Talk:IFDO

Critique of naming system

Recently, Flora introduced the idea of conceptualizing equal-step tunings in terms of various well-known mathematical means such as the geometric mean, arithmetic mean, and harmonic mean. To be clear, these were not novel tunings she proposed; rather, she introduced yet another way of identifying and reasoning about this existing equal-step family of tunings which have already been studied and named ad nauseum throughout xenharmonic history.

Flora wisely chose not to suggest identifying tunings by the "harmonic mean", due to the established and contradictory meaning of "harmonic" in our musical field. I have pushed back against her use of "arithmetic mean" for similar reasons; Shaahin Mohajeri introduced it to the xen lexicon in 2006 via his Arithmetic Division of the Octave (ADO) tuning (see here: https://yahootuninggroupsultimatebackup.github.io/makemicromusic/topicId_13427.html#13427), where Shaahin used the word as a reference to the mathematical concept of an arithmetic sequence/progression, which is a sequence formed by repeatedly adding a constant. So what Shaahin did was assume the constant was in terms of frequency. Another thing he did was use a word for an open sequence to describe a tuning which is a periodic division of an interval. In 2021, along with Billy Stiltner and Paul Erlich, I developed a naming system for equal-step tunings which handles both periodic divisions and open sequences. In this system, the former are "Equal Divisions", as in the overwhelming popular Equal Divisions of the Octave (EDO); the latter are "Arithmetic Sequences". So we borrowed Shaahin's use of "arithmetic", but repurposed it in a more appropriate way: for sequences, not divisions. And another thing we improved over Shaahin's concept was to treat pitch, not frequency, as the default musical resource; this agrees with the overwhelming popularity of EDOs, which are in fact equal pitch divisions of the octave. In short, we extended the established basic ideas of equal divisions and arithmetic sequences of pitch to the two other fundamental musical resources of frequency and length. We felt comfortable with the fact Shaahin's ADO made different assumptions than our system, because Shaahin was clearly not attempting to develop a full, internally-consistent system like we were (one need only look to his other tuning, EDL, which is neither consistent with ADO nor EDO). However, Flora does indeed seem to be attempting to develop a rivalling full internally-consistent system to the one we developed already. And my concern with Flora's proposal is that it uses the word "arithmetic" to describe tuning systems in this family but in a different sense than "arithmetic" has been being used historically. In the case of Shaahin's ADO, the two meanings happen to coincide. But when she extends the idea to "inverse-arithmetic mean", as a substitute for the even more unusable "harmonic mean", her meaning of "arithmetic" comes into conflict with our system's meaning. And we can see another negative consequence of Flora's proposal in that a page cropped up for "Arithmetic MOS", where CompactStar has gotten comfortable with treating "arithmetic" as synonymous with "of frequency" (see: https://en.xen.wiki/w/Talk:Arithmetic_MOS_scale).

Recently on the Discord server, CompactStar took a step toward generalizing Flora's concept of using mathematical means to describe equal-step tunings when he proposed the concept of an RD, where an interval is divided into equal parts according to the root mean square (RMS). Flora took the next step, showing that any power mean (p-mean) could be used in this way to define an equal-step tuning. The geometric mean is p = 0, the arithmetic mean is p = 1, the harmonic mean is p = -1, and the quadratic mean (same as RMS) is p = 2. So p = 0 corresponds with ED's/AS's of pitch, p = 1 corresponds with ED's/AS's of frequency, and p = -1 corresponds with ED's/AS's of length (of strings, resonating chambers, etc.). But this p = 2 corresponds with a new type of musical resource, on the other side of frequency yet from pitch. This is a novel tuning system and worth studying.

