 -6.264 +1.733 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2560/2187 = 345.5992{{c}}
: error map: {{val| 0.000 -10.757 -2.326 }}


{{Optimal ET sequence|legend=1| 7, 38c, 45c, 52, 59b, 66b }}
{{Optimal ET sequence|legend=1| 7, 38c, 45c, 52, 59b, 66b }}


[[Badness]] (Smith): 0.828
[[Badness]] (Sintel): 19.4


== Enipucrop ==
== Enipucrop ==
Enipucrop corresponds to {{nowrap|''n'' {{=}} 3/2}} and {{nowrap|''m'' {{=}} 3}}, and can be described as the 6b & 7 temperament. Its name is ''porcupine'' spelled backwards, because that is what this temperament is – it is porcupine, with the generator sharp of 2\7 such that the major and minor thirds switch places. The fifths are very flat, meaning that this is more of a melodic temperament than a harmonic one.
Enipucrop corresponds to {{nowrap| ''n'' {{=}} 3/2 }} and {{nowrap| ''m'' {{=}} 3 }}, and can be described as the {{nowrap| 6b & 7 }} temperament. Its name is ''porcupine'' spelled backwards, because that is what this temperament is – it is porcupine, with the generator sharp of 2\7 such that the major and minor thirds switch places. The fifths are very flat, meaning that this is more of a melodic temperament than a harmonic one.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~16/15 = 170.671
* [[WE]]: ~2 = 1210.1294{{c}}, ~16/15 = 174.5613{{c}}
* [[POTE]]: ~2 = 1200.000, ~16/15 = 173.101
: [[error map]]: {{val| +3.120 -6.264 +1.733 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 172.4620{{c}}
: error map: {{val| 0.000 -10.757 -2.326 }}


{{Optimal ET sequence|legend=1| 6b, 7 }}
{{Optimal ET sequence|legend=1| 6b, 7 }}


[[Badness]] (Smith): 0.1439
[[Badness]] (Sintel): 3.38


== Nadir ==
== Nadir ==
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~729/640 = 168.983
* [[WE]]: ~2 = 1202.743{{c}}, ~729/640 = 169.7633{{c}}
* [[CWE]]: ~2 = 1200.000, ~729/640 = 169.223
: [[error map]]: {{val| +2.743 -5.758 +1.900 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~729/640 = 169.2234{{c}}
: error map: {{val| 0.000 -9.625 -1.559 }}


{{Optimal ET sequence|legend=1| 7, 57c, 64, 71b, 78b, 85b }}
{{Optimal ET sequence|legend=1| 7, 57c, 64, 71b, 78b, 85b }}


[[Badness]] (Smith): 1.47
[[Badness]] (Sintel): 34.6


== Sixix (5-limit) ==
== Sixix (5-limit) ==
Line 283: Line 293:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~6/5 = 338.005
* [[WE]]: ~2 = 1200.9934{{c}}, ~6/5 = 338.6456{{c}}
: [[error map]]: {{val| 0.000 +8.020 -14.344 }}
: [[error map]]: {{val| +0.993 +7.797 -14.214 }}
* [[POTE]]: ~2 = 1200.000, ~6/5 = 338.365
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 338.1959{{c}}
: error map: {{val| 0.000 +6.217 -16.507 }}
: error map: {{val| 0.000 +7.065 -15.489 }}


{{Optimal ET sequence|legend=1| 7, 25, 32, 39c }}
{{Optimal ET sequence|legend=1| 7, 25, 32, 39c }}


[[Badness]] (Smith): 0.153088
[[Badness]] (Sintel): 3.59


== Absurdity (5-limit) ==
== Absurdity (5-limit) ==
: ''For extensions, see [[Porwell temperaments #Absurdity]].''
: ''For extensions, see [[Porwell temperaments #Absurdity]].''


