A proposal to simplify the notation of EDOs with bad fifths

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Re: A proposal to simplify the notation of EDOs with bad fifths

Post by volleo6144 »

Dave Keenan wrote: Wed Sep 30, 2020 10:54 am That shows up nicely, the cases where we will have to deal with overlaps. In the final product, I think we should always have the "b" EDO as the one that is partly obscured.
Yeah, it's just that the order I ended up having it draw the boxes was 5, 6, 7, 8, ..., 59, 60, 61, ..., 71, 72, and then all the b-edos in order... and drawing a box overlapping another box is obviously going to just get rid of that part of the other box. (See also how overlaps worked in my other tables.)
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Re: A proposal to simplify the notation of EDOs with bad fifths

Post by cmloegcmluin »

Well folks, I starred this thread months ago, and this morning set out to bring myself up to speed on it. I can't quite follow it, it turns out. I can usually eventually get the maths when dealing with microtonal stuff I desperately care about, but it doesn't come naturally to me.
Dave Keenan wrote: Mon Jul 13, 2020 10:27 am
volleo6144 wrote: Mon Jul 13, 2020 3:28 am ...
† This boundary is listed as the phidiant of 2 and 11, not of 9 and 2, in your table.
...
Good name, "phidiant". I have previously called it the "noble-mediant".
volleo6144 wrote: Mon Jul 13, 2020 1:09 pm Honestly, "noble-mediant" sounds better anyway. My choice of "phidiant" was ... something I came up with for a reason I can't tell anymore.
I like "phidiant"! It's especially nice if you pronounce φ as /fiː/ (same as "fee"), which I don't, but have nothing against it. I think I'd like to borrow that word for my writings on metallic MOS scales, if you don't mind. As it stands, the last paragraph of this section is an agonizing attempt to grapple with helpful terminology for this concept.

So I see y'all had some fun with hue balancing (as Dave and I did for that other most glorious of Sagittal diagrams, the JI precision levels diagram).

So what are the implications of this recent work? I see we have a new color, which won't have a member in the first 72 EDOs. When I look at @volleo6144's initial results of the function @Dave Keenan's describes here, it looks like the breaking points between the colors are somewhat irregular (because they hadn't been precisely defined yet), but one thing that's consistent is that each color (except the ones on the edges) have only one whole number within their bounds — except for one exception: cyan. But then I see that the new pink color was not introduced next to cyan, so that surprises me a bit. I looked at the numbers that define your new phidiant boundaries but I can't suss out the pattern to them. So that's the main area where I'm still pretty lost.

I understand from an email thread with Dave that meandered from a different subject to touch upon the Periodic table that we're now planning to include some second-best fifth notations in the chart, namely, ones which satisfy the threshold Dave described here.

Were there any other insights gleaned from this discussion that I haven't understood? Do these newly precise definitions of the color groups represent a step toward realizing this vision:
Dave Keenan wrote: Tue May 26, 2020 4:44 pm
cmloegcmluin wrote: Tue May 26, 2020 3:41 pm It's starting to come into focus. Would you say it'd be true that we could produce analogs to the Trojan capture zone diagram seen at the bottom of Figure 10 on page 19 of the Xenharmonikon (and also seen recently side-by-side with the JI precision level capture zones in SaBounds.gif on the consistent 37 thread, which was a treat!) for every color of the PT? So some Xn stacks (e.g. 22n and 27n) would share the same capture zones?
That's the hope. I tried to ensure that the choice of symbols for the EDOs in each colour would allow that. But since I had intimations of urgency in releasing the PT, I couldn't spend the time it would have taken to actually determine such capture zones. So I can't be sure. Very low priority.

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Re: A proposal to simplify the notation of EDOs with bad fifths

Post by volleo6144 »

cmloegcmluin wrote: Tue Nov 17, 2020 4:14 am I like "phidiant"! It's especially nice if you pronounce φ as /fiː/ (same as "fee"), which I don't, but have nothing against it. I think I'd like to borrow that word for my writings on metallic MOS scales, if you don't mind. As it stands, the last paragraph of this section is an agonizing attempt to grapple with helpful terminology for this concept.
Honestly, I totally didn't intend it to be actually used anywhere, but go ahead, I guess?

