Syntonic–chromatic equivalence continuum: Difference between revisions
Jump to navigation
Jump to search
No edit summary |
m Units. Cleanup. Style |
||
| Line 2: | Line 2: | ||
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| | * [[Mavila]] ({{nowrap| ''n'' {{=}} 1 }}) is generated by a fifth; | ||
* [[Dicot]] {{nowrap| | * [[Dicot]] ({{nowrap| ''n'' {{=}} 2 }}) splits its fifth in two; | ||
* [[Porcupine]] {{nowrap| | * [[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'' | 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| | * {{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 }} | ||
|- | |- | ||
| −4 | | −4 | ||
| Line 38: | Line 38: | ||
| [[Nethertone]] | | [[Nethertone]] | ||
| [[14348907/13107200]] | | [[14348907/13107200]] | ||
| {{ | | {{Monzo| -19 15 -2 }} | ||
|- | |- | ||
| −3 | | −3 | ||
| Line 44: | Line 44: | ||
| [[Deeptone]] a.k.a. tragicomical | | [[Deeptone]] a.k.a. tragicomical | ||
| [[177147/163840]] | | [[177147/163840]] | ||
| {{ | | {{Monzo| -15 11 -1 }} | ||
|- | |- | ||
| −2 | | −2 | ||
| Line 50: | Line 50: | ||
| [[Whitewood]] | | [[Whitewood]] | ||
| [[2187/2048]] | | [[2187/2048]] | ||
| {{ | | {{Monzo| -11 7 }} | ||
|- | |- | ||
| −1 | | −1 | ||
| Line 56: | Line 56: | ||
| [[Mavila]] | | [[Mavila]] | ||
| [[135/128]] | | [[135/128]] | ||
| {{ | | {{Monzo| -7 3 1 }} | ||
|- | |- | ||
| 0 | | 0 | ||
| Line 62: | Line 62: | ||
| [[Dicot]] | | [[Dicot]] | ||
| [[25/24]] | | [[25/24]] | ||
| {{ | | {{Monzo| -3 -1 2 }} | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 68: | Line 68: | ||
| [[Porcupine]] | | [[Porcupine]] | ||
| [[250/243]] | | [[250/243]] | ||
| {{ | | {{Monzo| 1 -5 3 }} | ||
|- | |- | ||
| 2 | | 2 | ||
| Line 74: | Line 74: | ||
| [[Tetracot]] | | [[Tetracot]] | ||
| [[20000/19683]] | | [[20000/19683]] | ||
| {{ | | {{Monzo| 5 -9 4 }} | ||
|- | |- | ||
| 3 | | 3 | ||
| Line 80: | Line 80: | ||
| [[Amity]] | | [[Amity]] | ||
| [[1600000/1594323]] | | [[1600000/1594323]] | ||
| {{ | | {{Monzo| 9 -13 5 }} | ||
|- | |- | ||
| 4 | | 4 | ||
| Line 86: | Line 86: | ||
| [[Gravity]] | | [[Gravity]] | ||
| [[129140163/128000000]] | | [[129140163/128000000]] | ||
| {{ | | {{Monzo| -13 17 -6 }} | ||
|- | |- | ||
| 5 | | 5 | ||
| Line 92: | Line 92: | ||
| [[Absurdity]] | | [[Absurdity]] | ||
| 10460353203/10240000000 | | 10460353203/10240000000 | ||
| {{ | | {{Monzo| -17 21 -7 }} | ||
|- | |- | ||
| … | | … | ||
| Line 103: | Line 103: | ||
| [[Meantone]] | | [[Meantone]] | ||
| [[81/80]] | | [[81/80]] | ||
| {{ | | {{Monzo| -4 4 -1 }} | ||
|} | |} | ||
We may invert the continuum by setting ''m'' such that {{nowrap | 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" | ||
| Line 121: | Line 121: | ||
| [[Shallowtone]] | | [[Shallowtone]] | ||
| [[295245/262144]] | | [[295245/262144]] | ||
| {{ | | {{Monzo| -18 10 1 }} | ||
|- | |- | ||
| 0 | | 0 | ||
| [[Whitewood]] | | [[Whitewood]] | ||
| [[2187/2048]] | | [[2187/2048]] | ||
| {{ | | {{Monzo| -11 7 }} | ||
|- | |- | ||
| 1 | | 1 | ||
| [[Meantone]] | | [[Meantone]] | ||
| [[81/80]] | | [[81/80]] | ||
| {{ | | {{Monzo| -4 4 -1 }} | ||
|- | |- | ||
| 2 | | 2 | ||
| [[Dicot]] | | [[Dicot]] | ||
| [[25/24]] | | [[25/24]] | ||
| {{ | | {{Monzo| -3 -1 2 }} | ||
|- | |- | ||
| 3 | | 3 | ||
| [[Enipucrop]] | | [[Enipucrop]] | ||
| [[1125/1024]] | | [[1125/1024]] | ||
| {{ | | {{Monzo| -10 2 3 }} | ||
|- | |- | ||
| … | | … | ||
| Line 151: | Line 151: | ||
| [[Mavila]] | | [[Mavila]] | ||
| [[135/128]] | | [[135/128]] | ||
| {{ | | {{Monzo| -7 3 1 }} | ||
|} | |} | ||