Absurdity corresponds to {{nowrap|''n'' {{=}} 7}}, and can be described as the 77 & 84 temperament, so named because it truly is an absurd temperament. The generator is ~81/80 and the period is ~800/729, which is (10/9)/(81/80).
Absurdity corresponds to {{nowrap| ''n'' {{=}} 7 }}, and can be described as the {{nowrap| 77 & 84 }} temperament, so named because it truly is an absurd temperament. The generator is ~81/80 and the period is ~800/729, which is (10/9)/(81/80). It tempers out the ''absurditon'', 10460353203/10240000000.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 306: Line 316:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~800/729 = 171.429, ~3/2 = 700.538 (~81/80 = 14.824)
* [[WE]]: ~800/729 = 171.4824{{c}}, ~3/2 = 700.4067{{c}} (~81/80 = 14.4772{{c}})
* [[POTE]]: ~800/729 = 171.429, ~3/2 = 700.187 (~81/80 = 14.473)
: [[error map]]: {{val| +0.376 -1.172 +0.836 }}
* [[CWE]]: ~800/729 = 171.4286{{c}}, ~3/2 = 700.3453{{c}} (~81/80 = 14.6310{{c}})
: error map: {{val| 0.000 +7.065 -15.489 }}


{{Optimal ET sequence|legend=1| 7, …, 70, 77, 84, 329, 413b, 497b }}
{{Optimal ET sequence|legend=1| 7, …, 70, 77, 84, 329, 413b, 497b }}


[[Badness]] (Smith): 0.341202
[[Badness]] (Sintel): 8.00


== Sevond (5-limit) ==
== Sevond (5-limit) ==
: ''For extensions, see [[Keemic temperaments #Sevond]].''
: ''For extensions, see [[Keemic temperaments #Sevond]].''


Sevond is a fairly obvious temperament; it just equates a stack of seven ~10/9's with ~2/1, hence the period is ~10/9. One generator from 5\7 puts you at ~3/2, two generators from 2\7 puts you at ~5/4. This corresponds to {{nowrap|''n'' {{=}} 7/2}} and {{nowrap|''m'' {{=}} 7/5}} and can be described as the {{nowrap|56 &amp; 63}} temperament.  
Sevond is a fairly obvious temperament; it just equates a stack of seven ~10/9's with ~2/1, hence the period is ~10/9. One generator from 5\7 puts you at ~3/2, two generators from 2\7 puts you at ~5/4. This corresponds to {{nowrap|''n'' {{=}} 7/2}} and {{nowrap|''m'' {{=}} 7/5}} and can be described as the {{nowrap| 56 & 63 }} temperament.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 323: Line 335:


{{Mapping|legend=1| 7 0 -6 | 0 1 2 }}
{{Mapping|legend=1| 7 0 -6 | 0 1 2 }}
: mapping generators: ~10/9, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~10/9 = 171.429, ~3/2 = 705.526 (~250/243 = 19.812)
* [[WE]]: ~10/9 = 171.3446{{c}}, ~3/2 = 705.9421{{c}} (~250/243 = 20.5637{{c}})
* [[POTE]]: ~10/9 = 171.429, ~3/2 = 706.288 (~250/243 = 20.574)
: [[error map]]: {{val| -0.588 +3.399 -3.673 }}
* [[CWE]]: ~10/9 = 171.4286{{c}}, ~3/2 = 705.9119{{c}} (~250/243 = 20.1977{{c}})
: error map: {{val| 0.000 +3.957 -3.061 }}


{{Optimal ET sequence|legend=1| 7, 42, 49, 56, 119 }}
{{Optimal ET sequence|legend=1| 7, …, 42, 49, 56, 119 }}


[[Badness]] (Smith): 0.339335
[[Badness]] (Sintel): 7.96


== Seville ==
== Seville ==
Seville is similar to sevond, but provides a less complex avenue to 5, but this time at the cost of accuracy. One generator from 5\7 puts you at ~3/2, and one generator from 2\7 puts you at ~5/4. This corresponds to {{nowrap|''n'' {{=}} 7/3}} and {{nowrap|''m'' {{=}} 7/4}}.  
Seville is similar to sevond, but provides a less complex avenue to 5, but this time at the cost of accuracy. One generator from 5\7 puts you at ~3/2, and one generator from 2\7 puts you at ~5/4. This corresponds to {{nowrap| ''n'' {{=}} 7/3 }} and {{nowrap| ''m'' {{=}} 7/4 }}.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 340: Line 356:


{{Mapping|legend=1| 7 0 5 | 0 1 1 }}
{{Mapping|legend=1| 7 0 5 | 0 1 1 }}
: mapping generators: ~125/108, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~125/108 = 171.429, ~3/2 = 710.606 (~25/24 = 24.891)
* [[WE]]: ~125/108 = 171.7481{{c}}, ~3/2 = 707.7272{{c}} (~25/24 = 20.7346{{c}})
* [[POTE]]: ~125/108 = 171.429, ~3/2 = 706.410 (~25/24 = 20.696)
: [[error map]]: {{val| +2.237 +8.009 -17.609 }}
* [[CWE]]: ~125/108 = 171.4286{{c}}, ~3/2 = 708.3739{{c}} (~25/24 = 22.6596{{c}})
: error map: {{val| 0.000 +6.419 -20.797 }}


