Xenharmonic Wiki

Magic family: Difference between revisions

← Older editNewer edit →
VisualWikitext
Cleanup; improve layout; edo lists review
m Cleanup (1/3)
Line 1: Line 1:
__FORCETOC__
{{interwiki
<span style="display: block; text-align: right;">[[de:Magische_Temperaturen|Deutsch]]</span>
| de = Magische Temperaturen
 
| en = Magic family
A magic temperament is optimal, for some searches, in the 9-limit. It has slightly higher complexity than [[Meantone family|meantone]] and is also closer to just intonation. It is the simplest rank 2 temperament that tunes every 9-limit interval better than is possible in [[12edo]]. Properties may depend on tuning and extension.
| es =
| ja =
}}
The '''magic family''' of temperaments tempers out [[3125/3072]], the small diesis or magic comma. A magic temperament is optimal, for some searches, in the 9-odd-limit. It has slightly higher complexity than [[Meantone family|meantone]] and is also closer to just intonation. It is the simplest rank 2 temperament that tunes every 9-limit interval better than is possible in [[12edo]]. Properties may depend on tuning and extension.


The most prominent deficiency of magic temperaments is that they lack [[Rothenberg propriety|proper]] or nearly-proper MOS scales in the 5 to 10 note "diatonic" region.
The most prominent deficiency of magic temperaments is that they lack [[Rothenberg propriety|proper]] or nearly-proper MOS scales in the 5 to 10 note "diatonic" region.


= Five limit magic =
= Five limit magic =
The 5-limit parent comma for the magic family is [[3125/3072]], the small diesis or magic comma. Its monzo is {{monzo| -10 -1 5 }}, and flipping that yields &lt;&lt;5 1 -10|| for the wedgie. This tells us the generator is a major third, and that to get to the interval class of fifths will require five of these. In fact, (5/4)<sup>5</sup> = 3 × 3125/3072. 13\41 is a highly recommendable generator, though 19\60, the [[optimal patent val]] generator, also makes a lot of sense and using [[19edo]] or [[22edo]] is always possible.
The 5-limit parent comma for the magic family is [[3125/3072]], the small diesis or magic comma. Its monzo is {{monzo| -10 -1 5 }}, and flipping that yields {{multival| 5 1 -10 }} for the [[wedgie]]. This tells us the generator is a major third, and that to get to the interval class of fifths will require five of these. In fact, (5/4)<sup>5</sup> = 3 × 3125/3072. 13\41 is a highly recommendable generator, though 19\60, the [[optimal patent val]] generator, also makes a lot of sense and using [[19edo]] or [[22edo]] is always possible.
 
[[Comma]]: 3125/3072
 
5-limit minimax


[&lt;1 0 0|, &lt;0 1 0|, &lt;2 1/5 0|]
Subgroup: 2.3.5


[[Eigenmonzo]]s: 2, 3
[[Comma list]]: 3125/3072


valid range: [360.000, 400.000] (10 to 3)
[[Mapping]]: [{{val| 1 0 2 }}, {{val| 0 5 1 }}]


nice range: [378.910, 386.314]
Mapping generators: ~2, ~5/4
 
strict range: [378.910, 386.314]


[[POTE generator]]: ~5/4 = 380.058
[[POTE generator]]: ~5/4 = 380.058


[[Algebraic generator]]: Terzbirat, the positive root of 9x<sup>2</sup> - 8x - 4 = (4 + 2√13)/9; approximately 380.3175 [[cent]]s.
[[Minimax tuning]]:  
* [[5-odd-limit]]
: [{{monzo| 1 0 0 }}, {{monzo| 0 1 0 }}, {{monzo| 2 1/5 0 }}]
: [[Eigenmonzo]]s: 2, 3


Map: [&lt;1 0 2|, &lt;0 5 1|]
[[Tuning ranges]]:
* valid range: [360.000, 400.000] (3\10 to 1\3)
* nice range: [378.910, 386.314]
* strict range: [378.910, 386.314]


Mapping generators: 2, 5/4
[[Algebraic generator]]: Terzbirat, the positive root of 9''x''<sup>2</sup> - 8''x'' - 4 = (4 + 2√13)/9; approximately 380.3175 [[cent]]s.


{{EDOs|legend=1| 16, 19, 41, 60, 221c, 281c }}
{{Val list|legend=1| 19, 41, 60, 221c, 281c }}


== Seven limit children ==
== Seven-limit extensions ==
The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. [[875/864]], the keemic comma, gives magic, and [[525/512]], Avicenna's enharmonic diesis, gives his annoying brother muggles. Both use the major third as a generator.
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. [[875/864]], the keemic comma, gives magic, and [[525/512]], Avicenna's enharmonic diesis, gives his annoying brother muggles. Both use the major third as a generator.


