Hemimean family: Difference between revisions

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The '''hemimean family''' of temperaments are rank-3 temperaments tempering out [[3136/3125]].  
The '''hemimean family''' of [[temperament]]s are [[rank-3 temperament]]s [[tempering out]] [[3136/3125]].  


The hemimean comma, 3136/3125, is the ratio between the diesis and the tritonic diesis, or jubilisma; that is, (128/125)/(50/49).
The hemimean comma, 3136/3125, is the ratio between the [[126/125|septimal semicomma (126/125)]] and the [[225/224|septimal kleisma (225/224)]]. This fact alone makes hemimean a very notable rank-3 temperament, as any non-meantone tuning of hemimean will split the [[81/80|syntonic comma (81/80)]] into two equal parts, each representing 126/125~225/224.  


If we write this equivalently using S-expressions we obtain ([[128/125|S4/S5]])/([[50/49|S5/S7]]) which can be rearranged to [[16/15|S4]]*[[49/48|S7]]/[[25/24|S5]]<sup>2</sup>.
Other equivalences characteristic to hemimean are [[128/125]]~[[50/49]] and [[49/45]]~([[25/24]])<sup>2</sup>.  
 
Then we can optionally replace S4 with a nontrivial equivalent S-expression, S4 = [[36/35|S6]]*[[49/48|S7]]*[[64/63|S8]]; substituting this in and simplifying yields:
 
S6*S7*S8*S7/S5<sup>2</sup> from which we can obtain an alternative equivalence 3136/3125 = ([[49/45]])/([[25/24]])<sup>2</sup>.


== Hemimean ==
== Hemimean ==
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[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]
* [[7-odd-limit|7-]] and [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 6/5 0 0 2/5 }}, {{monzo| 0 0 0 1 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 6/5 0 0 2/5 }}, {{monzo| 0 0 0 1 }}]
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.3.7
: [[eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.3.7


{{Optimal ET sequence|legend=1| 12, 19, 31, 68, 80, 87, 99, 217, 229, 328, 347, 446, 675c }}
{{Optimal ET sequence|legend=1| 12, 19, 31, 68, 80, 87, 99, 217, 229, 328, 347, 446, 675c }}
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=== Hemimean orion ===
=== Hemimean orion ===
As the second generator of hemimean, [[28/25]], is close to [[19/17]], and as the latter is the [[mediant]] of [[10/9]] and [[9/8]] (which combine to make [[5/4]] and whose difference is [[81/80|S9]]), it is natural to extend hemimean to the 2.3.5.7.17.19 subgroup by tempering out ([[28/25]])/([[19/17]]) = [[476/475]], or equivalently stated, the [[semiparticular]] (5/4)/(19/17)<sup>2</sup> = [[1445/1444]]. Notice 3136/3125 = (476/475)([[2128/2125]]) and that 2128/2125 = ([[1216/1215]])([[1701/1700]]), so it makes sense to temper out 1216/1215 and/or 1701/1700 as well. This temperament finds the harmonic 17 and 19 at (+5, +1) and (+5, +2), respectively, with virtually no additional error.  
As the second generator of hemimean, [[28/25]], is close to [[19/17]], and as the latter is the [[mediant]] of [[10/9]] and [[9/8]], it is natural to extend hemimean to the 2.3.5.7.17.19 subgroup by tempering out ([[28/25]])/([[19/17]]) = [[476/475]], or equivalently stated, the [[semiparticular]] (5/4)/(19/17)<sup>2</sup> = [[1445/1444]]. Notice 3136/3125 = (476/475)([[2128/2125]]) and that 2128/2125 = ([[1216/1215]])([[1701/1700]]), so it makes sense to temper out 1216/1215 and/or 1701/1700 as well. This temperament finds the harmonic 17 and 19 at (+5, +1) and (+5, +2), respectively, with virtually no additional error.
 
The S-expression-based comma list is { [[1216/1215|S16/S18]], [[1445/1444|S17/S19]], [[1701/1700|S18/S20]](, ([[136/135|S16*S17]])/([[190/189|S19*S20]]) = [[476/475|S16/S18 * S17/S19 * S18/S20]]) }.


Subgroup: 2.3.5.7.17
Subgroup: 2.3.5.7.17
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==== 2.3.5.7.17.19 subgroup ====
==== 2.3.5.7.17.19 subgroup ====
Subgroup: 2.3.5.7.17.19
Subgroup: 2.3.5.7.17.19
S-expression-based comma list: { [[1216/1215|S16/S18]], [[1445/1444|S17/S19]], [[1701/1700|S18/S20]](, ([[136/135|S16*S17]])/([[190/189|S19*S20]]) = [[476/475|S16/S18 * S17/S19 * S18/S20]]) }.


