Kleismic: Difference between revisions

From Xenharmonic Reference
← Older edit
VisualWikitext
mNo edit summary
No edit summary
 
(12 intermediate revisions by 2 users not shown)
Line 10: Line 10:
| Odd limit 2 = 2.3.5.13 15 | Mistuning 2 = 2.35 | Complexity 2 = 15
| Odd limit 2 = 2.3.5.13 15 | Mistuning 2 = 2.35 | Complexity 2 = 15
}}
}}
[[File:Kleismic11.png|thumb|Kleismic 11-note MOS]]
'''Kleismic''', [15 & 19], is a high-accuracy temperament (usually seen in its basic form as a 2.3.5 temperament) that equates a stack of six 6/5 minor thirds to one 3/1. Via a [[canonical extension|structurally induced]] extension to 2.3.5.13, it equates three 6/5's to one semitwelfth [[26/15]] and equates 25/24 to 26/25 and 27/26.


'''Kleismic''', 2.3.5.13[15 & 19], is a high-accuracy 2.3.5 temperament that equates a stack of six 6/5 minor thirds to one 3/1. Via a [[canonical extension|structurally induced]] extension to 2.3.5.13, it equates three 6/5's to one semitwelfth [[26/15]] and equates 25/24 to 26/25 and 27/26.
Kleismic harmony is naturally based on splitting 5/3 into 4/3 and 5/4 thus making 3:4:5 or 12:15:18 triads. Indeed, Kleismic[15] (4L11s) can be constructed from the [[generator sequence]] GS(3:4:5)[19] by tempering out four kleismas.
 
Kleismic, despite generating a heptatonic scale, is not particularly usefully a 7-form temperament; this is because 3/2 is an imperfect sixth rather than a perfect fifth. In the 11-form, however, it is much better as 5/4 and 4/3 are mapped to the same degree, resulting in a dichotomy of [0 4 8]/11 triads similar to the [0 2 4]/7 ones found in fifth-centric temperaments.
 
Kleismic can be seen as a counterpart to [[Diaschismic]], which as a 2.3.5.17 temperament splits 9/8 into two 16/15s that are also 17/16~18/17 - Diaschismic splits 9/8 into two; Kleismic splits it into three. The intersection of both temperaments in the 5-limit is [[34edo]], which splits 9/8 into six steps; for that reason the 34edo step is called the sextula.
 
Kleismic contains [[interordinal]] intervals, because 3/1 is split into an even number of parts. Because it equates 9/8 to three 25/24s, the inframinor and ultramajor third in [[alpha-dicot]] tuning are separated by 9/8, such that in kleismic augmenting or diminishing a 5/4 or 6/5 respectively by 25/24 results in an interordinal.
== Interval chain ==
In the following table, odd harmonics 1–15 are labeled in '''bold'''.  


Kleismic harmony is naturally based on splitting 5/3 into 4/3 and 5/4 thus making 3:4:5 or 12:15:18 triads. Indeed, Kleismic[15] (4L11s) can be constructed from the [[generator sequence]] GS(3:4:5)[19] by tempering out four kleismas.
{| class="wikitable sortable center-1 right-2"
! #
! Cents*
! class="unsortable" | Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
|-
| 1
| 317.1
| 6/5
|-
| 2
| 634.2
| 13/9, 36/25
|-
| 3
| 951.3
| 26/15
|-
| 4
| 68.4
| 25/24, 26/25, 27/26
|-
| 5
| 385.5
| '''5/4'''
|-
| 6
| 702.6
| '''3/2'''
|-
| 7
| 1019.6
| 9/5
|-
| 8
| 136.7
| 13/12, 27/25
|-
| 9
| 453.8
| 13/10
|-
| 10
| 770.9
| 25/16, 39/25
|-
| 11
| 1088.0
| '''15/8'''
|-
| 12
| 205.1
| '''9/8'''
|-
| 13
| 522.2
| 27/20
|-
| 14
| 839.3
| '''13/8'''
|-
| 15
| 1156.4
| 39/20
|-
| 16
| 273.5
| 75/64
|-
| 17
| 590.6
| 45/32
|-
| 18
| 907.7
| 27/16
|-
| 19
| 24.7
| 65/64, 81/80
|}
<nowiki/>* In 2.3.5.13-subgroup [[CWE tuning]], octave reduced
 