...Yes, I pronounce it as /fiː/, and I totally thought that /fiː/ was, like, by far the most common way to pronounce it (after all, it's how φι is pronounced in modern Greek, and there aren't any sounds that are absent in English like the Chinese1 <q> /ʨʰ/), but I guess not?
I can't suss out the pattern to [your new phidiant boundaries].
I'm not really sure there is one, apart from that they create Fibonacci-like sequences that alternate between colors: 66°113, 7°12, 73°125, 80°137, 153°262, 233°399. The particular sequences we chose actually end up moving 59 from gold to green, and keep all the other ones (below 72) in the same colors as before.
Nobody wrote:Mon Aug 15, 29227704433 3:03 am Gotta have a quote here, but I can't bother finding where the quote was from. Also I'm from the future.
Come to think of it, what is the pattern behind the colors in the JI chart?

1 Mandarin. There's more than one Chinese language, and the orthography is the same among them (and is also a part of Japanese and, to a lesser extent, Korean and Vietnamese), which is unfortunate, because it means that "Chinese" often gets conflated with Mandarin Chinese.
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Re: A proposal to simplify the notation of EDOs with bad fifths

Post by cmloegcmluin »

volleo6144 wrote: Tue Nov 17, 2020 11:45 am Honestly, I totally didn't intend it to be actually used anywhere, but go ahead, I guess?
If you really don't like it, I can also just not use it :) Or at least not credit you if you'd rather not associate with it.
...Yes, I pronounce it as /fiː/, and I totally thought that /fiː/ was, like, by far the most common way to pronounce it (after all, it's how φι is pronounced in modern Greek, and there aren't any sounds that are absent in English like the Chinese1 <q> /ʨʰ/), but I guess not?
Yeah, I dunno. And its wiktionary entry says the Ancient Greek (from which it eventually entered English) was pronounced like "fay"!
I can't suss out the pattern to [your new phidiant boundaries].
I'm not really sure there is one, apart from that they create Fibonacci-like sequences that alternate between colors: 66°113, 7°12, 73°125, 80°137, 153°262, 233°399. The particular sequences we chose actually end up moving 59 from gold to green, and keep all the other ones (below 72) in the same colors as before.
Okay, glad I'm not the only one then. Yeah, I'm comfortable enough with the concept and motive for the phi-based boundaries, actually. But I just don't know if there's a pattern to the integers they're noble mediants of. Maybe Dave can shed light on that.
Come to think of it, what is the pattern behind the colors in the JI chart?
I could copy and paste some stuff Dave wrote me, but perhaps he'd rather share it himself in case he wants to rework the thoughts to better suit this context.

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Re: A proposal to simplify the notation of EDOs with bad fifths

Post by Dave Keenan »

cmloegcmluin wrote: Tue Nov 17, 2020 4:14 am I like "phidiant"! It's especially nice if you pronounce φ as /fiː/ (same as "fee"), which I don't, but have nothing against it. I think I'd like to borrow that word for my writings on metallic MOS scales, if you don't mind. As it stands, the last paragraph of this section is an agonizing attempt to grapple with helpful terminology for this concept.
I think "phidiant" or "ϕediant" is a clever term, with mnemonic value in an exposition on the subject of noble mediants, but I'm wary of jargon, since it tends to make a field impenetrable to newcomers. So I recommend confining it to such an exposition and using "noble-mediant" elsewhere.

I am reminded that I still owe you a review of the rest of that HUGE SCARY metallic page. :o
So I see y'all had some fun with hue balancing (as Dave and I did for that other most glorious of Sagittal diagrams, the JI precision levels diagram).
I think of it more as lightness balancing, while hues are made maximally distinct, and chroma is maximised within the previous two constraints (except in the cases of grey and gold).
So what are the implications of this recent work? I see we have a new color, which won't have a member in the first 72 EDOs. When I look at @volleo6144's initial results of the function @Dave Keenan's describes here, it looks like the breaking points between the colors are somewhat irregular (because they hadn't been precisely defined yet), but one thing that's consistent is that each color (except the ones on the edges) have only one whole number within their bounds — except for one exception: cyan.
It took me a while to figure out that you meant "only one whole group-number". I had never noticed that pattern, probably because it has no musical or notational significance as far as I can tell. Unlike the chemistry Periodic Table, the "group number" here is mainly about geometry — where to plot something on a diagram so that very little space is wasted. Whereas the colours are about grouping EDOs that can have a common apotome-fraction notation stack. This depends on the relative sizes and apotome-slopes of the commas corresponding to various single-shaft sagittals.
But then I see that the new pink color was not introduced next to cyan, so that surprises me a bit.
The new pink colour was introduced because I found some EDOs in that region, namely 91 and 103 — see the noble colour boundaries — that I couldn't see any prospect of notating with either an orange (trojan) or a yellow (super-meantone?) notation.
I looked at the numbers that define your new phidiant boundaries but I can't suss out the pattern to them. So that's the main area where I'm still pretty lost.
The colour boundaries are still provisional, but defining them allowed volleo to create informative charts like this:


And by defining them as simple noble fractions of an octave we guaranteed that no EDO would ever be on a boundary and that small EDOs would be as far as possible from a boundary.

