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:

{Table revised and moved down}

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)

Mostly added this intro to the page, though I made the change of including 75/52 as a good rational interpretation. Lériendil (talk) 05:20, 3 May 2025 (UTC)

I was thinking about ~75/52 a while back, but didn't pursue it because of the triple compounding in the numerator (including 5♯.5♭), but I'll look into it some more to see how well it performs. Might as well also do this for ~49/34 (now only double compounding in the numerator, as ...7♯.7♭... or ...7.49...).

And what do you think about putting the (so far just one) temperament in the MOS tuning spectrum table?

Added: Lucius Chiaraviglio (talk) 07:59, 3 May 2025 (UTC)

Last modified: Lucius Chiaraviglio (talk) 08:55, 3 May 2025 (UTC)

Proposed revised table

{Moved here from original post}

Copying in the temperaments listed on the soft half of the 2L 15s table, and adding another column to this table to match that one:

Scale tree and tuning spectrum of 17L 2s

Generator(edo)							Cents		Step ratio		Comments
							Bright	Dark	L:s	Hardness
10\19							631.579	568.421	1:1	1.000	Equalized 17L 2s
						69\131	632.061	567.939	7:6	1.167
					59\112		632.143	567.857	6:5	1.200
						108\205	632.195	567.805	11:9	1.222
				49\93			632.258	567.742	5:4	1.250	Pycnic
						137\260	632.308	567.692	14:11	1.273
					88\167		632.335	567.665	9:7	1.286
						127\241	632.365	567.635	13:10	1.300
			39\74				632.432	567.568	4:3	1.333	Supersoft 17L 2s Liese (as 74d)
						146\277	632.491	567.509	15:11	1.364
					107\203		632.512	567.488	11:8	1.375
						175\332	632.530	567.470	18:13	1.385
				68\129			632.558	567.442	7:5	1.400
						165\313	632.588	567.412	17:12	1.417
					97\184		632.609	567.391	10:7	1.429
						126\239	632.636	567.364	13:9	1.444
		29\55					632.727	567.273	3:2	1.500	Soft 17L 2s
						135\256	632.812	567.188	14:9	1.556
					106\201		632.836	567.164	11:7	1.571
						183\347	632.853	567.147	19:12	1.583
				77\146			632.877	567.123	8:5	1.600
						202\383	632.898	567.102	21:13	1.615
					125\237		632.911	567.089	13:8	1.625
						173\328	632.927	567.073	18:11	1.636
			48\91				632.967	567.033	5:3	1.667	Semisoft 17L 2s Liesel (as 91ceef)
						163\309	633.010	566.990	17:10	1.700
					115\218		633.028	566.972	12:7	1.714
						182\345	633.043	566.957	19:11	1.727
				67\127			633.071	566.929	7:4	1.750
						153\290	633.103	566.897	16:9	1.778
					86\163		633.129	566.871	9:5	1.800
						105\199	633.166	566.834	11:6	1.833
	19\36						633.333	566.667	2:1	2.000	Basic 17L 2s Scales with tunings softer than this are proper
						104\197	633.503	566.497	11:5	2.200
					85\161		633.540	566.460	9:4	2.250
						151\286	633.566	566.434	16:7	2.286
				66\125			633.600	566.400	7:3	2.333
						179\339	633.628	566.372	19:8	2.375
					113\214		633.645	566.355	12:5	2.400
						160\303	633.663	566.337	17:7	2.429
			47\89				633.708	566.292	5:2	2.500	Semihard 17L 2s
						169\320	633.750	566.250	18:7	2.571
					122\231		633.766	566.234	13:5	2.600
						197\373	633.780	566.220	21:8	2.625
				75\142			633.803	566.197	8:3	2.667
						178\337	633.828	566.172	19:7	2.714
					103\195		633.846	566.154	11:4	2.750
						131\248	633.871	566.129	14:5	2.800
		28\53					633.962	566.038	3:1	3.000	Hard 17L 2s
						121\229	634.061	565.939	13:4	3.250	Alphatricot/Alphatrident
					93\176		634.091	565.909	10:3	3.333
						158\299	634.114	565.886	17:5	3.400
				65\123			634.146	565.854	7:2	3.500
						167\316	634.177	565.823	18:5	3.600
					102\193		634.197	565.803	11:3	3.667
						139\263	634.221	565.779	15:4	3.750
			37\70				634.286	565.714	4:1	4.000	Superhard 17L 2s
						120\227	634.361	565.639	13:3	4.333
					83\157		634.395	565.605	9:2	4.500
						129\244	634.426	565.574	14:3	4.667
				46\87			634.483	565.517	5:1	5.000
						101\191	634.555	565.445	11:2	5.500
					55\104		634.615	565.385	6:1	6.000
						64\121	634.711	565.289	7:1	7.000
9\17							635.294	564.706	1:0	→ ∞	Collapsed 17L 2s

Note that adding a column to the table is really needed to represent where Alphatricot hits its stride — some of the Alphatricot variants even hit their stride higher, but adding 2 more columns (initially tried this) starts bringing up EDOs for which no page exists yet.

While adding links to the temperaments in the 2L 15s MOS tuning spectrum table, I noticed that Liesel seems misplaced — even though the largest EDO listed in its optimal ET sequence is 91edo, it is 91edo with a bunch of warts, including the 'f' wart when the 13th harmonic has under 30% relative error, and the largest patent val listed is 36edo. For now, I have reproduced its analogous position in the 17L 2s MOS tuning spectrum table, but this seems like it should be changed to appear at 2/1 (36edo).

As a temporary solution I added a note to the scale tree about all those warts, and added Liese temperament with a similar note; going to make this live shortly.

Added: Lucius Chiaraviglio (talk) 09:24, 4 May 2025 (UTC)
Last modified: Lucius Chiaraviglio (talk) 07:50, 12 May 2025 (UTC)