Talk:Low harmonic entropy linear temperaments/WikispacesArchive

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HE values?

It would be nice if you actually gave some sort of quantification for each temperament to compare the various temperaments. Did you leave them out because you didn't want people to see that meantone totally slaughters every other temperament in terms of minimizing HE?

- igliashon August 17, 2011, 06:02:04 PM UTC-0700

On the contrary, I couldn't be more comfortable with the result that meantone is the clear winner for a broad range of HE fineness. I would have been disturbed if it had come out any other way.

The meantone pentatonic and diatonic scales are totally beautiful, and in my opinion this research gives a great theoretical explanation why they're the most popular scales in the world - they "totally slaughter" at minimizing HE.

But it really does depend on HE fineness as well as scale size. If your HE is fine enough and you demand a scale with more than 7 notes, then the pajara decatonic scale is the winner, not meantone chromatic. This is also a satisfying and unsurprising result to me.

The reason I don't give actual HE values is because there isn't just one HE value per temperament. They depend on averaging method, scale size, and HE fineness. So even though I decided on one averaging method (a specific trimmed mean), there are still 12 different numbers assigned to meantone: pentatonic, diatonic, and chromatic scales all have values for coarse, medium, fine, and extra fine.

But I do think it's important to mention in the article that meantone is the clear overall winner, so thanks for pointing that out.

- keenanpepper August 19, 2011, 07:09:39 PM UTC-0700

I agree with Igs that the results presentation is a little unclear. A temperament is listed if it wins for some number of notes and s level, right? And the parens tell us what s level a temperament wins for?

Also, this is octave-equivalent 2HE, right? That seems important to mention.

Finally, I recall there being some beautiful graphs when you posted this to tuning-math. Any reason you don't include them here?

- clumma August 19, 2011, 08:25:13 PM UTC-0700

I think the graphs should be included too, especially if you can't give values to each scale.

Also, I really think you're over-extending the theory if you're treating it as explaining the popularity of the diatonic scale. That's like suggesting that the reason Christianity is so popular is because the quality of writing in the Bible is so good; it is, in other words, ignoring the socio-political forces that have shaped the current state of the world. For instance, had Islam dominated Europe rather than Christianity, maqam rast might be more popular than the diatonic scale.

Also, I'm curious: if you think that success at minimizing HE is the major determinant of a scale's popularity, how do you expect anyone to have any success with microtonality? Won't everything just be "worse" than meantone? It seems like this way of thinking leads inexorably to some amount of pessimism, yet I know you are interested in non-meantone tunings. How do you reconcile that?

- igliashon August 19, 2011, 11:09:37 PM UTC-0700


I added miracle because it probably ought to have appeared; if it really didn't, look a little harder in terms of accuracy, and it should. If it really doesn't, take it out, but please explain what the hell is going on when you do.

- genewardsmith August 16, 2011, 05:28:06 PM UTC-0700

I'm sure that if I cranked up the HE fineness a little more that both miracle and also valentine would show up.

- keenanpepper August 16, 2011, 06:05:30 PM UTC-0700


Since low harmonic entropy doesn't seem to be giving results much different than previous badness figures (nor would we expect it to) I've made some changes on that basis. If someone thinks they are wrong, I hope they will provide actual numbers when reverting.

- genewardsmith August 16, 2011, 05:21:47 PM UTC-0700

That's totally cool with me. None of this is all that well-defined mathematically (depends what HE function you use, what kind of average...); I mostly want it to use as a list of really worthwile temperaments I personally want to explore more.

- keenanpepper August 16, 2011, 06:04:21 PM UTC-0700


Why was slendric removed from the list? Any particular reason?

- Sarzadoce August 16, 2011, 05:09:43 PM UTC-0700

"Slendric" at present is being used as the name of the rank 3 temperament tempering out 1029/1024; a full 7-limit temperament with a generator of 8/7 is propbably mothra or rodan.

- genewardsmith August 16, 2011, 05:19:33 PM UTC-0700

I don't think it makes sense to label the entry as either "mothra" or "rodan" because no intervals outside the 2.3.7 subgroup have a low enough complexity to contribute significantly to the harmonic entropy. So I want to call it something that doesn't imply any mapping for 5 at all. (This is one of my all-time favorite temperaments, so I'm interested what to call it.)

I was just calling it "slendric" because that's the name on this page:

Is there some other name I should use? Maybe "gamelismic" or something?

- keenanpepper August 16, 2011, 05:57:05 PM UTC-0700

I suggest we call the 2.3.7 subgroup temperament "slendric" and the rank three temperament "gamilismic", and quit using the two as synonyms. Sound good?

- genewardsmith August 16, 2011, 06:02:15 PM UTC-0700

One reason I thought of mothra is that you weren't calling it "fine", but fine or extra-fine it would seemingly have to be if it's just the 2.3.7 subgroup.

- genewardsmith August 16, 2011, 06:05:09 PM UTC-0700

Your proposed names sound great to me. I'm about to start a whole article on slendric.

Not sure what you mean about it having to be fine if it's a subgroup - slendric also appears as a strong minimum for quite coarse HE functions - mostly because it has a bunch of 4/3s and few of the other intervals are exceptionally rough (basically only the two-generator interval, ~21/16).

- keenanpepper August 16, 2011, 06:12:25 PM UTC-0700

"Fine" seems to corrspond to more accuracy and "coarse" with less, and slendric is highly accurate.

- genewardsmith August 16, 2011, 06:23:13 PM UTC-0700

Yeah, but it's also really simple. A 4:6:7 chord has complexity 3, and the 11-note MOS contains the entire tonality diamond. That's why it shows up even for coarse HE.

- keenanpepper August 16, 2011, 06:35:33 PM UTC-0700

Added a slendric entry on Chromatic pairs.

- genewardsmith August 16, 2011, 06:38:40 PM UTC-0700