There is no obvious pattern to the boundaries because they are located near points where one must jump from one comma's curve to another, to maintain a valid approximation of some apotome-fraction, as the fifth size varies, on this chart of comma apotome-fractions

Consider the simplest apotome fraction, the 1/2-apotome. Follow the "#=2" blue line on the Periodic Table. Notice that the 1/2-apotome symbol changes from :(|\: to :/|\: to :/|): . Then find the curves for (|\. /|\ and /|) on the above linked chart (fifth-sizes are flipped horizontally relative to the PT), and notice how switching from one to another at certain fifth sizes allows one to stay between 0.25 and 0.75 apotomes in height.
I understand from an email thread with Dave that meandered from a different subject to touch upon the Periodic table that we're now planning to include some second-best fifth notations in the chart, namely, ones which satisfy the threshold Dave described here.
That is correct. Here's a mockup of such a table and here are the required second-best notations, which, if you have no objection, can be considered to be completed, and can be entered into your "sagittal as software". The two fractions of an octave whose noble-mediant corresponds to each boundary can also be entered.
Were there any other insights gleaned from this discussion that I haven't understood? Do these newly precise definitions of the color groups represent a step toward realizing [the vision of analogs of the Trojan capture zones for every color]?
The short answer is "Yes". The long answer is: It may not be necessary, or wise, to take it all the way to defining capture zones. It may be sufficient to define which simple fractions of the apotome (corresponding to EDO degrees) a given symbol represents. e.g. for some colour, a given symbol might cover 3/5, 2/3 and 3/4-apotome. But there might be a different symbol for 4/6-apotome even though we normally think of 4/6 = 2/3. There's an example in the blue region of the PT, where :/|\: is 1/2, 2/4, 2/5, 3/7 and 3/8-apotome, but not 3/6 or 4/8-apotome. Instead, :(|\: is 3/6 and 4/8 apotome.

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Re: A proposal to simplify the notation of EDOs with bad fifths

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volleo6144 wrote: Tue Nov 17, 2020 11:45 am I'm not really sure there is [a pattern in the phidiant boundaries], apart from that they create Fibonacci-like sequences that alternate between colors: 66°113, 7°12, 73°125, 80°137, 153°262, 233°399.
True. There's no pattern, but I hope some of the (fuzzy) reasons I've given above make sense.
Nobody wrote:Mon Aug 15, 29227704433 3:03 am Gotta have a quote here, but I can't bother finding where the quote was from. Also I'm from the future.
: D
Come to think of it, what is the pattern behind the colors in the JI chart?
It's just map colouring. Making sure no two "countries" that share a border have the same colour, while not having to use too many different colours. And trying to make neighbours maximally distinct. Unlike the PT, there is no significance at all to any set of all the symbols with the same background colour on the JI notation chart.

I believe Douglas is working on an update of that chart, where the colours will carry a little more meaning.

@cmloegcmluin, please post whatever you've got from me on the subject.

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Re: A proposal to simplify the notation of EDOs with bad fifths

Post by Dave Keenan »

cmloegcmluin wrote: Tue Nov 17, 2020 3:52 pm Okay, glad I'm not the only one then. Yeah, I'm comfortable enough with the concept and motive for the phi-based boundaries, actually. But I just don't know if there's a pattern to the integers they're noble mediants of. Maybe Dave can shed light on that.
The ratios that they are noble mediants of, are fractions of an octave, each corresponding to the size of the fifth — for one EDO of each colour, on either side of the boundary. But I assume you want to know how I chose those two EDOs.

Way back before the PT was coloured and presented to the world by Dmitri Mendeleev (deceased), I made a preliminary attempt, described somewhere in this topic, to derive consistent notations for various regions, descending to EDOs greater than 100, with 12 or more division per apotome. This allowed me to give the original tentative boundaries in cents, to the nearest 0.1 of a cent.

