Reversed meantone: Difference between revisions

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<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">'''Reversed meantone (b11&amp;b19, 3.2.5 subgroup)'''</div>
<div style="line-height:1.6;">'''Reversed meantone (b11&amp;b19, 3.2.5 subgroup)'''
<div class="mw-collapsible-content">
Subgroup: 3.2.5
Subgroup: 3.2.5


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<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"><div style="line-height:1.6;">'''Reversed meantone (b11&amp;b19, 3.2.5.7 subgroup)'''</div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"><div style="line-height:1.6;">'''Reversed meantone (b11&amp;b19, 3.2.5.7 subgroup)'''
<div class="mw-collapsible-content">
Subgroup: 3.2.5.7
Subgroup: 3.2.5.7


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<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">'''Reversed meantone (b11de&amp;b19e, 3.2.5.7.11 subgroup)'''</div>
<div style="line-height:1.6;">'''Reversed meantone (b11de&amp;b19e, 3.2.5.7.11 subgroup)'''
<div class="mw-collapsible-content">
Subgroup: 3.2.5.7.11
Subgroup: 3.2.5.7.11


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<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">'''Reversed meantone (c16&amp;c28, 5.2.3 subgroup)'''</div>
<div style="line-height:1.6;">'''Reversed meantone (c16&amp;c28, 5.2.3 subgroup)'''
<div class="mw-collapsible-content">
Subgroup: 5.2.3 (3/2.5)
Subgroup: 5.2.3 (3/2.5)


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[[POTE generator]]: ~3/2 = 696.5784, ~10/9 = 192.5712
[[POTE generator]]: ~3/2 = 696.5784, ~10/9 = 192.5712


[[TOP tuning|TOP generator]]s: ~3/2 = 697.0491, ~10/9 = [[Tel:192.7014|192.7014]]
[[TOP tuning|TOP generator]]s: ~3/2 = 697.0491, ~10/9 = 192.7014


[[Optimal GPV sequence]]: {{Val list| c16, c28, c44 }}
[[Optimal GPV sequence]]: {{Val list| c16, c28, c44 }}
</div></div>
</div></div>


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"><div style="line-height:1.6;">'''Reversed meantone (c12d&amp;c16dee, 5.2.3.7 subgroup)'''</div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"><div style="line-height:1.6;">'''Reversed meantone (c12d&amp;c16dee, 5.2.3.7 subgroup)'''Subgroup: 5.2.3.7
<div class="mw-collapsible-content">
Subgroup: 5.2.3.7
 
[[Comma list]]: 64/63, 81/80
[[Comma list]]: 64/63, 81/80


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[[TOP tuning|TOP generator]]s: ~3/2 = 697.2159, ~10/9 = 202.3697
[[TOP tuning|TOP generator]]s: ~3/2 = 697.2159, ~10/9 = 202.3697


[[Optimal GPV sequence]]: {{Val list| c12d, c16dee, c28, c40, c52d, c64 }})
[[Optimal GPV sequence]]: {{Val list| c12d, c16dee, c28, c40, c52d, c64 }}
</div></div>
</div></div>


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">'''Reversed meantone (c12de&amp;c16dee, 5.2.3.7.11 subgroup)'''</div>
<div style="line-height:1.6;">'''Reversed meantone (c12de&amp;c16dee, 5.2.3.7.11 subgroup)'''
<div class="mw-collapsible-content">
Subgroup: 5.2.3.7.11
Subgroup: 5.2.3.7.11  


[[Comma list]]: 64/63, 81/80, 99/98  
[[Comma list]]: 64/63, 81/80, 99/98  
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</div></div>  
</div></div>  
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">'''Reversed meantone (q4&amp;q7, 3/2.2.3.5 subgroup)'''</div>
<div style="line-height:1.6;">'''Reversed meantone (q4&amp;q7, 3/2.2.3.5 subgroup)'''
<div class="mw-collapsible-content">
Subgroup: 3/2.2.3.5
Subgroup: 3/2.2.3.5


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<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"><div style="line-height:1.6;">'''Reversed meantone (q3&amp;q7, 3/2.2.3.5.7 subgroup)'''</div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"><div style="line-height:1.6;">'''Reversed meantone (q3&amp;q7, 3/2.2.3.5.7 subgroup)'''
<div class="mw-collapsible-content">
Subgroup: 3/2.2.3.5.7
Subgroup: 3/2.2.3.5.7


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<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">'''Reversed meantone (q7e&amp;q11, 3/2.2.3.5.7.11 subgroup)'''</div>
<div style="line-height:1.6;">'''Reversed meantone (q7e&amp;q11, 3/2.2.3.5.7.11 subgroup)'''
<div class="mw-collapsible-content">
Subgroup: 3/2.2.3.5.7.11
Subgroup: 3/2.2.3.5.7.11


