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17L 2s: Difference between revisions

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m Undo revision 109959 by Fredg999 (talk) Merge not needed, my mistake
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→Scale tree: Add note about warts on Liesel, and add Liese with similar note; also add another column to scale tree due to large size of MOS (and consistency with the scale tree for 15L 2s); remove Todo:complete table
 
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{{Infobox MOS}}
{{MOS intro}} From a regular temperament theory perspective, this scale is notable for corresponding to the mega chromatic scale of the [[Alphatricot family]] temperaments. Three bright generators can be interpreted to stack to [[3/1]], but unfortunately, the generator of 17L 2s itself does not have a very convenient rational representation, since the simple ratio [[23/16]] is off-scale flat, as is (just barely) the compound ratio [[36/25]], while the prime-over-compound ratio [[13/9]] is off-scale sharp. Using very high prime harmonics/subharmonics, we can make the interpretations of [[~]][[62/43]] (bright generator) or ~[[43/31]] (dark generator); the aforementioned Alphatricot family uses the highly compound ~[[59049/40960]] as a generator; and probably the best rational that falls within the scale is ~[[75/52]], three of which differ from 3/1 by the 0.2-cent comma of [[140625/140608]], the catasma.


{| class="wikitable"
A pitfall of the use of compound harmonics and subharmonics in a generator is that they multiply the effect of shifts in mapping of their respective primes with scale hardness — for instance, ~59049/40960 only maps correctly within a narrow step ratio range close to 10:3, while ~36/25 fails to map correctly even for several EDOs close to the soft end of the scale's tuning spectrum (as does the simpler but flatter ~23/16); the even simpler ~13/9 (off-scale sharp) is likewise affected. Using such generators outside of a narrow subset of the EDOs supporting the scale depends upon direct approximation of a compound harmonic and/or subharmonic such as 9 or 25. This is awkward when one also needs to use a component harmonic as specified in the patent vals of the EDOs, thus requiring the use of nonstandard conditional subgroup temperaments such as 2.3♯.3♭.5 and 2.3.5♯.5♭ (or 2.3.9.5 and 2.3.5.25), with provision of a rule specifying when to use the direct approximation as opposed to the patent val mapping.
! colspan="9" | generator
! | L
! | s
! | L/s
! | gen (cents)
! | comment
|-
| | 8\17
| |
| |
| |
| |
| |
| |
|
|
| | 1
| | 0
| |
| | 564.706
| |
|-
| |
| |
| |
| |
| |
| |
| |
|
|73\155
| | 9
| | 1
| | 9.000
| | 565.161
| |
|-
| |
| |
| |
| |
| |
| |
| |
|65\138
|
| | 8
| | 1
| | 8.000
| | 565.217
| |
|-
| |
| |
| |
| |
| |
| |
| |57\121
|
|
| | 7
| | 1
| | 7.000
| | 565.289
| |
|-
| |
| |
| |
| |
| |
| | 49\104
| |
|
|
| | 6
| | 1
| | 6.000
| | 565.385
| |