So in going to nobles, using a spreadsheet, I started with the smallest numbered EDO of each colour and calculated the noble mediant of their fifths (as octave fractions) then checked to see if it was close enough to the original cent value, and excluded and included the right EDOs according to PT. If not, I went to the next EDO on whichever side was necessary to move the boundary in the right direction. Repeat until happy.

Oh. I also checked that the difference of the crossed products was 1, to ensure it actually was noble.

59 changed its colour, because it could do so without changing its notation, and this gave a consistent boundary for all the other EDOs on either side of that boundary. It was a toss-up how to colour 59 when I made the table. I now have a reason to think I got it wrong.

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Re: A proposal to simplify the notation of EDOs with bad fifths

Post by cmloegcmluin »

Dave Keenan wrote: Tue Nov 17, 2020 4:15 pm I think "phidiant" or "ϕediant" is a clever term, with mnemonic value in an exposition on the subject of noble mediants, but I'm wary of jargon, since it tends to make a field impenetrable to newcomers. So I recommend confining it to such an exposition and using "noble-mediant" elsewhere.
I agree. What a dilemma though: trying to improve communication for those already indoctrinated without it coming at the expense of initiates. I think since material on the forum tends to be more random-access, we should definitely lean further away from jargon when we can.
I am reminded that I still owe you a review of the rest of that HUGE SCARY metallic page. :o
No rush. Just excited to see what I missed or messed up, to learn even more.


It took me a while to figure out that you meant "only one whole group-number". I had never noticed that pattern, probably because it has no musical or notational significance as far as I can tell.
Sorry about that! I certainly should have been clearer and used the terminology y'all had been using, but it was such a strong pattern I assumed it was by design!
Unlike the chemistry Periodic Table, the "group number" here is mainly about geometry — where to plot something on a diagram so that very little space is wasted. Whereas the colours are about grouping EDOs that can have a common apotome-fraction notation stack.
Whoa okay. Having not implemented my own Periodic table in code (yet...) I didn't realize that the group number was not the same as color.

I'm pretty sure if you sorted all the EDO's group numbers, there'd be no back-and-forth of colors at the boundaries, right? I'm sure there's a smarter math word for this concept, like mutual monotonicity or something... hopefully y'all get what I'm asking...

So have we just stopped speaking about group number now? That's just an implementation detail, little more than a value that would get multiplied by an x-scale to plot the chart? Now that we have the noble mediants to bound the colors.
And by defining them as simple noble fractions of an octave we guaranteed that no EDO would ever be on a boundary and that small EDOs would be as far as possible from a boundary.
I do at least get that idea pretty well, having studied continued fractions earlier this year.
Dave Keenan wrote: Tue Nov 17, 2020 5:00 pm The ratios that they are noble mediants of, are fractions of an octave, each corresponding to the size of the fifth — for one EDO of each colour, on either side of the boundary. But I assume you want to know how I chose those two EDOs.
Yes, that is exactly what I was trying to ask.
So in going to nobles, using a spreadsheet, I started with the smallest numbered EDO of each colour and calculated the noble mediant of their fifths (as octave fractions) then checked to see if it was close enough to the original cent value, and excluded and included the right EDOs according to PT. If not, I went to the next EDO on whichever side was necessary to move the boundary in the right direction. Repeat until happy.
Got it. Thank you for explaining.
This depends on the relative sizes and apotome-slopes of the commas corresponding to various single-shaft sagittals.
But then I see that the new pink color was not introduced next to cyan, so that surprises me a bit.
The new pink colour was introduced because I found some EDOs in that region, namely 91 and 103 — see the noble colour boundaries — that I couldn't see any prospect of notating with either an orange (trojan) or a yellow (super-meantone?) notation.
There is no obvious pattern to the boundaries because they are located near points where one must jump from one comma's curve to another, to maintain a valid approximation of some apotome-fraction, as the fifth size varies, on this chart of comma apotome-fractions
Okay, that makes sense. I had assumed that these colors were recognizing something very close to textures such as are found in the harmonic entropy charts, projective tuning space, Stern-Brocot tree, etc. those sorts of cold, hard mathematical fractally underpinnings. So in the scheme of things, there are a ton of variables and sagittations building up into those decisions. They're a ways away from mathematical fundament. They're "fuzzy" as you say.
Consider the simplest apotome fraction, the 1/2-apotome. Follow the "#=2" blue line on the Periodic Table. Notice that the 1/2-apotome symbol changes from :(|\: to :/|\: to :/|): . Then find the curves for (|\. /|\ and /|) on the above linked chart (fifth-sizes are flipped horizontally relative to the PT), and notice how switching from one to another at certain fifth sizes allows one to stay between 0.25 and 0.75 apotomes in height.
First of all, that's really helpful explanation. Thanks for that. I really am a visual learner...