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[[Optimal GPV sequence]]: {{Val list| q7e, q11, q18, q25 }}
[[Optimal GPV sequence]]: {{Val list| q7e, q11, q18, q25 }}
</div></div>
</div></div>


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">'''Reversed meantone (q16&amp;q25, 5/2.2.3.5 subgroup)'''</div>
<div style="line-height:1.6;">'''Reversed meantone (q16&amp;q25, 5/2.2.3.5 subgroup)'''
<div class="mw-collapsible-content">
Subgroup: 5/2.2.3.5
Subgroup: 5/2.2.3.5


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[[Optimal GPV sequence]]: {{Val list| q16, q25, q41 }}
[[Optimal GPV sequence]]: {{Val list| q16, q25, q41 }}
</div></div>  
</div></div>  
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"><div style="line-height:1.6;">'''Reversed meantone (q16&amp;q25, 5/2.2.3.5.7 subgroup)'''</div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"><div style="line-height:1.6;">'''Reversed meantone (q16&amp;q25, 5/2.2.3.5.7 subgroup)'''
<div class="mw-collapsible-content">
Subgroup: 5/2.2.3.5.7
Subgroup: 5/2.2.3.5.7


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<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">'''Reversed meantone (q16&amp;q41, 5/2.2.3.5.7.11 subgroup)'''</div>
<div style="line-height:1.6;">'''Reversed meantone (q16&amp;q41, 5/2.2.3.5.7.11 subgroup)'''
<div class="mw-collapsible-content">
Subgroup: 5/2.2.3.5.7.11
Subgroup: 5/2.2.3.5.7.11


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<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">'''Reversed meantone (q9&amp;q14, 5/3.2.3.5 subgroup)'''</div>
<div style="line-height:1.6;">'''Reversed meantone (q9&amp;q14, 5/3.2.3.5 subgroup)'''
<div class="mw-collapsible-content">
Subgroup: 5/3.2.3.5
Subgroup: 5/3.2.3.5


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[[Optimal GPV sequence]]: {{Val list| q9, q14, q23 }}
[[Optimal GPV sequence]]: {{Val list| q9, q14, q23 }}
</div></div>  
</div></div>  
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"><div style="line-height:1.6;">'''Reversed meantone (q9&amp;q14d, 5/3.2.3.5.7 subgroup)'''</div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;"><div style="line-height:1.6;">'''Reversed meantone (q9&amp;q14d, 5/3.2.3.5.7 subgroup)'''
<div class="mw-collapsible-content">
Subgroup: 5/3.2.3.5.7
Subgroup: 5/3.2.3.5.7


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<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">'''Reversed meantone (q9&amp;q14, 5/3.2.3.5.7.11 subgroup)'''</div>
<div style="line-height:1.6;">'''Reversed meantone (q9&amp;q14, 5/3.2.3.5.7.11 subgroup)'''
<div class="mw-collapsible-content">
Subgroup: 5/3.2.3.5.7.11
Subgroup: 5/3.2.3.5.7.11


[[Comma list]]: 81/80, 126/125, 100/99
[[Comma list]]: 81/80, 126/125, 100/99


[[Gencom]]: [5/3 ; 81/80 126/125 100/99]
[[Gencom]]: [5/3 10/9; 81/80 126/125 100/99]


[[Mapping|Sval mapping]]: [{{val| 1 2 3 4 4 6}}, {{val| 0 -3 -4 -4 -1 -6}}]
[[Mapping|Sval mapping]]: [{{val| 1 2 3 4 4 6}}, {{val| 0 -3 -4 -4 -1 -6}}]

Revision as of 06:04, 10 June 2022

Reversed meantone is a temperament which tempers out the 41-limit comma 82/81.

Properties

As meantone is based on the syntonic comma, 81/80, tempering the fifth flat, tempering 82/81 instead results in a sharper fifth, and a major third equivalent to the 41st harmonic instead of the 5th, so it might as well be called reverse meantone. As a very high limit interval, however, that 41/32 is far less recognizable as an interval than meantone’s 5/4, and would more likely be heard as a flat 9/7. Additionally, the 41st is very delicate, and mistuning by several cents destroys it, so if its use is intended as more than a joke exact quarter comma tempering is best, although 39edo does a fair job.

Related to this idea, 162/161 is a 23-limit comma (specifically 161 = 7 × 23), and 163/162 being prime would indeed be ridiculous.

The more well known 64/63 comma equates 9/8 with 8/7 instead of 10/9, which also results in a sharper fifth, and the major third is equivalent to 9/7.