|-
| |
| |
| |
| |
| |
| |
| |90\191
|
|
| | 11
| | 2
| | 5.500
| | 565.445
| |
|-
| |
| |
| |
| |
| | 41\87
| |
| |
|
|
| | 5
| | 1
| | 5.000
| | 565.517
| |
|-
| |
| |
| |
| |
| |
| | 74\157
| |
|
|
| | 9
| | 2
| | 4.500
| | 565.605
| |
|-
| |
| |
| |
| | 33\70
| |
| |
| |
|
|
| | 4
| | 1
| | 4.000
| | 565.714
| |
|-
| |
| |
| |
| |
| |
| | 91\193
| |
|
|
| | 11
| | 3
| | 3.667
| | 565.803
| |
|-
| |
| |
| |
| |
| | 58\123
| |
| |
|
|
| | 7
| | 2
| | 3.500
| | 565.854
| |
|-
| |
| |
| |
| |
| |
| | 83\176
| |
|
|
| | 10
| | 3
| | 3.333
| | 565.909
| |
|-
| |
| |
| | 25\53
| |
| |
| |
| |
|
|
| | 3
| | 1
| | 3.000
| | 566.038
| |
|-
| |
| |
| |
| |
| |
| | 92\195
| |
|
|
| | 11
| | 4
| | 2.750
| | 566.154
| |
|-
| |
| |
| |
| |
| | 67\142
| |
| |
|
|
| | 8
| | 3
| | 2.667
| | 566.197
| |
|-
| |
| |
| |
| | 42\89
| |
| |
| |
|
|
| | 5
| | 2
| | 2.500
| | 566.292
| |
|-
| |
| |
| |
| |
| | 59\125
| |
| |
|
|
| | 7
| | 3
| | 2.333
| | 566.400
| |
|-
| |
| |
| |
| |
| |
| | 76\161
| |
|
|
| | 9
| | 4
| | 2.250
| | 566.460
| |
|-
| |
| |
| |
| |
| |
| | 93\197
| |
|
|
| | 11
| | 5
| | 2.200
| | 566.497
| |
|-
| |
| | 17\36
| |
| |
| |
| |
| |
|
|
| | 2
| | 1
| | 2.000
| | 566.667
| |
|-
| |
| |
| |
| |
| |
| | 94\199
| |
|
|
| | 11
| | 6
| | 1.833
| | 566.834
| |
|-
| |
| |
| |
| |
| |
| | 77\163
| |
|
|
| | 9
| | 5
| | 1.800
| | 566.871
| |
|-
| |
| |
| |
| |
| | 60\127
| |
| |
|
|
| | 7
| | 4
| | 1.750
| | 566.929
| |
|-
| |
| |
| |
| | 43\91
| |
| |
| |
|
|
| | 5
| | 3
| | 1.667
| | 567.033
| |
|-
| |
| |
| |
| |
| | 69\146
| |
| |
|
|
| | 8
| | 5
| | 1.600
| | 567.123
| |
|-
| |
| |
| | 26\55
| |
| |
| |
| |
|
|
| | 3
| | 2
| | 1.500
| | 567.273
| |
|-
| |
| |
| |
| |
| |
| | 87\184
| |
|
|
| | 10
| | 7
| | 1.429
| | 567.391
| |
|-
| |
| |
| |
| |
| | 61\129
| |
| |
|
|
| | 7
| | 5
| | 1.400
| | 567.442
| |
|-
| |
| |
| |
| | 35\74
| |
| |
| |
|
|
| | 4
| | 3
| | 1.333
| | 567.568
| |
|-
| |
| |
| |
| |
| |
| | 79\167
| |
|
|
| | 9
| | 7
| | 1.286
| | 567.665
| |
|-
| |
| |
| |
| |
| | 44\93
| |
| |
|
|
| | 5
| | 4
| | 1.250
| | 567.742
| |
|-
| |
| |
| |
| |
| |
| | 53\112
| |
|
|
| | 6
| | 5
| | 1.200
| | 567.857
| |
|-
| |
| |
| |
| |
| |
| |
| |62\131
|
|
| | 7
| | 6
| | 1.167
| | 567.939
| |
|-
| |
| |
| |
| |
| |
| |
| |
|71\150
|
| | 8
| | 7
| | 1.143
| | 568.000
| |
|-
| |
| |
| |
| |
| |
| |
| |
|
|80\169
| | 9
| | 8
| | 1.125
| | 568.047
| |
|-
| | 9\19
| |
| |
| |
| |
| |
| |
|
|
| | 1
| | 1
| | 1.000
| | 568.421
| |
|}


[[Category:Abstract MOS patterns]]
== Scale properties ==
[[category:todo:intro]]
{{TAMNAMS use}}
[[category:todo:expand]]
 
=== Intervals ===
{{MOS intervals}}
 
=== Generator chain ===
{{MOS genchain}}
 
=== Modes ===
{{MOS mode degrees}}
 
== Scale tree ==
{{MOS tuning spectrum
| Depth = 6
| 5/4 = [[Pycnic]]
| 4/3 = [[Liese]] (as 74d)
| 5/3 = [[Liesel]] (as 91ceef)
| 13/4 = [[Alphatricot family#Alphatrident|Alphatricot/Alphatrident]]
}}
 
{{Stub}}
Retrieved from "https://en.xen.wiki/w/17L_2s"