Magic: Difference between revisions

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'''Magic''' is a 2.3.5 temperament that equates a stack of five 5/4 major thirds to one 3/1. It also equates 25/24 to 128/125, shrinking the difference between 5/4 and 6/5.
'''Magic''' (19 & 22) is a 2.3.5 temperament that equates a stack of five 5/4 major thirds to one 3/1. It also equates 25/24 to 128/125, shrinking the difference between 5/4 and 6/5. It is a 3-cluster temperament, as indicated by the edo join (22 - 19 = 3).
 
== Interval chain ==
In the following table, odd harmonics 1–15 and their inverses are in '''bold'''.
{| class="wikitable center-1 right-2"
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
|-
| 1
| 380.5
| '''5/4'''
|-
| 2
| 760.9
| 14/9
|-
| 3
| 1141.4
| 27/14, 31/16
|-
| 4
| 321.8
| 6/5, 29/24
|-
| 5
| 702.3
| '''3/2'''
|-
| 6
| 1082.7
| '''15/8''', 28/15
|-
| 7
| 263.2
| 7/6
|-
| 8
| 643.7
| 36/25
|-
| 9
| 1024.1
| 9/5, 29/16
|-
| 10
| 204.6
| '''9/8'''
|-
| 11
| 585.0
| 7/5
|-
| 12
| 965.5
| '''7/4'''
|-
| 13
| 145.9
| 35/32
|}
<nowiki/>* In 7-limit CWE tuning


== Extensions ==
== Extensions ==
{{adv|Magic divides ~16/15 in half (into two 25/24's), so it easily extends to prime 31 (at +3 generators) by tempering out S31 {{=}} 961/960. Since in 7-limit Magic, the 16/15 is also a 15/14, we can also extend 2.3.5.7.31 Magic to add prime 29 (at +9 generators) by equating 16/15 to 31/29.}}
Magic divides ~16/15 in half (into two 25/24's), so it easily extends to prime 31 (at +3 generators) by tempering out S31 {{=}} 961/960. Since in 7-limit Magic, the 16/15 is also a 15/14, we can also extend 2.3.5.7.31 Magic to add prime 29 (at +9 generators) by equating 16/15 to 31/29.


{{Adv|This can be seen from:}}
{{Adv|This can be seen from the following [[S-expression]] for the Magic comma:}}


  3125/3072
  3125/3072
Line 27: Line 93:
  = (S15*S28*S29*S30)^2*S31
  = (S15*S28*S29*S30)^2*S31


{{Adv|making the above 2.3.5.7.29.31 extension a [[canonical extension|structurally induced extension]].}}
{{Adv|[[canonical extension|structurally inducing]] the above 2.3.5.7.29.31 extension.}}
 
== List of patent vals ==
The following patent vals support 2.3.5 Magic. Vals that are contorted in 2.3.5 are not included.
{| class="wikitable sortable"
! Edo
!Extension to 7|| Generator || Fifth
|-
||3
| ||400.000||800.000
|-
||25
| ||384.000||720.000
|-
||22
|19 & 22||381.818||709.091
|-
||107
| ||381.308||706.542
|-
||85
|19 & 22||381.176||705.882
|-
||63
|19 & 22||380.952||704.762
|-
||104
| ||380.769||703.846
|-
||41
|19 & 22||380.488||702.439
|-
||60
|19 & 22||380.000||700.000
|-
||79
| ||379.747||698.734
|-
||19
|19 & 22||378.947||694.737
|-
||35
| ||377.143||685.714
|-
||16
| ||375.000||675.000
|}


{{cat|Temperaments}}{{Navbox regtemp}}
{{cat|Temperaments}}{{Navbox regtemp}}

Latest revision as of 23:03, 14 March 2026

Magic
Subgroups 2.3.5, 2.3.5.7
Reduced mapping ⟨1; 5 1 12]
ET join 19 & 22
Generators (CWE) ~5/4 = 380.5¢
MOS scales 3L 4s, 3L 7s, …, 3L 16s, 19L 3s
Ploidacot alpha-pentacot
Comma basis 3125/3072 (5-limit);
225/224, 245/243 (7-limit)
Minimax error 5-odd-limit: 5.9¢;
9-odd-limit: 5.9¢
Target scale size 5-odd-limit: 7 notes;
9-odd-limit: 13 notes

Magic (19 & 22) is a 2.3.5 temperament that equates a stack of five 5/4 major thirds to one 3/1. It also equates 25/24 to 128/125, shrinking the difference between 5/4 and 6/5. It is a 3-cluster temperament, as indicated by the edo join (22 - 19 = 3).