Temperament data

Reversed meantone (5&12, 2.3.41 subgroup)

Subgroup: 2.3.41

Comma list: 82/81

Gencom: [2 4/3; 82/81]

Sval mapping: [⟨1 2 7], ⟨0 -1 -4]]

POTE generator: ~4/3 = 494.5086

TOP generators: ~2 = 1199.6961, ~4/3 = 494.3834

Optimal GPV sequence: Template:Val list


Reversed meantone (5&12, 2.3.7.41 subgroup)

Subgroup: 2.3.7.41

Comma list: 64/63, 82/81

Gencom: [2 4/3; 64/63 82/81]

Sval mapping: [⟨1 2 2 7], ⟨0 -1 2 -4]]

POTE generator: ~4/3 = 490.0323

TOP generators: ~2 = 1197.2342, ~4/3 = 488.9029

Optimal GPV sequence: Template:Val list


Reversed meantone (5&12, 2.3.7.11.41 subgroup)

Subgroup: 2.3.7.11.41

Comma list: 64/63, 82/81, 99/98

Gencom: [2 4/3; 64/63 82/81 99/98]

Sval mapping: [⟨1 2 2 1 7], ⟨0 -1 2 6 -4]]

POTE generator: ~4/3 = 492.1787

TOP generators: ~2 = 1197.9683, ~4/3 = 491.3454

Optimal GPV sequence: Template:Val list

Non-octave

As meantone is based within the octave and on the syntonic comma, 81/80, tempering the fifth flat, and the thirds to be simple rational intervals, basing it within the third or fifth harmonic or the perfect fifth or the major tenth, sixth or third instead still may temper 81/80, even if it might as well be called reverse meantone. Of these, the major third of 5/4 is the least recognizable as an interval, though still very recognizable. Additionally, the major third of 5/4 is a very narrow interval to base a temperament within, so if its use is intended as more than a joke it might as well be the augmented temperament 4ned5/2.

Temperament data

Reversed meantone (b11&b19, 3.2.5 subgroup)

Subgroup: 3.2.5

Comma list: 81/80

Gencom: [3 3/2; 81/80]

Sval mapping: [⟨1 1 0], ⟨0 -1 4]]

POTE generator: ~3/2 = 698.3375

TOP generators: ~3 = 1898.4460, ~3/2 = 697.0491

Optimal GPV sequence: Template:Val list


Reversed meantone (b11&b19, 3.2.5.7 subgroup)

Subgroup: 3.2.5.7

Comma list: 64/63, 81/80

Gencom: [3 3/2; 64/63 81/80]

Sval mapping: [⟨1 1 0 3], ⟨0 -1 4 -13]]

POTE generator: ~3/2 = 698.5001

TOP generators: ~3 = 1898.4580, ~3/2 = 697.2159

Optimal GPV sequence: Template:Val list

Reversed meantone (b11de&b19e, 3.2.5.7.11 subgroup)

Subgroup: 3.2.5.7.11

Comma list: 64/63, 81/80, 99/98

Gencom: [2 3/2; 64/63 81/80 99/98]

Sval mapping: [⟨1 1 0 4 7], ⟨0 -1 4 -6 -13]]

POTE generator: ~3/2 = 702.7738

TOP generators: ~3 = 1893.9031, ~3/2 = 699.7986

Optimal GPV sequence: Template:Val list

Reversed meantone (c16&c28, 5.2.3 subgroup)

Subgroup: 5.2.3 (3/2.5)

Comma list: 81/80

Gencom: [3/2 10/9; 81/80]

Sval mapping: [⟨4 2 3], ⟨0 -1 —1]]

POTE generator: ~3/2 = 696.5784, ~10/9 = 192.5712

TOP generators: ~3/2 = 697.0491, ~10/9 = 192.7014

Optimal GPV sequence: Template:Val list

Reversed meantone (c12d&c16dee, 5.2.3.7 subgroup)Subgroup: 5.2.3.7

Comma list: 64/63, 81/80

Gencom: [3/2 10/9; 64/63 81/80]

Sval mapping: [⟨4 2 3 6], ⟨0 -1 —1 -4]]

POTE generator: ~3/2 = 696.5784, ~10/9 = 201.7001

TOP generators: ~3/2 = 697.2159, ~10/9 = 202.3697

Optimal GPV sequence: Template:Val list

Reversed meantone (c12de&c16dee, 5.2.3.7.11 subgroup)

Subgroup: 5.2.3.7.11

Comma list: 64/63, 81/80, 99/98

Gencom: [3/2 10/9; 64/63 81/80 99/98]

Sval mapping: [⟨4 2 3 6 8], ⟨0 -1 —1 -4 -7]]