= Magic =
= Magic =
Line 41: Line 44:
Magic tempers out not only 3125/3072 and 875/864, but also [[225/224]], [[245/243]], and [[10976/10935]]. [[41edo]] is a good magic tuning, and 19 or 22 note MOS are possible scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1.
Magic tempers out not only 3125/3072 and 875/864, but also [[225/224]], [[245/243]], and [[10976/10935]]. [[41edo]] is a good magic tuning, and 19 or 22 note MOS are possible scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1.


Magic, with its accurate fifths, works well with [[9-odd-limit]] harmony. It's more accurate than [[Meantone family|meantone]] and simpler than [[Garibaldi temperament|garibaldi]]. It's a little tricky to work with because in it fifths are a relatively complex interval and it doesn't naturally work with scales of around seven notes to the octave. Its wedgie is &lt;&lt;5 1 12 -10 5 25||.
Magic, with its accurate fifths, works well with [[9-odd-limit]] harmony. It is more accurate than [[meantone]] and simpler than [[Garibaldi temperament|garibaldi]]. It is a little tricky to work with because in its fifths are a relatively complex interval and it does not naturally work with scales of around seven notes to the octave.  


225/224 is the [[Marvel temperaments|marvel]] comma. Because the augmented triad is the simplest triad in magic temperaments, it is especially significant in magic temperament.
225/224 is the [[Marvel temperaments|marvel]] comma. Because the augmented triad is the simplest triad in magic temperaments, it is especially significant in magic temperament.


245/243, the [[Sensamagic clan|sensamagic]] comma, leads to another essentially tempered 9-limit triad with two thirds approximating 9/7 and the other 6/5. It also divides the approximate 3/2 into two steps of 7/6 and one of 10/9.  
245/243, the [[Sensamagic clan|sensamagic]] comma, leads to another essentially tempered 9-odd-limit triad with two thirds approximating 9/7 and the other 6/5. It also divides the approximate 3/2 into two steps of 7/6 and one of 10/9.  


By adding [[100/99]] to the list of commas, magic can be extended to an 11-limit version, &lt;&lt;5 1 12 -8 ... ||. For this, [[104edo]] provides an excellent tuning, as it does also for the rank three temperaments tempering out 100/99 with 225/224, 245/243 or 875/864. Septimage (see below) is also an excellent 11-limit magic tuning.
By adding [[100/99]] to the list of commas, magic can be extended to an 11-limit version, {{multival| 5 1 12 -8 … }}. For this, [[104edo]] provides an excellent tuning, as it does also for the rank-3 temperaments tempering out 100/99 with 225/224, 245/243 or 875/864. Septimage (see below) is also an excellent 11-limit magic tuning.


Commas: 225/224, 245/243
Subgroup: 2.3.5.7


7 and 9 limit minimax
[[Comma list]]: 225/224, 245/243


[|1 0 0 0&gt;, |0 1 0 0&gt;, |2 1/5 0 0&gt;, |-1 12/5 0 0&gt;]
[[Mapping]]: [{{val| 1 0 2 -1 }}, {{val| 0 5 1 12 }}]


[[Eigenmonzo]]s: 2, 3
Mapping generators: ~2, ~5/4


valid range: [378.947, 381.818] (19 to 22)
{{Multival|legend=1| 5 1 12 -10 5 25 }}


nice range: [378.910, 386.314]
[[POTE generator]]: ~5/4 = 380.352


strict range: [378.947, 381.818]
[[Minimax tuning]]:
* 7- and [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 2 1/5 0 0 }}, {{monzo| -1 12/5 0 0 }}]
: [[Eigenmonzo]]s: 2, 3


[[POTE generator]]: 380.352
[[Tuning ranges]]:  
* valid range: [378.947, 381.818] (6\19 to 7\22)
* nice range: [378.910, 386.314]
* strict range: [378.947, 381.818]


Algebraic generators: Tirzbirat or Septimage, the real root of 5x^5+4x-20, 380.7604 cents.
[[Algebraic generator]]: Tirzbirat or Septimage, the real root of 5''x''<sup>5</sup> + 4''x'' - 20, 380.7604 cents.