| Line 190: | Line 190: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 | * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 689.879{{c}} | ||
* [[CWE]]: ~2 | * [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 689.312{{c}} | ||
{{Optimal ET sequence|legend=1| 7, 33, 40, 47, 87b }} | {{Optimal ET sequence|legend=1| 7, 33, 40, 47, 87b }} | ||
| Line 200: | Line 200: | ||
: ''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 ( | 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 211: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~3/2 = 681.801 | * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 681.801{{c}} | ||
* [[CWE]]: ~2 = 1200.000, ~3/2 = 682.662 | * [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 682.662{{c}} | ||
{{Optimal ET sequence|legend=1| 7, 30b, 37b, 44b, 51b, 58bc, 65bbc }} | {{Optimal ET sequence|legend=1| 7, 30b, 37b, 44b, 51b, 58bc, 65bbc }} | ||
| Line 228: | Line 228: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~2560/2187 = 345.946 | * [[CTE]]: ~2 = 1200.000{{c}}, ~2560/2187 = 345.946{{c}} | ||
* [[CWE]]: ~2 = 1200.000, ~2560/2187 = 345.599 | * [[CWE]]: ~2 = 1200.000{{c}}, ~2560/2187 = 345.599{{c}} | ||
{{Optimal ET sequence|legend=1| 7, 38c, 45c, 52, 59b, 66b }} | {{Optimal ET sequence|legend=1| 7, 38c, 45c, 52, 59b, 66b }} | ||
| Line 236: | Line 236: | ||
== 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 | ||
| Line 247: | Line 247: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~16/15 = 170.671 | * [[CTE]]: ~2 = 1200.000{{c}}, ~16/15 = 170.671{{c}} | ||
* [[POTE]]: ~2 = 1200.000, ~16/15 = 173.101 | * [[POTE]]: ~2 = 1200.000{{c}}, ~16/15 = 173.101{{c}} | ||
{{Optimal ET sequence|legend=1| 6b, 7 }} | {{Optimal ET sequence|legend=1| 6b, 7 }} | ||
| Line 264: | Line 264: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~729/640 = 168.983 | * [[CTE]]: ~2 = 1200.000{{c}}, ~729/640 = 168.983{{c}} | ||
* [[CWE]]: ~2 = 1200.000, ~729/640 = 169.223 | * [[CWE]]: ~2 = 1200.000{{c}}, ~729/640 = 169.223{{c}} | ||
{{Optimal ET sequence|legend=1| 7, 57c, 64, 71b, 78b, 85b }} | {{Optimal ET sequence|legend=1| 7, 57c, 64, 71b, 78b, 85b }} | ||
| Line 283: | Line 283: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~6/5 = 338.005 | * [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 338.005{{c}} | ||
: [[error map]]: {{val| 0.000 +8.020 -14.344 }} | : [[error map]]: {{val| 0.000 +8.020 -14.344 }} | ||
* [[POTE]]: ~2 = 1200.000, ~6/5 = 338.365 | * [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 338.365{{c}} | ||
: error map: {{val| 0.000 +6.217 -16.507 }} | : error map: {{val| 0.000 +6.217 -16.507 }} | ||
| Line 295: | Line 295: | ||
: ''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). It tempers out the ''absurditon'', 10460353203/10240000000. | 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 306: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~800/729 = 171.429, ~3/2 = 700.538 (~81/80 = 14.824) | * [[CTE]]: ~800/729 = 171.429{{c}}, ~3/2 = 700.538{{c}} (~81/80 = 14.824{{c}}) | ||
* [[POTE]]: ~800/729 = 171.429, ~3/2 = 700.187 (~81/80 = 14.473) | * [[POTE]]: ~800/729 = 171.429{{c}}, ~3/2 = 700.187{{c}} (~81/80 = 14.473{{c}}) | ||
{{Optimal ET sequence|legend=1| 7, …, 70, 77, 84, 329, 413b, 497b }} | {{Optimal ET sequence|legend=1| 7, …, 70, 77, 84, 329, 413b, 497b }} | ||
| Line 316: | Line 316: | ||
: ''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 & | 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 325: | Line 325: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~10/9 = 171.429, ~3/2 = 705.526 (~250/243 = 19.812) | * [[CTE]]: ~10/9 = 171.429{{c}}, ~3/2 = 705.526{{c}} (~250/243 = 19.812{{c}}) | ||