{{Optimal ET sequence|legend=1| 7, 35b, 42c }}
{{Optimal ET sequence|legend=1| 7, 35b, 42c }}


[[Badness]] (Smith): 0.4377
[[Badness]] (Sintel): 10.3


== Artoneutral (5-limit) ==
== Artoneutral (5-limit) ==
: ''For extensions, see [[Hemifamity temperaments #Artoneutral]].''
: ''For extensions, see [[Hemifamity temperaments #Artoneutral]].''


5-limit artoneutral is generated by ~243/200 (or ~400/243), same as that of [[amity]] but sharper. This corresponds to {{nowrap|''n'' {{=}} 9/2}} and {{nowrap|''m'' {{=}} 9/7}} and can be described as the 80 & 87 temperament, though [[94edo]] is a notable tuning not appearing in the optimal ET sequence.  
5-limit artoneutral is generated by ~243/200 (or ~400/243), same as that of [[amity]] but sharper. This corresponds to {{nowrap| ''n'' {{=}} 9/2 }} and {{nowrap| ''m'' {{=}} 9/7 }} and can be described as the {{nowrap| 80 & 87 }} temperament, though [[94edo]] is a notable tuning not appearing in the optimal ET sequence.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 358: Line 378:
[[Comma list]]: {{monzo| 14 -22 9 }}
[[Comma list]]: {{monzo| 14 -22 9 }}


{{Mapping|legend=1| 1 8 18 | 0 -9 -22 }}
{{Mapping|legend=1| 1 -1 -4 | 0 9 22 }}


: mapping generators: ~2, ~400/243
: mapping generators: ~2, ~243/200


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~400/243 = 855.213
* [[WE]]: ~2 = 1199.7434{{c}}, ~243/200 = 344.7454{{c}}
* [[CWE]]: ~2 = 1200.000, ~400/243 = 855.196
: [[error map]]: {{val| -0.257 +1.010 -0.889 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/200 = 344.8041{{c}}
: error map: {{val| 0.000 +1.282 -0.624 }}


{{Optimal ET sequence|legend=1| 7, … 73, 80, 87 }}
{{Optimal ET sequence|legend=1| 7, … 73, 80, 87 }}


[[Badness]] (Smith): 0.348
[[Badness]] (Sintel): 8.17
 
== Geb (5-limit) ==
: ''For extensions, see [[Metric microtemperaments #Geb]] and [[Breedsmic temperaments #Osiris]].''
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| -31 43 -16 }}
 
{{Mapping|legend=1| 1 -3 -10 | 0 16 43 }}
 
: mapping generators: ~2, ~8000/6561
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0135{{c}}, ~8000/6561 = 343.8718{{c}}
: [[error map]]: {{val| +0.014 -0.047 +0.038 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8000/6561 = 343.8683{{c}}
: error map: {{val| 0.000 -0.062 +0.024 }}
 
{{Optimal ET sequence|legend=1| 7, …, 157, 164, 171, 506, 677, 848 }}
 
[[Badness]] (Sintel): 2.78


[[Category:7edo]]
[[Category:7edo]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]

Latest revision as of 11:48, 6 September 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The syntonic–chromatic equivalence continuum is a continuum of 5-limit temperaments which equate a number of syntonic commas (81/80) with the apotome (2187/2048). This continuum is theoretically interesting in that these are all 5-limit temperaments supported by 7edo.

All temperaments in the continuum satisfy (81/80)n ~ 2187/2048. Varying n results in different temperaments listed in the table below. It converges to meantone as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 7edo due to it being the unique equal temperament that tempers out both commas and thus tempers out all combinations of them. The just value of n is 5.2861…, and temperaments near this tend to be the most accurate ones.

2187/2048 is the characteristic 3-limit comma tempered out in 7edo, and has many advantages as a target. In each case, n equals the order of harmonic 5 in the corresponding comma, and equals the number of steps to obtain the interval class of harmonic 3 in the generator chain. For example:

  • Mavila (n = 1) is generated by a fifth;
  • Dicot (n = 2) splits its fifth in two;
  • Porcupine (n = 3) splits its fourth in three;
  • Etc.