== List of patent vals ==
:''Main article: [[Kleismic/Patent vals]]''


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

Latest revision as of 01:57, 2 June 2026

Kleismic
Subgroups 2.3.5, 2.3.5.13
Reduced mapping ⟨1; 6 5 14]
ET join 15 & 19
Generators (CWE) ~6/5 = 317.1¢
MOS scales 3L 1s, 4L 3s, 4L 7s, 4L 11s, 15L 4s
Ploidacot alpha-hexacot
Comma basis 15625/15552 (2.3.5);
325/324, 625/624 (2.3.5.13)
Minimax error 5-odd-limit: 1.35¢;
2.3.5.13 15-odd-limit: 2.35¢
Target scale size 5-odd-limit: 7 notes;
2.3.5.13 15-odd-limit: 15 notes
Kleismic 11-note MOS

Kleismic, [15 & 19], is a high-accuracy temperament (usually seen in its basic form as a 2.3.5 temperament) that equates a stack of six 6/5 minor thirds to one 3/1. Via a structurally induced extension to 2.3.5.13, it equates three 6/5's to one semitwelfth 26/15 and equates 25/24 to 26/25 and 27/26.

Kleismic harmony is naturally based on splitting 5/3 into 4/3 and 5/4 thus making 3:4:5 or 12:15:18 triads. Indeed, Kleismic[15] (4L11s) can be constructed from the generator sequence GS(3:4:5)[19] by tempering out four kleismas.

Kleismic, despite generating a heptatonic scale, is not particularly usefully a 7-form temperament; this is because 3/2 is an imperfect sixth rather than a perfect fifth. In the 11-form, however, it is much better as 5/4 and 4/3 are mapped to the same degree, resulting in a dichotomy of [0 4 8]/11 triads similar to the [0 2 4]/7 ones found in fifth-centric temperaments.

Kleismic can be seen as a counterpart to Diaschismic, which as a 2.3.5.17 temperament splits 9/8 into two 16/15s that are also 17/16~18/17 - Diaschismic splits 9/8 into two; Kleismic splits it into three. The intersection of both temperaments in the 5-limit is 34edo, which splits 9/8 into six steps; for that reason the 34edo step is called the sextula.

Kleismic contains interordinal intervals, because 3/1 is split into an even number of parts. Because it equates 9/8 to three 25/24s, the inframinor and ultramajor third in alpha-dicot tuning are separated by 9/8, such that in kleismic augmenting or diminishing a 5/4 or 6/5 respectively by 25/24 results in an interordinal.

Interval chain

In the following table, odd harmonics 1–15 are labeled in bold.

# Cents* Approximate ratios
0 0.0 1/1
1 317.1 6/5
2 634.2 13/9, 36/25
3 951.3 26/15
4 68.4 25/24, 26/25, 27/26
5 385.5 5/4
6 702.6 3/2
7 1019.6 9/5
8 136.7 13/12, 27/25
9 453.8 13/10
10 770.9 25/16, 39/25
11 1088.0 15/8
12 205.1 9/8
13 522.2 27/20
14 839.3 13/8
15 1156.4 39/20
16 273.5 75/64
17 590.6 45/32
18 907.7 27/16
19 24.7 65/64, 81/80

* In 2.3.5.13-subgroup CWE tuning, octave reduced

List of patent vals

Main article: Kleismic/Patent vals


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 Mabilic (alpha-triseph) • Orgone (trimech) • Didacus (diseph)
No-octaves Sensamagic (monogem)
Exotemperaments Dicot • Mavila • Father
Higher-rank
Rank-3 Hemifamity • Marvel • Parapyth
Retrieved from "https://xenreference.com/wiki/index.php?title=Kleismic&oldid=7393"