In my opinion, originally, Flora's idea of using mathematical means to describe these popular equal-step tunings was unnecessarily obscure, in comparison with the system I helped design a couple years ago which is much more directly accessible to practicing musicians, through its use of the three actual physical/psychoacoustic resources of frequency, pitch, and length. However, this new idea which leverages mathematical power means to explore new tunings, and describe this family of tunings along a smooth continuum — e.g. there's potential for tunings defined somewhere between equal divisions of frequency and pitch like 1/2MD, or between pitch and length e.g. (-1/2)MD — finally justifies Flora's concept. And so, if mathematical means are to be used in this way, then I would strongly prefer use of the power mean's power in the name, rather than "arithmetic" and the new coinage "inverse-arithmetic" (which is a bit strained anyway) as in AD and ID. I hope that in consideration of all of this, Flora and CompactStar will consider switching to refer to AD's as 1MD's and ID's as (-1)MD's. --Cmloegcmluin (talk) 19:06, 27 March 2023 (UTC)

A few points to add:

1. I disagree I used the term arithmetic differently from the sense of Shaahin Mohajeri (2006), altho I admit it differs from Billy Stiltner et al (2021). As one way to see it, frequency is implied as the default measurement – but more appropriately the measurement is determined by the specific word in the place. Using arithmetic for anything other than frequency would bug me, just as harmonic mean bugs us since its defined in terms of length.
2. I suppose the letter M stands for mean, such that the full name is m-mean division? Not that I disapprove of a change of the page title, at this point I hesitate to say it's the right word choice. Specifically, whenever there's a mean there's also a sequence, including the fractional-order ones, since the mean is really an element of a special case of a sequence. So this concept is based on not exactly means but sequences. I'd consider the full name is order-m division or order-m sequence division.
3. I believe m = 1 and m = -1 deserve verbal names (even tho they're disputed) in addition to the formal one, just as m = 0 does.

FloraC (talk) 10:40, 28 March 2023 (UTC)

1. I'm sorry, but you are simply wrong to disagree with my point that you used "arithmetic" differently than Shaahin. Frequency being implied as the default measurement is the only valid way to see this. If you check the link provided, it is absolutely clear that Shaahin intended it to refer to arithmetic sequences/progressions, and not to the arithmetic mean. I'm sorry that using arithmetic for anything other than frequency bugs you, but this is the way it has been done since 2006 and I hope that even if it doesn't work for you that you will respect those for whom it has been working well.
2. Yes, the M stands for mean. It's possible that you're using "sequence" and "division" differently than they are used in the existing system; I cannot understand what a "sequence division" could be in terms of what has already been established, since those are two mutually exclusive tuning types: one open-ended, the other periodic. I wasn't explicit about this in my previous post, but this application of power means to equal-step tunings applies well for both divisions and sequences. That is, I would expect that an EFD (equal frequency division) would be equivalent to a 1MD (1-mean division), and an AFS (arithmetic frequency sequence) would be equivalent to a 1MS (1-mean sequence); similarly, an ELD (equal length division) would be equivalent to a (-1)MD ((-1)-mean division) and an ALS (arithmetic length sequence) would be equivalent to a (-1)MS ((-1)-mean sequence). The same for E(P)D = 0MD and APS = 0MS, too.
3. I think AMD for "arithmetic mean division" as a substitute for 1MD, GMD for "geometric mean division" as a substitute for 0MD, and HMD for "harmonic mean division" as a substitute for (-1)MD, would all be fine. Using simply AD and HD (or ID) is not acceptable, because it is incompatible with ED, implying that in an ED, the frequencies of pitches are related to their immediate neighbors by the "equal mean" of their frequencies; however, the "equal mean" does not exist, so this is not true. The established system simply assumes pitch to have naturally been the default resource, and so an ED is an E(P)D, and this was extended to EFD and ELD.

--Cmloegcmluin (talk) 18:23, 28 March 2023 (UTC)

Just becuz you so framed it isn't evidence that I be wrong. You showed it yourself that Shaahin's ado was originally intended to be measured in frequency, where as the extension I made involves viewing the word as indicating the scale of measurement, as in AD vs ED vs ID. The other problem, which also has to do with your second point, is that you don't realize the mean is an element of a special case of the sequence/progression, so my extension isn't based exactly on the means, but a uniform of both. FloraC (talk) 03:48, 29 March 2023 (UTC)

Adding my grain of salt: I don't think it's wrong to have both a systematic naming system for theorists who want to get everything covered unambiguously and, in parallel, a set of everyday terms for musicians. I view cmloegcmluin's system of harmonotonic tunings as a systematic naming system, whereas the term "IDO" tries to follow the footsteps of "EDO" as a more colloquial term (i.e. not as unambiguous, but understood by most in context). I personally don't like "IDO" because I find it much harder to understand compared to "subharmonic series segment" (used by Sevish and probably more before/after him) or "undertone scale" (used by Andrew Heathwaite and probably more before/after him), while also not being as explicit as cmloegcmluin's systematic terminology. There are also shorthand notations like 24::12 (pronounced as "subharmonics 24 to 12") that convey more effectively the idea of "12ido" that aren't much longer or harder to say than "12ido". As for fractional-order tuning systems, I think it's better to start by developing a systematic naming system, if only because it might not turn out useful or necessary to make it musician-friendly later. --Fredg999 (talk) 01:11, 29 March 2023 (UTC)