Alright, but it can't be a simple, flat value (expressible as cents) where it cuts over from :(|\: to :/|\:, right? Because if I look at the #=4 line or the #=6 line, and instead look at their 2nd or 3rd symbols respectively, which are also symbols for the half apotome, the cut-offs are in different places. For the #=2 line, all we can tell is that we switch from :(|\: to :/|\: somewhere between +18.0 and +3.9 (between 10-EDO and 17-EDO). For the #=4 line, we can narrow it down to somewhere between +9.2 and +3.9 (between 27-EDO and 34-EDO). For the #=6 line, we can narrow it down to somewhere between ... wait a second ... +3.9 and +1.5 (between 51-EDO and 58-EDO). So there has to be more going on.
It may not be necessary, or wise, to take it all the way to defining capture zones. It may be sufficient to define which simple fractions of the apotome (corresponding to EDO degrees) a given symbol represents. e.g. for some colour, a given symbol might cover 3/5, 2/3 and 3/4-apotome. But there might be a different symbol for 4/6-apotome even though we normally think of 4/6 = 2/3. There's an example in the blue region of the PT, where :/|\: is 1/2, 2/4, 2/5, 3/7 and 3/8-apotome, but not 3/6 or 4/8-apotome. Instead, :(|\: is 3/6 and 4/8 apotome.
And maybe this 3/6 ≠ 4/8 business is what starts to explain that?
Here's a mockup of such a table and here are the required second-best notations, which, if you have no objection, can be considered to be completed, and can be entered into your "sagittal as software". The two fractions of an octave whose noble-mediant corresponds to each boundary can also be entered.
The code as of yet has nary a mention of EDOs. But it will soon. I don't know what you mean in this last sentence, though.


Dave Keenan wrote: Tue Nov 17, 2020 4:39 pm I believe Douglas is working on an update of that chart, where the colours will carry a little more meaning.
A little more meaning is right. The symbols which are apotome complements of each other (the ones above the half apotome mirror) will have the same color. Other than that, it'll be a similar meaningless but visually helpful cylcing of maximally distinct colors.
@cmloegcmluin, please post whatever you've got from me on the subject.
Alright here's what I have. Enjoy, everyone! viewtopic.php?p=2840#p2840

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Re: A proposal to simplify the notation of EDOs with bad fifths

Post by Dave Keenan »

cmloegcmluin wrote: Wed Nov 18, 2020 7:30 am
Dave Keenan wrote: Tue Nov 17, 2020 4:15 pm I think "phidiant" or "ϕediant" is a clever term, with mnemonic value in an exposition on the subject of noble mediants, but I'm wary of jargon, since it tends to make a field impenetrable to newcomers. So I recommend confining it to such an exposition and using "noble-mediant" elsewhere.
I agree. What a dilemma though: trying to improve communication for those already indoctrinated without it coming at the expense of initiates. I think since material on the forum tends to be more random-access, we should definitely lean further away from jargon when we can.
Just to be clear. I think that "phidiant" or similar should only appear in that part of your metallic article where you're first explaining the concept of a noble-mediant, but you should use "noble-mediant" everywhere else in the article.

I think it's fine for us to save typing by inventing shorter names for things we are discussing in depth at a particular time, but these should not be used in articles intended for a wider audience, or as a permanent record. Even in our own discussions such short terms should be considered ephemeral, like ice sculptures. If we come back to the topic months or years later we shouldn't assume such terms are remembered.
Whoa okay. Having not implemented my own Periodic table in code (yet...) I didn't realize that the group number was not the same as color.
Yeah. No relation whatsoever. Sorry. :)
I'm pretty sure if you sorted all the EDO's group numbers, there'd be no back-and-forth of colors at the boundaries, right? I'm sure there's a smarter math word for this concept, like mutual monotonicity or something... hopefully y'all get what I'm asking...
Oh sure! Because colour (or rather one numbering thereof) is monotonic with fifth size, and group number is also monotonic with fifth-size, it follows that colour is also monotonic with group number.
So have we just stopped speaking about group number now? That's just an implementation detail, little more than a value that would get multiplied by an x-scale to plot the chart?
You got it.
Now that we have the noble mediants to bound the colors.
I don't see group number as ever having been used to bound the colours, although I can see why you might have thought so.
Okay, that makes sense. I had assumed that these colors were recognizing something very close to textures such as are found in the harmonic entropy charts, projective tuning space, Stern-Brocot tree, etc. those sorts of cold, hard mathematical fractally underpinnings. So in the scheme of things, there are a ton of variables and sagittations building up into those decisions. They're a ways away from mathematical fundament. They're "fuzzy" as you say.
You got it.
Alright, but it can't be a simple, flat value (expressible as cents) where it cuts over from :(|\: to :/|\:, right? Because if I look at the #=4 line or the #=6 line, and instead look at their 2nd or 3rd symbols respectively, which are also symbols for the half apotome, the cut-offs are in different places. For the #=2 line, all we can tell is that we switch from :(|\: to :/|\: somewhere between +18.0 and +3.9 (between 10-EDO and 17-EDO). For the #=4 line, we can narrow it down to somewhere between +9.2 and +3.9 (between 27-EDO and 34-EDO). For the #=6 line, we can narrow it down to somewhere between ... wait a second ... +3.9 and +1.5 (between 51-EDO and 58-EDO). So there has to be more going on.