Comma list: 476/475, 1216/1215, 1445/1444
Comma list: 476/475, 1216/1215, 1445/1444
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=== Semiorion ===
=== Semiorion ===
Semiorion is an alternative subgroup extension of lower complexity, which splits the octave into two.  
Semiorion is an alternative subgroup extension of lower complexity, which splits the octave into two. The S-expression-based comma list is {[[289/288|S17]], [[361/360|S19]], [[1216/1215|S16/S18]](, [[1701/1700|S18/S20]], [[476/475]] = [[2128/2125|S16/S20]] * [[1445/1444|S17/S19]])}.  


Subgroup: 2.3.5.7.17
Subgroup: 2.3.5.7.17
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==== 2.3.5.7.17.19 subgroup ====
==== 2.3.5.7.17.19 subgroup ====
Subgroup: 2.3.5.7.17.19
Subgroup: 2.3.5.7.17.19
S-expression-based comma list: {[[289/288|S17]], [[361/360|S19]], [[1216/1215|S16/S18]](, [[1701/1700|S18/S20]], [[476/475]] = [[2128/2125|S16/S20]] * [[1445/1444|S17/S19]])}


Comma list: 289/288, 361/360, 476/475
Comma list: 289/288, 361/360, 476/475

Revision as of 13:18, 9 February 2024

The hemimean family of temperaments are rank-3 temperaments tempering out 3136/3125.

The hemimean comma, 3136/3125, is the ratio between the septimal semicomma (126/125) and the septimal kleisma (225/224). This fact alone makes hemimean a very notable rank-3 temperament, as any non-meantone tuning of hemimean will split the syntonic comma (81/80) into two equal parts, each representing 126/125~225/224.

Other equivalences characteristic to hemimean are 128/125~50/49 and 49/45~(25/24)2.

Hemimean

Subgroup: 2.3.5.7

Comma list: 3136/3125 (hemimean)

Mapping: [⟨1 0 0 -3], ⟨0 1 0 0], ⟨0 0 2 5]]

mapping generators: ~2, ~3, ~56/25

Mapping to lattice: [⟨0 0 2 5], ⟨0 1 0 0]]

Lattice basis:

28/25 length = 0.5055, 3/2 length = 1.5849
Angle (28/25, 3/2) = 90 degrees

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.9550, ~28/25 = 193.6499

Minimax tuning:

[[1 0 0 0⟩, [0 1 0 0⟩, [6/5 0 0 2/5⟩, [0 0 0 1⟩]
Eigenmonzo (unchanged-interval) basis: 2.3.7

Optimal ET sequence: 12, 19, 31, 68, 80, 87, 99, 217, 229, 328, 347, 446, 675c

Badness: 0.160 × 10-3

Complexity spectrum: 5/4, 7/5, 4/3, 6/5, 8/7, 7/6, 9/8, 10/9, 9/7

Projection pairs: 5 3136/625 7 68841472/9765625 to 2.3.25/7

Hemimean orion

As the second generator of hemimean, 28/25, is close to 19/17, and as the latter is the mediant of 10/9 and 9/8, it is natural to extend hemimean to the 2.3.5.7.17.19 subgroup by tempering out (28/25)/(19/17) = 476/475, or equivalently stated, the semiparticular (5/4)/(19/17)2 = 1445/1444. Notice 3136/3125 = (476/475)(2128/2125) and that 2128/2125 = (1216/1215)(1701/1700), so it makes sense to temper out 1216/1215 and/or 1701/1700 as well. This temperament finds the harmonic 17 and 19 at (+5, +1) and (+5, +2), respectively, with virtually no additional error.

The S-expression-based comma list is { S16/S18, S17/S19, S18/S20(, (S16*S17)/(S19*S20) = S16/S18 * S17/S19 * S18/S20) }.

Subgroup: 2.3.5.7.17

Comma list: 1701/1700, 3136/3125

Sval mapping: [⟨1 0 0 -3 -5], ⟨0 1 0 0 5], ⟨0 0 2 5 1]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.1960, ~28/25 = 193.6548

Optimal ET sequence: 12, 19g, 31g, …, 87, 99, 217, 229, 316, 328h, 446, 545c, 873cg

Badness: 0.573

2.3.5.7.17.19 subgroup

Subgroup: 2.3.5.7.17.19

Comma list: 476/475, 1216/1215, 1445/1444

Sval mapping: [⟨1 0 0 -3 -5 -6], ⟨0 1 0 0 5 5], ⟨0 0 2 5 1 2]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~19/17 = 193.647

Optimal ET sequence: 12, 19gh, 31gh, …, 87, 99, 118, 210gh, 217, 229, 328h, 446

Badness: 0.456

Semiorion

Semiorion is an alternative subgroup extension of lower complexity, which splits the octave into two. The S-expression-based comma list is {S17, S19, S16/S18(, S18/S20, 476/475 = S16/S20 * S17/S19)}.

