Very low accuracy temperaments: Difference between revisions
→Quint: add error maps |
Delete antonym, antony, and geryon, from the precedent of deletion of {7/6}. Perhaps {9/8} should be the threshold for tempering out superparticulars |
||
| (15 intermediate revisions by 4 users not shown) | |||
| Line 1: | Line 1: | ||
{{Technical data page}} | {{Technical data page}} | ||
Below are listed some '''very low accuracy temperaments'''. Temperaments with exceedingly low accuracy such as these tend to [[tempering out|temper out]] very large intervals such as [[9/8]], [[10/9]], [[32/27]], or [[15/14]], equating wildly different interval sizes with [[semitone (interval size measure)|semitone]]-level or even greater [[error]]s | Below are listed some '''very low accuracy temperaments'''. Temperaments with exceedingly low accuracy such as these tend to [[tempering out|temper out]] very large intervals such as [[9/8]], [[10/9]], [[32/27]], or [[15/14]], equating wildly different interval sizes with [[semitone (interval size measure)|semitone]]-level or even greater [[error]]s, and often swapping the sizes of simple ratios compared to just intonation. As a result, all of them are right on or even beyond the edge of what can be sensibly called a temperament at all; that is to say, they are [[exotemperament]]s. | ||
== Antitonic == | == Antitonic == | ||
| Line 31: | Line 31: | ||
Comma list: 9/8, 15/14 | Comma list: 9/8, 15/14 | ||
{{Mapping|legend=0| 2 3 0 1 | 0 0 1 1 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 46: | Line 46: | ||
Comma list: 9/8, 12/11, 15/14 | Comma list: 9/8, 12/11, 15/14 | ||
{{Mapping|legend=0| 2 3 0 1 7 | 0 0 1 1 0 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 61: | Line 61: | ||
Comma list: 9/8, 11/10, 15/14 | Comma list: 9/8, 11/10, 15/14 | ||
{{Mapping|legend=0| 2 3 0 1 2 | 0 0 1 1 1 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 76: | Line 76: | ||
Comma list: 9/8, 35/32 | Comma list: 9/8, 35/32 | ||
{{Mapping|legend=0| 2 3 0 10 | 0 0 1 -1 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 91: | Line 91: | ||
Comma list: 9/8, 12/11, 35/32 | Comma list: 9/8, 12/11, 35/32 | ||
{{Mapping|legend=0| 2 3 0 10 7 | 0 0 1 -1 0 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 109: | Line 109: | ||
{{Mapping|legend=1| 3 5 0 | 0 0 1 }} | {{Mapping|legend=1| 3 5 0 | 0 0 1 }} | ||
: mapping generators: ~4/3, ~5 | : mapping generators: ~4/3, ~5 | ||
| Line 124: | Line 123: | ||
== Antonian == | == Antonian == | ||
{{Main| Antonian }} | {{Main| Antonian }} | ||
This temperament family is characterized by the [[color notation|yo 2nd]] ([[10/9]]) being tempered out. It identifies [[3/2]] with [[5/3]], [[4/3]] with [[6/5]], and [[5/4]] with [[9/8]]. | This temperament family is characterized by the [[color notation|yo 2nd]] ([[10/9]]) being tempered out. It identifies [[3/2]] with [[5/3]], [[4/3]] with [[6/5]], and [[5/4]] with [[9/8]]. | ||
| Line 149: | Line 149: | ||
Comma list: 10/9, 15/14 | Comma list: 10/9, 15/14 | ||
{{Mapping|legend=0| 1 0 -1 -2 | 0 1 2 3 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 158: | Line 158: | ||
Badness (Sintel): 0.606 | Badness (Sintel): 0.606 | ||
=== Brutus === | === Brutus === | ||
| Line 194: | Line 164: | ||
Comma list: 10/9, 28/25 | Comma list: 10/9, 28/25 | ||
{{Mapping|legend=0| 1 0 -1 -4 | 0 1 2 4 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 211: | Line 181: | ||
Comma list: 10/9, 35/32 | Comma list: 10/9, 35/32 | ||
{{Mapping|legend=0| 1 0 -1 6 | 0 1 2 -2 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 228: | Line 198: | ||
Comma list: 10/9, 49/45 | Comma list: 10/9, 49/45 | ||
{{Mapping|legend=0| 2 0 -2 -1 | 0 1 2 2 }} | |||
: mapping generators: ~7/5, ~3 | : mapping generators: ~7/5, ~3 | ||
| Line 246: | Line 216: | ||
