Syntonic–chromatic equivalence continuum: Difference between revisions

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.  

* [[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;  

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.

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

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

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

| {{Monzo| -11 7 }}

| {{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.  

| {{Monzo| -18 10 1 }}

| {{Monzo| -11 7 }}

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

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

| {{Monzo| -10 2 3 }}

| {{Monzo| -7 3 1 }}

* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 689.879{{c}}

* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 689.312{{c}}

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.

* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 681.801{{c}}

* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 682.662{{c}}

* [[CTE]]: ~2 = 1200.000{{c}}, ~2560/2187 = 345.946{{c}}

* [[CWE]]: ~2 = 1200.000{{c}}, ~2560/2187 = 345.599{{c}}

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.

* [[CTE]]: ~2 = 1200.000{{c}}, ~16/15 = 170.671{{c}}

* [[POTE]]: ~2 = 1200.000{{c}}, ~16/15 = 173.101{{c}}

* [[CTE]]: ~2 = 1200.000{{c}}, ~729/640 = 168.983{{c}}

* [[CWE]]: ~2 = 1200.000{{c}}, ~729/640 = 169.223{{c}}

* [[CTE]]: ~2 = 1200.000{{c}}, ~6/5 = 338.005{{c}}

* [[POTE]]: ~2 = 1200.000{{c}}, ~6/5 = 338.365{{c}}

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.

* [[CTE]]: ~800/729 = 171.429{{c}}, ~3/2 = 700.538{{c}} (~81/80 = 14.824{{c}})

* [[POTE]]: ~800/729 = 171.429{{c}}, ~3/2 = 700.187{{c}} (~81/80 = 14.473{{c}})

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.  

* [[CTE]]: ~10/9 = 171.429{{c}}, ~3/2 = 705.526{{c}} (~250/243 = 19.812{{c}})

* [[POTE]]: ~10/9 = 171.429{{c}}, ~3/2 = 706.288{{c}} (~250/243 = 20.574{{c}})

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 }}.  

* [[CTE]]: ~125/108 = 171.429{{c}}, ~3/2 = 710.606{{c}} (~25/24 = 24.891{{c}})

* [[POTE]]: ~125/108 = 171.429{{c}}, ~3/2 = 706.410{{c}} (~25/24 = 20.696{{c}})

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.  

* [[CTE]]: ~2 = 1200.000{{c}}, ~400/243 = 855.213{{c}}

* [[CWE]]: ~2 = 1200.000{{c}}, ~400/243 = 855.196{{c}}