Subgroup: 2.3.5.7.17

Comma list: 289/288, 3136/3125

Sval mapping: [⟨2 0 0 -6 5], ⟨0 1 0 0 1], ⟨0 0 2 5 0]]

sval mapping generators: ~17/12, ~3, ~56/25

Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.3471, ~28/25 = 193.6499

Optimal ET sequence: 12, 30d, 38d, 50, 62, 68, 106d, 118, 248g, 316g

Badness: 1.095

2.3.5.7.17.19 subgroup

Subgroup: 2.3.5.7.17.19

Comma list: 289/288, 361/360, 476/475

Mapping: [⟨2 0 0 -6 5 3], ⟨0 1 0 0 1 1], ⟨0 0 2 5 0 1]]

Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.509, ~28/25 = 193.669

Optimal ET sequence: 12, …, 50, 68, 106d, 118, 248g, 316g

Badness: 0.569

Belobog

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125

Mapping: [⟨1 0 0 -3 -9], ⟨0 1 0 0 2], ⟨0 0 2 5 8]]

mapping generators: ~2, ~3, ~56/25

Mapping to lattice: [⟨0 -2 2 5 4], ⟨0 -1 0 0 -2]]

Lattice basis:

28/25 length = 0.3829, 16/15 length = 1.1705
Angle (28/25, 16/15) = 93.2696

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.7205, ~28/25 = 193.5545

Minimax tuning:

[[1 0 0 0 0⟩, [27/22 6/11 -5/22 -3/11 5/22⟩, [24/11 -4/11 -2/11 2/11 2/11⟩, [27/11 -10/11 -5/11 5/11 5/11⟩, [24/11 -4/11 -13/11 2/11 13/11⟩]
Eigenmonzo (unchanged-interval) basis: 2.9/7.11/5

Optimal ET sequence: 12, 19e, 31, 68e, 87, 99e, 118, 130, 217, 248

Badness: 0.609 × 10-3

Projection pairs: 5 3136/625 7 68841472/9765625 11 1700108992512/152587890625 to 2.3.25/7

Scales: belobog31

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1001/1000, 3136/3125

Mapping: [⟨1 0 0 -3 -9 15], ⟨0 1 0 0 2 -2], ⟨0 0 2 5 8 -7]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.8219, ~28/25 = 193.5816

Optimal ET sequence: 31, 43, 56, 74, 87, 118, 130, 217, 248, 347e, 378, 465, 595e

Badness: 1.11 × 10-3

Bellowblog

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 625/624

Mapping: [⟨1 0 0 -3 -9 -4], ⟨0 1 0 0 2 -1], ⟨0 0 2 5 8 8]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.5667, ~28/25 = 193.2493

Optimal ET sequence: 12f, 19e, 31, 56, 68e, 87, 118, 186ef, 205d

Badness: 1.26 × 10-3

Siebog

Subgroup: 2.3.5.7.11

Comma list: 540/539, 3136/3125

Mapping: [⟨1 0 0 -3 8], ⟨0 1 0 0 3], ⟨0 0 2 5 -8]]

mapping generators: ~2, ~3, ~56/25

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.1636, ~28/25 = 193.8645

Minimax tuning:

[[1 0 0 0 0⟩, [0 1 0 0 0⟩, [8/5 3/5 1/5 0 -1/5⟩, [1 3/2 1/2 0 -1/2⟩, [8/5 3/5 -4/5 0 4/5⟩]
Eigenmonzo (unchanged-interval) basis: 2.3.11/5

Optimal ET sequence: 12e, 18e, 19, 31, 68e, 80, 99e, 130, 210e, 241, 340ce, 371ce, 470cdee, 501cde, 581cdee, 711ccdee

Badness: 0.870 × 10-3

Triglav

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125

Mapping: [⟨1 0 2 2 1], ⟨0 1 2 5 2], ⟨0 0 -4 -10 -1]]

mapping generators: ~2, ~3, ~18/11

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.2875, ~18/11 = 854.3132

Optimal ET sequence: 24d, 31, 80, 87, 111, 118, 198, 316, 514c, 545c

Badness: 0.819 × 10-3

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