At n = 7, the corresponding temperament splits the octave into seven instead, as after a stack of seven syntonic commas, both the orders of 3 and 5 are multiples of 7 again.

If we let k = n − 2 so that k = 0 means n = 2, k = −1 means n = 1, etc. then the continuum corresponds to (81/80)k = 25/24. Some prefer this way of conceptualising it because:

  • 25/24 is the classic chromatic semitone, notable in the 5-limit as the difference between 5/4 and 6/5, so this shifted continuum could also logically be termed the "syntonic–chromatic equivalence continuum". This means that at k = 0, 5/4 and 6/5 are mapped to the same interval while 81/80 becomes independent of 25/24 (meaning 81/80 may or may not be tempered) because the relation becomes (81/80)0 ~ 1/1 ~ 25/24.
  • k = 1 and upwards up to a point represent temperaments with the potential for reasonably good accuracy as equating at least one 81/80 with 25/24 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be gravity (k = 4), with the only exception being meantone (n = k = ∞). Temperaments corresponding to k = 0, −1, −2, … are comparatively low-accuracy to the point of developing various intriguing structures and consequences.
  • 25/24 is the simplest ratio to be tempered in the continuum, the second simplest being 81/80 at unsigned infinity, which together are the two smallest 5-limit superparticular intervals and the only superparticular intervals in the continuum.
Temperaments with integer n
k n Temperament Comma
Ratio Monzo
−5 −3 Nadir 1162261467/1048576000 [-23 19 -3⟩
−4 −2 Nethertone 14348907/13107200 [-19 15 -2⟩
−3 −1 Deeptone a.k.a. tragicomical 177147/163840 [-15 11 -1⟩
−2 0 Whitewood 2187/2048 [-11 7⟩
−1 1 Mavila 135/128 [-7 3 1⟩
0 2 Dicot 25/24 [-3 -1 2⟩
1 3 Porcupine 250/243 [1 -5 3⟩
2 4 Tetracot 20000/19683 [5 -9 4⟩
3 5 Amity 1600000/1594323 [9 -13 5⟩
4 6 Gravity 129140163/128000000 [-13 17 -6⟩
5 7 Absurdity 10460353203/10240000000 [-17 21 -7⟩
… … … …
∞ ∞ Meantone 81/80 [-4 4 -1⟩

We may invert the continuum by setting m such that 1/m + 1/n = 1. This may be called the mavila–chromatic equivalence continuum, which is essentially the same thing. The just value of m is 1.2333… The mavila comma is both larger and more complex than the syntonic comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.

Temperaments with integer m
m Temperament Comma
Ratio Monzo
−1 Shallowtone 295245/262144 [-18 10 1⟩
0 Whitewood 2187/2048 [-11 7⟩
1 Meantone 81/80 [-4 4 -1⟩
2 Dicot 25/24 [-3 -1 2⟩
3 Enipucrop 1125/1024 [-10 2 3⟩
… … … …
∞ Mavila 135/128 [-7 3 1⟩
Temperaments with fractional n and m (and thus also k)
n k m Temperament Comma
7/3 = 2.3 1/3 = 0.3 7/4 = 1.75 Seville [-5 -7 7⟩
5/2 = 2.5 1/2 = 0.5 5/3 = 1.6 Sixix [-2 -6 5⟩
7/2 = 3.5 3/2 = 1.5 7/5 = 1.4 Sevond [6 -14 7⟩
9/2 = 4.5 5/2 = 2.5 9/7 = 1.285714 Artoneutral [14 -22 9⟩
21/4 = 5.25 13/4 = 3.25 21/17 = 1.235… Brahmagupta [40 -56 21⟩
37/7 = 5.285714 37/7 = 3+2/7 37/30 = 1.23 Raider [71 -99 37⟩
16/3 = 5.3 10/3 = 3.3 16/13 = 1.230769 Geb [-31 43 -16⟩
11/2 = 5.5 7/2 = 3.5 11/9 = 1.2 Undetrita [-22 30 -11⟩

Deeptone a.k.a. tragicomical

Main article: Deeptone

Deeptone is generated by a fifth, which is typically sharper than in 7edo but flatter than in flattone. The ~5/4 is reached by eleven fifths octave-reduced, which is an augmented third (C–E♯).

