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:

* [[Mavila]] ({{nowrap| ''n'' {{=}} 1 }}) is generated by a fifth;

* [[Dicot]] ({{nowrap| ''n'' {{=}} 2 }}) splits its fifth in two;

* [[Porcupine]] ({{nowrap| ''n'' {{=}} 3 }}) splits its fourth in three;

 

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''

! rowspan="2" | ''k''

! rowspan="2" | ''n''

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

| {{Monzo| -19 15 -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 }}

 

 

| {{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 }}

| 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 }}

 

 

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♯).

 

 

 

 

* [[WE]]: ~2 = 1203.7664{{c}}, ~3/2 = 691.1525{{c}}

 

 

[[Badness]] (Sintel): 9.44

 

: ''For extensions, see [[Mint temperaments #Shallowtone]].''

 

 

 

 

 

* [[WE]]: ~2 = 1206.3211{{c}}, ~3/2 = 686.6308{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 682.6617{{c}}

 

 

[[Badness]] (Sintel): 15.6

 

== Nethertone ==

[[Subgroup]]: 2.3.5

 

 

 

* [[WE]]: ~2 = 1203.1196{{c}}, ~2560/2187 = 346.2857{{c}}

: [[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 }}

 

[[Badness]] (Sintel): 19.4

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

[[Comma list]]: 1125/1024

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

[[Optimal tuning]]s:  

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

: [[error map]]: {{val| +10.129 -5.380 -16.932 }}

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

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

[[Badness]] (Sintel): 3.38

== Nadir ==

[[Subgroup]]: 2.3.5

[[Comma list]]: 1162261467/1048576000

: mapping generators: ~2, ~729/640

: [[error map]]: {{val| +2.743 -5.758 +1.900 }}

: error map: {{val| 0.000 -9.625 -1.559 }}

{{Optimal ET sequence|legend=1| 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 generators: ~2, ~6/5

: [[error map]]: {{val| +0.993 +7.797 -14.214 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 338.1959{{c}}

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

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

[[Badness]] (Sintel): 3.59

== 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.

[[Subgroup]]: 2.3.5

[[Comma list]]: 10460353203/10240000000

{{Mapping|legend=1| 7 0 -17 | 0 1 3 }}

[[Optimal tuning]]s:

* [[WE]]: ~800/729 = 171.4824{{c}}, ~3/2 = 700.4067{{c}} (~81/80 = 14.4772{{c}})

: [[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}})

 

 

[[Badness]] (Sintel): 8.00

 

== Sevond (5-limit) ==

 

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.

 

 

 

 

: [[error map]]: {{val| -0.588 +3.399 -3.673 }}

: error map: {{val| 0.000 +3.957 -3.061 }}

 

 

[[Badness]] (Sintel): 7.96

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

 

 

 

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

 

 

 

 

 

 

 

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

 

 

 

 

 

 

 

 

 

 

 

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

[[Subgroup]]: 2.3.5

[[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}}

: [[error map]]: {{val| +0.060 -0.204 +0.161 }}

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

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

[[Badness]] (Sintel): 4.09