I don't see how inverse-arithmetic division could be described as more "colloquial" than equal division of length. To me it seems impossible to argue that this is more of an "everyday term for musicians", since ID references a recently made-up variation on a mathematical mean which most musicians do not know in the first place, while ELD references the physical property of string or resonating chamber length that practicing musicians actually physically deal with. The EFD/E(P)D/ELD system is both the unambiguous and comprehensive system for theorists as well as the more accessible system for music makers, and it has slight (2 years) historical precedence. --Cmloegcmluin (talk) 02:17, 29 March 2023 (UTC)

In case I wasn't clear, I wasn't stating that IDO is more colloquial than EDL/ELD, but rather that IDO seems derived and inspired by EDO, which is more colloquial than its systematic counterpart EPDO. I agree that "equal length division" and "equal frequency division" are clear enough to be used directly by musicians, especially when using irrational numbers. As for (sub)harmonic series segment, now that I think about it, I believe they cover only a subset of otonal/utonal divisions/sequences (e.g. 4:7:10:...:25:28 is not a harmonic series segment, but it is an OD/OS), so unless I'm mistaken, it might be good to keep both terms around to describe the different concepts. --Fredg999 (talk) 03:01, 29 March 2023 (UTC)

Ah, sorry, I see what you mean now. However, now I don't see how ID "seems derived and inspired by" ED any more than EFD and ELD are derived and inspired by E(P)D. --Cmloegcmluin (talk) 03:43, 29 March 2023 (UTC)

I have documented the history here: ADO#History, and also updated the IDO page accordingly. Because this way of using "arithmetic" is contentious:

a) I have suggested that the pages Arithmetic interval chain and Arithmetic MOS scale) be withdrawn from the main wiki and changed to user pages, at least for now (see: Talk:Arithmetic interval chain and Talk:Arithmetic MOS scale).

b) I have added ODO and UDO as additional equivalent bolded terms to all of the recently created ADO and IDO pages, and created redirect pages to them. However, I don't think this is quite enough; I believe that the redirects should be swapped so that the ODO and UDO pages are the destination pages (and accordingly that the order of the bolded terms be switched on each page so that ODO and UDO take precedence, and ODO/UDO used afterward throughout each page).

c) I also suggest that the ADO and IDO infoboxes be changed to the ODO and UDO infoboxes.

I have not taken those actions yet, however, because I did not create all these pages myself, and I want to respect CompactStar's authorship and give them a chance to weigh in on the matter. I'm sorry this is a lot of changes and suggested further changes, but I was just not available to spend my time and energy on this issue while it was beginning to balloon out of control these past couple months. --Cmloegcmluin (talk) 18:43, 30 March 2023 (UTC)

I strongly oppose moving ado to odo and ido to udo. If anything, efdo and eldo would be acceptable. As I said in Talk:Arithmetic tuning, OD and UD shouldn't be used to identify tuning systems. FloraC (talk) 05:29, 31 March 2023 (UTC)

Sorry, but I still don't really understand your concern about OD and UD. On the Talk page you mention (Talk:Arithmetic tuning) you said something about "lack of orthogonality". When I pressed you to explain this idea, you wrote: "I explained the lack of orthogonality as 'one specification is often encompassed by another', and so by having orthogonality our specifications would be minimal and disjoint from each other. Note that this is not abouting deprecating the concepts and the names, only how we specify individual tuning systems." I'm sorry I dropped that conversational thread, and I'm really sorry about this too, but I still can't figure out what you're trying to say here. Could you try explaining in another way, please?

