Syntonic–chromatic equivalence continuum: Difference between revisions

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)^''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 [[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:

* 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.

 

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)^''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 {{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)⁰ ~ 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'' {{=}} ∞ }}). 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.

|+ style="font-size: 105%;" | Temperaments with integer ''n''

| −5

| −3

| [[Nadir]]

| {{Monzo| -23 19 -3 }}

| −4

| −2

| [[Nethertone]]

| {{Monzo| -19 15 -2 }}

| −3

| −1

| [[Deeptone]] a.k.a. tragicomical

| {{Monzo| -15 11 -1 }}

| −2

| {{Monzo| -11 7 }}

| −1

| {{Monzo| -7 3 1 }}

| {{Monzo| -3 -1 2 }}

| {{Monzo| 1 -5 3 }}

| {{Monzo| 5 -9 4 }}

| {{Monzo| 9 -13 5 }}

| {{Monzo| -13 17 -6 }}

| {{Monzo| -17 21 -7 }}

| {{Monzo| -4 4 -1 }}

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"

|+ style="font-size: 105%;" | Temperaments with integer ''m''

| {{Monzo| -11 7 }}

| {{Monzo| -4 4 -1 }}

| {{Monzo| -3 -1 2 }}

| {{Monzo| -10 2 3 }}

| {{Monzo| -7 3 1 }}

|+ style="font-size: 105%;" | Temperaments with fractional ''n'' and ''m'' (and thus also ''k'')

! ''n'' !! ''k'' !! ''m'' !! Temperament !! Comma

| 7/3 = 2.{{overline|3}} || 1/3 = 0.{{overline|3}} || 7/4 = 1.75 || [[Seville]] || {{monzo| -5 -7 7 }}

| 5/2 = 2.5 || 1/2 = 0.5 || 5/3 = 1.{{overline|6}} || [[Sixix]] || {{monzo| -2 -6 5 }}

| 7/2 = 3.5 || 3/2 = 1.5 || 7/5 = 1.4 || [[Sevond]] || {{monzo| 6 -14 7 }}

| 9/2 = 4.5 || 5/2 = 2.5 || 9/7 = 1.{{overline|285714}} || [[Artoneutral (temperament)|Artoneutral]] || {{monzo| 14 -22 9 }}

| 21/4 = 5.25 || 13/4 = 3.25 || 21/17 = 1.235… || [[Brahmagupta]] || {{monzo| 40 -56 21 }}

| 37/7 = 5.{{overline|285714}} || 37/7 = 3+2/7 || 37/30 = 1.2{{overline|3}} || [[Raider]] || {{monzo| 71 -99 37 }}

| 16/3 = 5.{{overline|3}} || 10/3 = 3.{{overline|3}} || 16/13 = 1.{{overline|230769}} || [[Geb]] || {{monzo| -31 43 -16 }}

| 11/2 = 5.5 || 7/2 = 3.5 || 11/9 = 1.{{overline|2}} || [[Undetrita]] || {{monzo| -22 30 -11 }}

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.

{{Mapping|legend=1| 1 2 2 | 0 -3 2 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1210.1294{{c}}, ~16/15 = 174.5613{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 172.4620{{c}}

: error map: {{val| 0.000 -19.341 -41.390 }}

[[Badness]] (Sintel): 3.38

 

 

 

 

 

 

 

 

 

 

 

: error map: {{val| 0.000 +7.065 -15.489 }}

 

 

== Absurdity (5-limit) ==

: ''For extensions, see [[Porwell temperaments #Absurdity]].''

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.

 

: [[error map]]: {{val| +0.376 -1.172 +0.836 }}

 

 

 

 

 

 

 

{{Mapping|legend=1| 7 0 -6 | 0 1 2 }}

 

[[Optimal tuning]]s:

: [[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 }}

 

 

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

 

: mapping generators: ~125/108, ~3

[[Optimal tuning]]s:

* [[WE]]: ~125/108 = 171.7481{{c}}, ~3/2 = 707.7272{{c}} (~25/24 = 20.7346{{c}})

: [[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}})

{{Optimal ET sequence|legend=1| 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 {{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.  

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

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.7434{{c}}, ~243/200 = 344.7454{{c}}

: [[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 }}

[[Badness]] (Sintel): 8.17

== Geb (5-limit) ==

: ''For extensions, see [[Metric microtemperaments #Geb]] and [[Breedsmic temperaments #Osiris]].''

[[Comma list]]: {{monzo| -31 43 -16 }}

{{Mapping|legend=1| 1 -3 -10 | 0 16 43 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0135{{c}}, ~8000/6561 = 343.8718{{c}}

* [[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

== Undetrita (5-limit) ==

: ''For extensions, see [[Hemimean clan #Undetrita]].''

[[Comma list]]: {{monzo| -22 30 -11 }}

{{Mapping|legend=1| 1 0 -2 | 0 11 30 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0596{{c}}, ~177147/160000 = 172.8864{{c}}

* [[CWE]]: ~2 = 1200.000{{c}}, ~177147/160000 = 172.8804{{c}}

: error map: {{val| 0.000 -0.271 +0.098 }}

{{Optimal ET sequence|legend=1| 7, …, 111, 118, 1541, 1659, 1777, 1895b, …, 2839b, 2957b }}

[[Badness]] (Sintel): 4.09