* [[POTE]]: ~10/9 = 171.429, ~3/2 = 706.288 (~250/243 = 20.574) | * [[POTE]]: ~10/9 = 171.429{{c}}, ~3/2 = 706.288{{c}} (~250/243 = 20.574{{c}}) | ||
{{Optimal ET sequence|legend=1| 7, 42, 49, 56, 119 }} | {{Optimal ET sequence|legend=1| 7, 42, 49, 56, 119 }} | ||
| Line 333: | Line 333: | ||
== 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 342: | Line 342: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~125/108 = 171.429, ~3/2 = 710.606 (~25/24 = 24.891) | * [[CTE]]: ~125/108 = 171.429{{c}}, ~3/2 = 710.606{{c}} (~25/24 = 24.891{{c}}) | ||
* [[POTE]]: ~125/108 = 171.429, ~3/2 = 706.410 (~25/24 = 20.696) | * [[POTE]]: ~125/108 = 171.429{{c}}, ~3/2 = 706.410{{c}} (~25/24 = 20.696{{c}}) | ||
{{Optimal ET sequence|legend=1| 7, 35b, 42c }} | {{Optimal ET sequence|legend=1| 7, 35b, 42c }} | ||
| Line 352: | Line 352: | ||
: ''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 363: | Line 363: | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1200.000, ~400/243 = 855.213 | * [[CTE]]: ~2 = 1200.000{{c}}, ~400/243 = 855.213{{c}} | ||
* [[CWE]]: ~2 = 1200.000, ~400/243 = 855.196 | * [[CWE]]: ~2 = 1200.000{{c}}, ~400/243 = 855.196{{c}} | ||
{{Optimal ET sequence|legend=1| 7, … 73, 80, 87 }} | {{Optimal ET sequence|legend=1| 7, … 73, 80, 87 }} | ||
Revision as of 10:50, 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.
| 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.
| 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⟩ |
| 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
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 ET sequence: 7, 33, 40, 47, 87b
Badness (Smith): 0.403
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 ET sequence: 7, 30b, 37b, 44b, 51b, 58bc, 65bbc
Badness (Smith): 0.666
Nethertone
Subgroup: 2.3.5
Comma list: 14348907/13107200
Mapping: [⟨1 1 -1], ⟨0 2 15]]
- mapping generators: ~2, ~2560/2187
Optimal ET sequence: 7, 38c, 45c, 52, 59b, 66b
Badness (Smith): 0.828
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
Badness (Smith): 0.1439
Nadir
Subgroup: 2.3.5
Comma list: 1162261467/1048576000
Mapping: [⟨1 2 5], ⟨0 -3 -19]]
- mapping generators: ~2, ~729/640
Optimal ET sequence: 7, 57c, 64, 71b, 78b, 85b
Badness (Smith): 1.47
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
- CTE: ~2 = 1200.000 ¢, ~6/5 = 338.005 ¢
- error map: ⟨0.000 +8.020 -14.344]
- POTE: ~2 = 1200.000 ¢, ~6/5 = 338.365 ¢
- error map: ⟨0.000 +6.217 -16.507]
Optimal ET sequence: 7, 25, 32, 39c
Badness (Smith): 0.153088
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
- CTE: ~800/729 = 171.429 ¢, ~3/2 = 700.538 ¢ (~81/80 = 14.824 ¢)
- POTE: ~800/729 = 171.429 ¢, ~3/2 = 700.187 ¢ (~81/80 = 14.473 ¢)
Optimal ET sequence: 7, …, 70, 77, 84, 329, 413b, 497b
Badness (Smith): 0.341202
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]]
- CTE: ~10/9 = 171.429 ¢, ~3/2 = 705.526 ¢ (~250/243 = 19.812 ¢)
- POTE: ~10/9 = 171.429 ¢, ~3/2 = 706.288 ¢ (~250/243 = 20.574 ¢)
Optimal ET sequence: 7, 42, 49, 56, 119
Badness (Smith): 0.339335
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]]
- CTE: ~125/108 = 171.429 ¢, ~3/2 = 710.606 ¢ (~25/24 = 24.891 ¢)
- POTE: ~125/108 = 171.429 ¢, ~3/2 = 706.410 ¢ (~25/24 = 20.696 ¢)
Optimal ET sequence: 7, 35b, 42c
Badness (Smith): 0.4377
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 8 18], ⟨0 -9 -22]]
- mapping generators: ~2, ~400/243
Optimal ET sequence: 7, … 73, 80, 87
Badness (Smith): 0.348