Subgroup: 2.3.5

Comma list: 177147/163840

Mapping: [⟨1 0 -15], ⟨0 1 11]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1203.7664 ¢, ~3/2 = 691.1525 ¢
error map: ⟨+3.766 -7.036 +1.298]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 689.3122 ¢
error map: ⟨0.000 -12.643 -3.879]

Optimal ET sequence: 7, 33, 40, 47, 54b

Badness (Sintel): 9.44

Shallowtone (5-limit)

For extensions, see Mint temperaments #Shallowtone.

Shallowtone is generated by a fifth, which is typically sharper than in mavila but flatter than in 7edo. The ~5/4 is reached by minus ten fifths octave-reduced, which is an augmented third (C–E𝄪) in melodic antidiatonic notation and a diminished third (C–E𝄫) in harmonic antidiatonic notation.

Subgroup: 2.3.5

Comma list: 295245/262144

Mapping: [⟨1 0 18], ⟨0 1 -10]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1206.3211 ¢, ~3/2 = 686.6308 ¢
error map: ⟨+6.321 -9.003 -2.053]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 682.6617 ¢
error map: ⟨0.000 -19.293 -12.931]

Optimal ET sequence: 7, 30b, 37b, 44b, 51b, 58bc, 65bbc

Badness (Sintel): 15.6

Nethertone

Subgroup: 2.3.5

Comma list: 14348907/13107200‎‎

Mapping: [⟨1 1 -1], ⟨0 2 15]]

mapping generators: ~2, ~2560/2187

Optimal tunings:

  • WE: ~2 = 1203.1196 ¢, ~2560/2187 = 346.2857 ¢
error map: ⟨+3.120 -6.264 +1.733]
  • CWE: ~2 = 1200.0000 ¢, ~2560/2187 = 345.5992 ¢
error map: ⟨0.000 -10.757 -2.326]

Optimal ET sequence: 7, 38c, 45c, 52, 59b, 66b

Badness (Sintel): 19.4

Enipucrop

Enipucrop corresponds to n = 3/2 and m = 3, and can be described as the 6b & 7 temperament. Its name is porcupine spelled backwards, because that is what this temperament is – it is porcupine, with the generator sharp of 2\7 such that the major and minor thirds switch places. The fifths are very flat, meaning that this is more of a melodic temperament than a harmonic one.

Subgroup: 2.3.5

Comma list: 1125/1024

Mapping: [⟨1 2 2], ⟨0 -3 2]]

mapping generators: ~2, ~16/15

Optimal tunings:

  • WE: ~2 = 1210.1294 ¢, ~16/15 = 174.5613 ¢
error map: ⟨+3.120 -6.264 +1.733]
  • CWE: ~2 = 1200.0000 ¢, ~16/15 = 172.4620 ¢
error map: ⟨0.000 -10.757 -2.326]

Optimal ET sequence: 6b, 7

Badness (Sintel): 3.38

Nadir

Subgroup: 2.3.5

Comma list: 1162261467/1048576000

Mapping: [⟨1 2 5], ⟨0 -3 -19]]

mapping generators: ~2, ~729/640

Optimal tunings:

  • WE: ~2 = 1202.743 ¢, ~729/640 = 169.7633 ¢
error map: ⟨+2.743 -5.758 +1.900]
  • CWE: ~2 = 1200.000 ¢, ~729/640 = 169.2234 ¢
error map: ⟨0.000 -9.625 -1.559]

Optimal ET sequence: 7, 57c, 64, 71b, 78b, 85b

Badness (Sintel): 34.6

Sixix (5-limit)

For extensions, see Archytas clan #Sixix.

Subgroup: 2.3.5

Comma list: 3125/2916

Mapping: [⟨1 3 4], ⟨0 -5 -6]]

mapping generators: ~2, ~6/5

Optimal tunings:

  • WE: ~2 = 1200.9934 ¢, ~6/5 = 338.6456 ¢
error map: ⟨+0.993 +7.797 -14.214]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 338.1959 ¢
error map: ⟨0.000 +7.065 -15.489]

Optimal ET sequence: 7, 25, 32, 39c

Badness (Sintel): 3.59

Absurdity (5-limit)

For extensions, see Porwell temperaments #Absurdity.

Absurdity corresponds to n = 7, and can be described as the 77 & 84 temperament, so named because it truly is an absurd temperament. The generator is ~81/80 and the period is ~800/729, which is (10/9)/(81/80). It tempers out the absurditon, 10460353203/10240000000.