And maybe this 3/6 ≠ 4/8 business is what starts to explain that?
Yes indeed. The rest of the explanation is why 1/2-apotome doesn't necessarily equal 2/4-apotome in this context. It's very simple. We try to go for the simplest comma that will do the job. But a comma that rounds to 1/2-apotome only needs to be between 0.5/2 and 1.5/2-apotome (0.25 to 0.75) while one that rounds to 2/4 must be between 1.5/4 and 2.5/4-apotome (0.375 to 0.625) and so is more tightly constrained. A simple comma may be inside the first range but outside the second, and so we must use a more complex comma for the second one.

As you know, there are many other considerations that feed into choosing a good EDO notation, that may upset the desire for strict Trojan-like capture zones.
Dave Keenan wrote: Here's a mockup of such a table and here are the required second-best notations, which, if you have no objection, can be considered to be completed, and can be entered into your "sagittal as software". The two fractions of an octave whose noble-mediant corresponds to each boundary can also be entered.
The code as of yet has nary a mention of EDOs. But it will soon. I don't know what you mean in this last sentence, though.
I guess I should have said: If you want to enter the colour boundaries into your software, it would be good to do it as a pair of rationals, or four integers, from which the noble-mediant can be computed as needed.
Alright here's what I have. Enjoy, everyone! viewtopic.php?p=2840#p2840
Thanks for that.

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Re: A proposal to simplify the notation of EDOs with bad fifths

Post by cmloegcmluin »

Dave Keenan wrote: Wed Nov 18, 2020 11:24 am Just to be clear. I think that "phidiant" or similar should only appear in that part of your metallic article where you're first explaining the concept of a noble-mediant, but you should use "noble-mediant" everywhere else in the article.

I think it's fine for us to save typing by inventing shorter names for things we are discussing in depth at a particular time, but these should not be used in articles intended for a wider audience, or as a permanent record. Even in our own discussions such short terms should be considered ephemeral, like ice sculptures. If we come back to the topic months or years later we shouldn't assume such terms are remembered.
That's much clearer. Thanks for expanding.
I don't see group number as ever having been used to bound the colours, although I can see why you might have thought so.
I didn't mean to imply that I thought groups ever had bounded the colors. My understanding was that before we had the noble mediants, we had nothing. We had just decided, up to 72-EDO, which ones were which color, and not worried about where the exact boundaries should be, until we wanted volleo6144 to build us a towering tornado of EDOs way past 72.
The rest of the explanation is why 1/2-apotome doesn't necessarily equal 2/4-apotome in this context. It's very simple. We try to go for the simplest comma that will do the job. But a comma that rounds to 1/2-apotome only needs to be between 0.5/2 and 1.5/2-apotome (0.25 to 0.75) while one that rounds to 2/4 must be between 1.5/4 and 2.5/4-apotome (0.375 to 0.625) and so is more tightly constrained. A simple comma may be inside the first range but outside the second, and so we must use a more complex comma for the second one.
Cool. Yeah, that totally makes sense. In fact, I had already thought about that as the likely explanation, but sometimes I just have too many tabs open and am trying to reply to too many things too fast that I don't take the time and care to get every thought out.
I guess I should have said: If you want to enter the colour boundaries into your software, it would be good to do it as a pair of rationals, or four integers, from which the noble-mediant can be computed as needed.
Got it. Thanks for clarifying.

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