One thing I'm confused about is that I thought recently on Discord you said you preferred OD and UD over EFD and ELD, though it was on the condition that they be able to apply to divisions of irrational intervals and therefore be non-JI (which I noted was an unacceptable condition). I only find it slightly less unacceptable to use EFD where OD is possible, because of the implication that an EFD must be non-JI. But maybe I'm just a bit lost because there have been so many subtleties to this discussion, and it has been fragmented in so many places, and it's taken place relatively slowly over the course of many years, so it's hard for me to keep everything straight. I have probably accidentally said inconsistent things here or there, so I sincerely apologize if you find that I wasted your time or confused you due to something like that.

Oh, but I did figure out what you meant by a "sequence division" in the discussion above. I think my confusion arose because we're using "sequence" in two different ways. The system I designed with Paul and Billy uses sequence as opposed to a division, i.e. open-ended as opposed to periodic. But you're using sequence as in arithmetic sequence, geometric sequence, harmonic sequence, etc. (which are closely related to arithmetic mean, geometric mean, harmonic mean, etc.) Sorry I didn't understand that immediately. So, I propose that since in your context, we can use "progression" as an exact synonym, we stick to using "progression" for that context, to avoid further miscommunication. Hopefully that works for you, too. --Cmloegcmluin (talk) 23:55, 31 March 2023 (UTC)

Hmmm, here's another way to speak of odo's lack of orthogonality: it contains redundant information, as both otonal and octave imply rationality and not becuz it has to be designed this way. An orthogonal system will have these kind of redundancies minimized, like that orthogonality in linear algebra, you know. In this case ado or efdo is sufficient to uniquely identify the object with rationality conveyed thru the divided interval itself, so it's advantageous. Using otonal will introduce a kind of perplexity in which we can find "grammatically" possible but "semantically" invalid combinations such as *odφ.

I can see how sequence means two different things. What I was calling the sequence was also known as "linspace" in some programming libraries, if that makes any sense to you. For the open-ended sequence, let's say progression from now on.

FloraC (talk) 14:54, 1 April 2023 (UTC)

I'm glad we're in agreement on the progression vs. sequence thing. Thanks.

And okay, I think I finally understand what you're saying now re: "orthogonality". Thank you for your patience with me. However, it doesn't dissuade me from seeing ODO as the more appropriate equivalence to ADO.

I've tried to use an extended analogy to deal with this issue. Now, this maybe isn't a perfect or exact analogy, nor is it any simpler than the music problem we're looking at, but perhaps a problem with an matching structure but different contents will give us a new opportunity to reason through our disagreement. Maybe it'll give us a clean foundation from where we can identity places our assumptions have diverged.

So suppose we have a situation where the easy, obvious, and popular things to care about are sets of circle centers. Then someone points out that we might consider ellipses, too, but when they start experimenting with this, they realize that technically what we should be interested in when we generalize circles like this are the lower of these shapes' foci, not their centers.

Then people come along to give the community systematic ways to name this stuff. For example, they could use n-LFSp: n-cardinality Lower Foci Sets of shapes which are somehow specified by p. Perhaps this p could be the proportion between the distance between foci and the radius, or something like that. And of course we could use a special case of C for Circle in place of however we'd specify p for circles (akin to how we use the special case of O for Octave in place of 2).

But here's where I think you and I would differ. If I understand your thinking correctly, then I expect you'd say that we should only use n-LFSp, including for the primary use case of circles: n-LFSC. On the other hand, I'd say that we should use n-CSC for Center Sets of Circles, and only revert to n-LFSp when experimenting with these more complex ellipses and their foci. Your problem with my n-CSC acronym would be twofold:

1. It contains a redundancy, because if we know we're using circles, then we already know their lower foci are their centers, so you think it's forbiddingly wasteful or confusing to say so.
2. It makes the naming system overall more complex, because we have two acronyms where we could get away with only one.

But I would counterargue:

1. Redundancy isn't good, I agree, but having some is not a deal-breaker. "Center" conveys much more than simply "circleness", and "circle" conveys much more than simply "centerness"; it's well worth the slight amount of overlap in their meanings for us to convey the full implications of both.
2. It's more important to give the primary use casers the clearest, simplest acronym, and let the experimenters deal with how to work around it. It's Dave's and my "don't FUSS" principle: don't foul up the simple stuff. In other words, what's more important than the simplicity of the system itself is the utilitarian simplicity of the average users' experience of the system. I accept the risk of n-CSp where someone mistakenly uses "centers" to refer to non-circular ellipses; that may confuse some of the advanced people, but that's on them; it doesn't bother me because it's not endangering the 99% of basic users who are doing the simple stuff with circles.