Map: [&lt;1 0 2 -1|, &lt;0 5 1 12|]
{{Val list|legend=1| 19, 41, 142cd, 183cd, 224cd }}
 
Mapping generators: 2, 5/4
 
{{EDOs|legend=1| 19, 41, 142cd, 183cd, 224cd }}


== 11-limit ==
== 11-limit ==
Line 78: Line 83:
Tempering 100/99 allows for a tritone substitution where the extended 5-limit tuning of a dominant seventh with a 9/5 above the root shares its tritone with an 8:10:11:12:16 chord rooted on the seventh of the original chord. (The tritone of the dominant seventh is (9/5)/(5/4) = 36/25. (16/11)/(36/25) = 100/99.)
Tempering 100/99 allows for a tritone substitution where the extended 5-limit tuning of a dominant seventh with a 9/5 above the root shares its tritone with an 8:10:11:12:16 chord rooted on the seventh of the original chord. (The tritone of the dominant seventh is (9/5)/(5/4) = 36/25. (16/11)/(36/25) = 100/99.)


Commas: 225/224, 245/243, 100/99
Subgroup: 2.3.5.7.11
 
valid range: [378.947, 381.818] (19 to 22)


nice range: [378.910, 386.314]
Comma list: 225/224, 245/243, 100/99


strict range: [378.947, 381.818]
Mapping: [{{val| 1 0 2 -1 6 }}, {{val| 0 5 1 12 -8 }}]


[[POTE generator]]: 380.696
POTE generator: ~5/4 = 380.696


Map: [&lt;1 0 2 -1 6|, &lt;0 5 1 12 -8|]
Tuning ranges:
* valid range: [378.947, 381.818] (6\19 to 7\22)
* nice range: [378.910, 386.314]
* strict range: [378.947, 381.818]


{{EDOs|legend=1| 19, 22, 41, 104, 145c }}
{{Val list|legend=1| 19, 22, 41, 104, 145c }}


Badness: 0.0204
Badness: 0.0204


=== 13-limit ===
=== 13-limit ===
Commas: 100/99, 105/104, 144/143, 196/195


valid range: [378.947, 381.818] (19 to 22f)
Subgroup: 2.3.5.7.11.13


nice range: [378.617, 386.314]
Comma list: 100/99, 105/104, 144/143, 196/195


strict range: [378.947, 381.818]
Mapping: [{{val| 1 0 2 -1 6 -2 }}, {{val| 0 5 1 12 -8 18 }}]


POTE generator: ~5/4 = 380.427
POTE generator: ~5/4 = 380.427


Map: [&lt;1 0 2 -1 6 -2|, &lt;0 5 1 12 -8 18|]
Tuning ranges:
* valid range: [378.947, 381.818] (6\19 to 7\22)
* nice range: [378.617, 386.314]
* strict range: [378.947, 381.818]


{{EDOs|legend=1| 19, 22f, 41, 265cdef }}
{{Val list|legend=1| 19, 22f, 41, 265cdef }}


Badness: 0.0215
Badness: 0.0215


=== Sorcery ===
=== Sorcery ===
Commas: 65/64, 78/77, 91/90, 100/99


valid range: 378.947 (19)
Subgroup: 2.3.5.7.11.13


nice range: [359.472, 386.314]
Comma list: 65/64, 78/77, 91/90, 100/99


strict range: 378.947
Mapping: [{{val| 1 0 2 -1 6 4 }}, {{val| 0 5 1 12 -8 -1 }}]


POTE generator: ~5/4 = 380.477
POTE generator: ~5/4 = 380.477


Map: [&lt;1 0 2 -1 6 4|, &lt;0 5 1 12 -8 -1|]
Tuning ranges:
* valid range: 378.947 (6\19)
* nice range: [359.472, 386.314]
* strict range: 378.947


{{EDOs|legend=1| 19, 22, 41f }}
{{Val list|legend=1| 19, 22, 41f }}


Badness: 0.0258
Badness: 0.0258


=== Necromancy ===
=== Necromancy ===
Commas: 100/99, 225/224, 245/243, 275/273


valid range: [380.488, 380.952] (41 to 63)
Subgroup: 2.3.5.7.11.13


nice range: [378.910, 386.314]
Comma list: 100/99, 225/224, 245/243, 275/273


strict range: [380.488, 380.952]
Mapping: [{{val| 1 0 2 -1 6 11 }}, {{val| 0 5 1 12 -8 -23 }}]


POTE generator: ~5/4 = 380.787
POTE generator: ~5/4 = 380.787


Map: [&lt;1 0 2 -1 6 11|, &lt;0 5 1 12 -8 -23|]
Tuning ranges:
* valid range: [380.488, 380.952] (13\41 to 20\63)
* nice range: [378.910, 386.314]
* strict range: [380.488, 380.952]


{{EDOs| 19f, 22, 41, 63, 104 }}
{{Val list|legend=1| 19f, 22, 41, 63, 104 }}


Badness: 0.0253
Badness: 0.0253
Retrieved from "https://en.xen.wiki/w/Magic_family"