Comma list: 10/9, 49/48 | Comma list: 10/9, 49/48 | ||
{{Mapping|legend=0| 1 0 -1 2 | 0 2 4 1 }} | |||
: mapping generators: ~2, ~7/4 | : mapping generators: ~2, ~7/4 | ||
| Line 283: | Line 253: | ||
Comma list: 10/9, 16/15, 22/21 | Comma list: 10/9, 16/15, 22/21 | ||
{{Mapping|legend=0| 3 5 7 0 2 | 0 0 0 1 1 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 294: | Line 264: | ||
== Quad == | == Quad == | ||
Quad is identical to 4et in the 5-limit, but has an independent generator for prime 7. | Quad is identical to [[4edo|4et]] in the 5-limit, but has an independent generator for prime 7. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 313: | Line 283: | ||
== Quint == | == Quint == | ||
Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is | Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is already quite accurately approximated by 5edo for its size, this temperament provides little improvement, so it is only provided below for bookkeeping purposes. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what [[blackwood]] does. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 320: | Line 290: | ||
{{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }} | {{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }} | ||
: Mapping generators: ~6/5, ~7 | : Mapping generators: ~6/5, ~7 | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[WE]]: ~6/5 = 236.416, ~7/4 = 1002.701 (~28/27 = 57. | * [[WE]]: ~6/5 = 236.416{{c}}, ~7/4 = 1002.701{{c}} (~28/27 = 57.038{{c}}) | ||
: [[error map]]: {{val| -17.921 -10.628 +50.676 -1.966 }} | : [[error map]]: {{val| -17.921 -10.628 +50.676 -1.966 }} | ||
* [[CWE]]: ~6/5 = 240.000, ~7/4 = 1005.135 (~28/27 = 45.135) | * [[CWE]]: ~6/5 = 240.000{{c}}, ~7/4 = 1005.135{{c}} (~28/27 = 45.135{{c}}) | ||
: error map: {{val| 0.000 +18.045 +93.686 +36.309 }} | : error map: {{val| 0.000 +18.045 +93.686 +36.309 }} | ||
{{Optimal ET sequence|legend=1| 5, 15ccd }} | {{Optimal ET sequence|legend=1| 5, 15ccd, 20cccd }} | ||
[[Badness]] (Sintel): 1.22 | [[Badness]] (Sintel): 1.22 | ||
== Malacoda == | == Malacoda == | ||
| Line 360: | Line 310: | ||
{{Mapping|legend=1| 1 0 3 2 | 0 2 -1 1 }} | {{Mapping|legend=1| 1 0 3 2 | 0 2 -1 1 }} | ||
: mapping generators: ~2, ~7/4 | : mapping generators: ~2, ~7/4 | ||
| Line 416: | Line 365: | ||
Comma list: 15/14, 22/21, 33/32 | Comma list: 15/14, 22/21, 33/32 | ||
{{Mapping|legend=0| 1 0 7 6 5 | 0 1 -3 -2 -1 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 446: | Line 395: | ||
== Plutus == | == Plutus == | ||
{{Distinguish| Pluto }} | |||
{{See also| Meantone family }} | {{See also| Meantone family }} | ||
| Line 469: | Line 419: | ||
Comma list: 15/14, 22/21, 81/80 | Comma list: 15/14, 22/21, 81/80 | ||
{{Mapping|legend=0| 1 0 -4 -5 -6 | 0 1 4 5 6 }} | |||
Optimal tunings: | Optimal tunings: | ||
| Line 478: | Line 428: | ||
Badness (Sintel): 1.08 | Badness (Sintel): 1.08 | ||
== Codex == | == Codex == | ||
| Line 558: | Line 451: | ||
== Oxygen == | == Oxygen == | ||
{{See also| Porcupine family }} | {{See also| Porcupine family }} | ||
Oxygen extends porcupine into the 7-limit by conflating 6/5 with 8/7. While this means it does not represent | Oxygen extends porcupine into the 7-limit by conflating 6/5 with 8/7, and conflating 10/9 with 7/6. While this means it does not represent 7-limit JI with any real accuracy, it is still of interest because its comma basis suggests potential utility to construct [[fokker block|Fokker blocks]]. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 574: | Line 467: | ||
{{Optimal ET sequence|legend=1| 1c, …, 6bcd, 7d }}* | {{Optimal ET sequence|legend=1| 1c, …, 6bcd, 7d }}* | ||
<nowiki/>*[[ | <nowiki/>* [[optimal patent val]]: [[8edo|8]] | ||
[[Badness]] (Sintel): 1.52 | [[Badness]] (Sintel): 1.52 | ||