POTE generator: ~3/2 = 696.5784, ~10/9 = 204.5471

TOP generators: ~3/2 = 699.7986, ~10/9 = 205.4927

Optimal GPV sequence: Template:Val list)

Reversed meantone (q4&q7, 3/2.2.3.5 subgroup)

Subgroup: 3/2.2.3.5

Comma list: 81/80

Gencom: [3/2 9/8; 81/80]

Sval mapping: [⟨1 2 3 4], ⟨0 -1 -1 0]]

POTE generator: ~9/8 = 198.7378

TOP generators: ~3/2 = 699.3710, ~9/8 = 198.0062

Optimal GPV sequence: Template:Val list


Reversed meantone (q3&q7, 3/2.2.3.5.7 subgroup)

Subgroup: 3/2.2.3.5.7

Comma list: 64/63, 81/80

Gencom: [3/2 9/8; 64/63 81/80]

Sval mapping: [⟨1 2 3 4 6], ⟨0 -1 -1 0 -4]]

POTE generator: ~9/8 = 205.3121

TOP generators: ~3/2 = 700.2305, ~9/8 = 204.8077

Optimal GPV sequence: Template:Val list


Reversed meantone (q7e&q11, 3/2.2.3.5.7.11 subgroup)

Subgroup: 3/2.2.3.5.7.11

Comma list: 64/63, 81/80, 99/98

Gencom: [3/2 9/8; 64/63 81/80 99/98]

Sval mapping: [⟨1 2 3 4 4 4], ⟨0 -1 -1 0 3 7]]

POTE generator: ~9/8 = 195.0236

TOP generators: ~3/2 = 698.2079, ~9/8 = 193.9826

Optimal GPV sequence: Template:Val list

Reversed meantone (q16&q25, 5/2.2.3.5 subgroup)

Subgroup: 5/2.2.3.5

Comma list: 81/80

Gencom: [5/2 3/2; 81/80]

Sval mapping: [⟨1 -1 -1 0], ⟨0 4 5 4]]

POTE generator: ~3/2 = 696.9241

TOP generators: ~5/2 = 1586.4184, ~3/2 = 696.9702

Optimal GPV sequence: Template:Val list

Reversed meantone (q16&q25, 5/2.2.3.5.7 subgroup)

Subgroup: 5/2.2.3.5.7

Comma list: 64/63, 81/80

Gencom: [5/2 3/2; 64/63 81/80]

Sval mapping: [⟨1 -1 -1 0 3], ⟨0 4 5 4 -2]]

POTE generator: ~3/2 = 696.7420

TOP generators: ~5/2 = 1587.1426, ~3/2 = 697.1060

Optimal GPV sequence: Template:Val list

Reversed meantone (q16&q41, 5/2.2.3.5.7.11 subgroup)

Subgroup: 5/2.2.3.5.7.11

Comma list: 81/80, 126:125, 176:175

Gencom: [5/2 3/2; 81/80 126:125 176:175]

Sval mapping: [⟨1 -1 -1 0 3 7], ⟨0 4 5 4 -2 -10]]

POTE generator: ~3/2 = 696.3353

TOP generators: ~5/2 = 1587.1426, ~3/2 = 697.2531

Optimal GPV sequence: Template:Val list

Reversed meantone (q9&q14, 5/3.2.3.5 subgroup)

Subgroup: 5/3.2.3.5

Comma list: 81/80

Gencom: [5/3 10/9; 81/80]

Val mapping: [⟨1 2 3 4], ⟨0 -3 -4 -4]]

POTE generator: ~10/9 = 189.0028

TOP generators: ~5/3 = 884.9987, ~10/9 = 189.1396

Optimal GPV sequence: Template:Val list

Reversed meantone (q9&q14d, 5/3.2.3.5.7 subgroup)

Subgroup: 5/3.2.3.5.7

Comma list: 126/125, 81/80

Gencom: [5/3 10/9; 126/125 81/80]

Sval mapping: [⟨1 2 3 4 6], ⟨0 -3 -4 -4 -10]]

POTE generator: ~10/9 = 192.2011

TOP generators: ~5/3 = 887.4720, ~10/9 = 192.8777

Optimal GPV sequence: Template:Val list

Reversed meantone (q9&q14, 5/3.2.3.5.7.11 subgroup)

Subgroup: 5/3.2.3.5.7.11

Comma list: 81/80, 126/125, 100/99

Gencom: [5/3 10/9; 81/80 126/125 100/99]

Sval mapping: [⟨1 2 3 4 4 6], ⟨0 -3 -4 -4 -1 -6]]

POTE generator: ~10/9 = 190.2709

TOP generators: ~5/3 = 886.3893, ~10/9 = 190.7078

Optimal GPV sequence: Template:Val list

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