Interval chain

In the following table, odd harmonics 1–15 and their inverses are in bold.

# Cents* Approximate ratios
0 0.0 1/1
1 380.5 5/4
2 760.9 14/9
3 1141.4 27/14, 31/16
4 321.8 6/5, 29/24
5 702.3 3/2
6 1082.7 15/8, 28/15
7 263.2 7/6
8 643.7 36/25
9 1024.1 9/5, 29/16
10 204.6 9/8
11 585.0 7/5
12 965.5 7/4
13 145.9 35/32

* In 7-limit CWE tuning

Extensions

Magic divides ~16/15 in half (into two 25/24's), so it easily extends to prime 31 (at +3 generators) by tempering out S31 = 961/960. Since in 7-limit Magic, the 16/15 is also a 15/14, we can also extend 2.3.5.7.31 Magic to add prime 29 (at +9 generators) by equating 16/15 to 31/29.

This can be seen from the following S-expression for the Magic comma: 

3125/3072
= (25/24)^2/(16/15)
= (25/24)*(25/24)/(32/31*31/30)
= (25/24)S25*S26*S27*S28*S29*S30/(32/31)
= (S25*S26*S27*S28*S29*S30)^2*S31
= (S15*S28*S29*S30)^2*S31

structurally inducing the above 2.3.5.7.29.31 extension.

List of patent vals

The following patent vals support 2.3.5 Magic. Vals that are contorted in 2.3.5 are not included.

Edo Extension to 7 Generator Fifth
3 400.000 800.000
25 384.000 720.000
22 19 & 22 381.818 709.091
107 381.308 706.542
85 19 & 22 381.176 705.882
63 19 & 22 380.952 704.762
104 380.769 703.846
41 19 & 22 380.488 702.439
60 19 & 22 380.000 700.000
79 379.747 698.734
19 19 & 22 378.947 694.737
35 377.143 685.714
16 375.000 675.000


View • Talk • EditRegular temperaments
Rank-2
Acot Blackwood (1/5-octave) • Whitewood (1/7-octave) • Compton (1/12-octave)
Monocot Meantone • Schismic • Gentle-fifth temperaments • Archy
Complexity 2 Diaschismic (diploid monocot) • Pajara (diploid monocot) • Injera (diploid monocot) • Rastmatic (dicot) • Mohajira (dicot) • Intertridecimal (dicot) • Interseptimal (alpha-dicot)
Complexity 3 Augmented (triploid) • Misty (triploid) • Slendric (tricot) • Porcupine (omega-tricot)
Complexity 4 Diminished (tetraploid) • Tetracot (tetracot) • Buzzard (alpha-tetracot) • Squares (beta-tetracot) • Negri (omega-tetracot)
Complexity 5-6 Magic (alpha-pentacot) • Amity (gamma-pentacot) • Kleismic (alpha-hexacot) • Miracle (hexacot)
Higher complexity Orwell (alpha-heptacot) • Sensi (beta-heptacot) • Octacot (octacot) • Wurschmidt (beta-octacot) • Valentine (enneacot) • Ammonite (epsilon-enneacot) • Myna (beta-decacot) • Ennealimmal (enneaploid dicot)
Straddle-3 A-Team (alter-tricot) • Machine (alter-monocot)
No-3 Trismegistus (alpha-triseph) • Orgone (trimech) • Didacus (diseph)
No-octaves Sensamagic (monogem)
Exotemperament Dicot • Mavila • Father
Higher-rank
Rank-3 Hemifamity • Marvel • Parapyth
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