Subgroup: 2.3.5

Comma list: 10460353203/10240000000

Mapping: [⟨7 0 -17], ⟨0 1 3]]

mapping generators: ~800/729, ~3

Optimal tunings:

  • WE: ~800/729 = 171.4824 ¢, ~3/2 = 700.4067 ¢ (~81/80 = 14.4772 ¢)
error map: ⟨+0.376 -1.172 +0.836]
  • CWE: ~800/729 = 171.4286 ¢, ~3/2 = 700.3453 ¢ (~81/80 = 14.6310 ¢)
error map: ⟨0.000 +7.065 -15.489]

Optimal ET sequence: 7, …, 70, 77, 84, 329, 413b, 497b

Badness (Sintel): 8.00

Sevond (5-limit)

For extensions, see Keemic temperaments #Sevond.

Sevond is a fairly obvious temperament; it just equates a stack of seven ~10/9's with ~2/1, hence the period is ~10/9. One generator from 5\7 puts you at ~3/2, two generators from 2\7 puts you at ~5/4. This corresponds to n = 7/2 and m = 7/5 and can be described as the 56 & 63 temperament.

Subgroup: 2.3.5

Comma list: 5000000/4782969

Mapping: [⟨7 0 -6], ⟨0 1 2]]

mapping generators: ~10/9, ~3

Optimal tunings:

  • WE: ~10/9 = 171.3446 ¢, ~3/2 = 705.9421 ¢ (~250/243 = 20.5637 ¢)
error map: ⟨-0.588 +3.399 -3.673]
  • CWE: ~10/9 = 171.4286 ¢, ~3/2 = 705.9119 ¢ (~250/243 = 20.1977 ¢)
error map: ⟨0.000 +3.957 -3.061]

Optimal ET sequence: 7, …, 42, 49, 56, 119

Badness (Sintel): 7.96

Seville

Seville is similar to sevond, but provides a less complex avenue to 5, but this time at the cost of accuracy. One generator from 5\7 puts you at ~3/2, and one generator from 2\7 puts you at ~5/4. This corresponds to n = 7/3 and m = 7/4.

Subgroup: 2.3.5

Comma list: 78125/69984

Mapping: [⟨7 0 5], ⟨0 1 1]]

mapping generators: ~125/108, ~3

Optimal tunings:

  • WE: ~125/108 = 171.7481 ¢, ~3/2 = 707.7272 ¢ (~25/24 = 20.7346 ¢)
error map: ⟨+2.237 +8.009 -17.609]
  • CWE: ~125/108 = 171.4286 ¢, ~3/2 = 708.3739 ¢ (~25/24 = 22.6596 ¢)
error map: ⟨0.000 +6.419 -20.797]

Optimal ET sequence: 7, 35b, 42c

Badness (Sintel): 10.3

Artoneutral (5-limit)

For extensions, see Hemifamity temperaments #Artoneutral.

5-limit artoneutral is generated by ~243/200 (or ~400/243), same as that of amity but sharper. This corresponds to n = 9/2 and m = 9/7 and can be described as the 80 & 87 temperament, though 94edo is a notable tuning not appearing in the optimal ET sequence.

Subgroup: 2.3.5

Comma list: [14 -22 9⟩

Mapping: [⟨1 -1 -4], ⟨0 9 22]]

mapping generators: ~2, ~243/200

Optimal tunings:

  • WE: ~2 = 1199.7434 ¢, ~243/200 = 344.7454 ¢
error map: ⟨-0.257 +1.010 -0.889]
  • CWE: ~2 = 1200.0000 ¢, ~243/200 = 344.8041 ¢
error map: ⟨0.000 +1.282 -0.624]

Optimal ET sequence: 7, … 73, 80, 87

Badness (Sintel): 8.17

Geb (5-limit)

For extensions, see Metric microtemperaments #Geb and Breedsmic temperaments #Osiris.

Subgroup: 2.3.5

Comma list: [-31 43 -16⟩

Mapping: [⟨1 -3 -10], ⟨0 16 43]]

mapping generators: ~2, ~8000/6561

Optimal tunings:

  • WE: ~2 = 1200.0135 ¢, ~8000/6561 = 343.8718 ¢
error map: ⟨+0.014 -0.047 +0.038]
  • CWE: ~2 = 1200.0000 ¢, ~8000/6561 = 343.8683 ¢
error map: ⟨0.000 -0.062 +0.024]

Optimal ET sequence: 7, …, 157, 164, 171, 506, 677, 848

Badness (Sintel): 2.78

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