Talk:17L 2s
Alphatricot/Alphatrident
Alphatricot/Alphatrident is in the vicinity of 176edo, but I can't figure out how to get it into the MOS tuning spectrum table (I know the template has a way to do it, because I've seen it elsewhere, but it's undocumented, and when I looked at the page source where I saw it elsewhere, it seemed unintuitive how it was used there).
Added: Lucius Chiaraviglio (talk) 06:04, 28 April 2025 (UTC)
I am going to try to see if this will work. Using this as a staging area to make sure I get the template parameters right rather than experiment on the real page and botch the whole thing:
| Generator(edo) | Cents | Step ratio | Comments | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Bright | Dark | L:s | Hardness | |||||||
| 10\19 | 631.579 | 568.421 | 1:1 | 1.000 | Equalized 17L 2s | |||||
| 59\112 | 632.143 | 567.857 | 6:5 | 1.200 | ||||||
| 49\93 | 632.258 | 567.742 | 5:4 | 1.250 | ||||||
| 88\167 | 632.335 | 567.665 | 9:7 | 1.286 | ||||||
| 39\74 | 632.432 | 567.568 | 4:3 | 1.333 | Supersoft 17L 2s | |||||
| 107\203 | 632.512 | 567.488 | 11:8 | 1.375 | ||||||
| 68\129 | 632.558 | 567.442 | 7:5 | 1.400 | ||||||
| 97\184 | 632.609 | 567.391 | 10:7 | 1.429 | ||||||
| 29\55 | 632.727 | 567.273 | 3:2 | 1.500 | Soft 17L 2s | |||||
| 106\201 | 632.836 | 567.164 | 11:7 | 1.571 | ||||||
| 77\146 | 632.877 | 567.123 | 8:5 | 1.600 | ||||||
| 125\237 | 632.911 | 567.089 | 13:8 | 1.625 | ||||||
| 48\91 | 632.967 | 567.033 | 5:3 | 1.667 | Semisoft 17L 2s | |||||
| 115\218 | 633.028 | 566.972 | 12:7 | 1.714 | ||||||
| 67\127 | 633.071 | 566.929 | 7:4 | 1.750 | ||||||
| 86\163 | 633.129 | 566.871 | 9:5 | 1.800 | ||||||
| 19\36 | 633.333 | 566.667 | 2:1 | 2.000 | Basic 17L 2s Scales with tunings softer than this are proper | |||||
| 85\161 | 633.540 | 566.460 | 9:4 | 2.250 | ||||||
| 66\125 | 633.600 | 566.400 | 7:3 | 2.333 | ||||||
| 113\214 | 633.645 | 566.355 | 12:5 | 2.400 | ||||||
| 47\89 | 633.708 | 566.292 | 5:2 | 2.500 | Semihard 17L 2s | |||||
| 122\231 | 633.766 | 566.234 | 13:5 | 2.600 | ||||||
| 75\142 | 633.803 | 566.197 | 8:3 | 2.667 | ||||||
| 103\195 | 633.846 | 566.154 | 11:4 | 2.750 | ||||||
| 28\53 | 633.962 | 566.038 | 3:1 | 3.000 | Hard 17L 2s | |||||
| 93\176 | 634.091 | 565.909 | 10:3 | 3.333 | Alphatricot/Alphatrident is around here | |||||
| 65\123 | 634.146 | 565.854 | 7:2 | 3.500 | ||||||
| 102\193 | 634.197 | 565.803 | 11:3 | 3.667 | ||||||
| 37\70 | 634.286 | 565.714 | 4:1 | 4.000 | Superhard 17L 2s | |||||
| 83\157 | 634.395 | 565.605 | 9:2 | 4.500 | ||||||
| 46\87 | 634.483 | 565.517 | 5:1 | 5.000 | ||||||
| 55\104 | 634.615 | 565.385 | 6:1 | 6.000 | ||||||
| 9\17 | 635.294 | 564.706 | 1:0 | → ∞ | Collapsed 17L 2s | |||||
Well, it seems to work despite me not being able to figure out how to tell MOS tuning spectrum what the MOS is (extracts from the page it is embedded in? — but this is a Talk page), but I had better leave this for a bit for review before committing it to the real page. (This is for review of the Comment entries I put in the table from the music theory perspective, as well as making sure that this displays properly on other people's computers.)
Proposed text to add to introduction section
From a regular temperament theory perspective, this scale is notable for corresponding to the mega chromatic scale of the Alphatricot family temperaments. Unfortunately, its generator does not have a convenient rational representation — the simple ratios 23/16 and even 36/25 are off-scale flat (although just barely in the case of 36/25, which is near just in the equalized endpoint 19edo), while the simple ratio 13/9 is off-scale sharp. The Alphatricot family uses ~59049/40960 as a generator.
Added: Lucius Chiaraviglio (talk) 06:19, 28 April 2025 (UTC)
Last modified: Lucius Chiaraviglio (talk) 19:24, 29 April 2025 (UTC)
- There are several points that should be discussed.
- With this phrasing, it will not be possible to express the generator of m-chromatic as 3/2.
- From a regular temperament theory perspective, in order for the generator to be 36/25, it is necessary to prohibit 6/5 in this case. For example, it would be a 2.3.25-subgroup hanson. (Hanson normally means alpha-hexacot.)
- I don't think the size of just interval that represent the generator of the temperament need to fall within an exact range. 2.3.25-restricted hanson's generator is …~36/25(631.28c)~625/432(639.39c)~…, and 2.3.25.13-restricted cata's generator is …~36/25~13/9(636.62c)~…, both of which seem to fit this mos.
- --Dummy index (talk) 13:30, 1 May 2025 (UTC)
- I also noticed that problem with 36/25 (unless you make a nonstandard subgroup notation extension that lets you use both a prime and a multiple of that prime or both flat and sharp versions of that prime, depending upon a simple selection rule). Probably should add a note about that to what I proposed above. With respect to fitting into the range, if you DON'T do that (and depending upon generator constitution, often even if you do), you end up with an awful lot of EDOs where the generator doesn't map correctly — for instance, both 23/16 and 13/9 have spotty mapping in this tuning table (although at least covering enough EDOs to be useful for a decent subset of it), while the Alphatricot generator doesn't map correctly for anything other than a very narrow band close to 53edo. I've been working on this under Musical Mad Science under my user page (but it's nowhere near ready to put here or on any other official page), and found that 62/43 maps correctly to almost everything (and the very small number of exceptions are candidates for wart rescue). Lucius Chiaraviglio (talk) 15:25, 1 May 2025 (UTC)
Proposed revised introduction section
17L 2s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 17 large steps and 2 small steps, repeating every octave. 17L 2s is related to 2L 7s, expanding it by 10 tones. Generators that produce this scale range from 631.6 ¢ to 635.3 ¢, or from 564.7 ¢ to 568.4 ¢. From a regular temperament theory perspective, this scale is notable for corresponding to the mega chromatic scale of the Alphatricot family temperaments.
Unfortunately, the generator of 17L 2s does not have a convenient rational representation without using very high prime harmonics/subharmonics, such as ~62/43 (bright generator) or ~43/31 (dark generator). (For comparison considering bright generators, 5L 2s is easily generated using ~3/2, and 11L 2s is easily generated using ~11/8.) For 17L 2s, 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; the aforementioned Alphatricot family uses the highly compound ~59049/40960 as a generator.
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.
Added: Lucius Chiaraviglio (talk) 15:14, 2 May 2025 (UTC)
Last modified: Lucius Chiaraviglio (talk) 20:03, 2 May 2025 (UTC)