So if 99% of people have been coming here to deal with and talk about circle centers — and don't have any interest in the fact that these circle centers are also lower foci, which is a more advanced concept — then n-CSC is the more appropriate acronym. And I'm saying that this is analogous to people coming here to deal with overtone-based tunings, which a very basic music concept — and having no particular interest or awareness about their steps being equal frequency amounts, which is a more advanced physics/math concept. This is why I co-designed the naming system which offers n-ODp for the ordinary musician interested in otonalities, and n-EFDp for the more advanced experimenters. --Cmloegcmluin (talk) 19:58, 1 April 2023 (UTC)

(I'm resetting the indentation here. Hope that's okay.)

Initial apology

Hi, Flora. I should begin with an apology. I have been complaining that you interpreted "arithmetic" in a different sense than Shaahin originally intended. While I still believe it's clear that Shaahin intended the meaning as in "arithmetic progression", and that you mean it as in "arithmetic mean", I can now see that these are essentially the same meaning, because the principles that relate one step to the next in an arithmetic progression are the same which relate the value between two adjacent steps via the arithmetic mean (and the same point about the connection between progressions and means can be said about geometric progressions/means, and harmonic progressions/means). Almost certainly you have attempted to convey this idea to me and others in the past weeks, and probably some people got it right away, but it took a while to penetrate my thick skull. I'm very sorry about that. Perhaps I was blinded by confusion on account of that issue where there were two different senses in which we were using the word "sequence"; we recently agreed to use "progression" for the first kind, as in the "arithmetic progressions" I've spent this paragraph dealing with, and "sequence" for the other kind, as in the sequences (open-ended) vs. divisions (periodic) distinction used in the naming system I co-designed.

And so, I can see that you, in fact, have not distorted or abused Shaahin's original thinking at all, as I originally argued. The truth is the other way around. Of the two of us, it is you who are properly extending Shaahin's intentions when he coined "ADO". Both you and my team use "arithmetic" in the same way he does, but on two other points, you go with Shaahin, while my team went against. I have already acknowledged these two other points above, but I'll repeat them here:

1. You've accepted his assumption of frequency as the default musical resource from which to make an arithmetic progression from / take arithmetic means of. We did not.
2. You've accepted his association of "divisions" with "arithmetic". We did not.

So I recognize that some recent edits I made to the ADO and IDO pages should be revised. Also, I retract my suggestion that your acronyms be forced to include 'M' for mean, as in AMDO and IMDO; I can now see that this is unnecessary, confusing, and doesn't solve my main problem with your system anyway. Sorry again.

ADO should not be extended

And so now I can come to my point. My primary concern is not whether or not you've properly extended Shaahin's concept. My concern is that Shaahin's concept should not have been extended. In particular, I believe the first of the two design choices Shaahin made — which your ADO/IDO system is perpetuating — is dangerously confusing. It's this first choice, i.e. assuming frequency, which is the major problem; the second choice, i.e. "arithmetic divisions', in minor in comparison. I will be spending most of my time in this post addressing the problem with this first choice.

In support of my argument, I conducted some research. I searched through available archives of public discussion of xenharmonic topics for all relevant occurrences of the word "arithmetic", and gathered over 200 of these. My sources included the Mills tuning list, Yahoo Groups!, Facebook, and Discord. My oldest datum is from 1995; I did not include results that were more recent than 1 year old, and I also avoided unfairly including results that were explicitly about the system that Paul, Billy, and I designed. My goal was to understand how people in the community since the internet age began have most frequency used this word when describing tunings or scales. Along the way, I learned a lot about the pre-internet historical usage of the term as well.

Here's a key piece of information. Perhaps you understood this already. This was news to me. Originally, "arithmetic" was understood to refer to string length, not to frequency as Shaahin and yourself are doing. The concept of frequency — the period of vibrations in air — was not known to the ancient Greeks. The resource that was immediately visually able to be apprehended by them was the length of the vibrating string. This is why we experience confusion today where the harmonic mean relates frequencies of a subharmonic series rather than the harmonic series: ever since the discovery of frequency, it has been more intuitive for us modern thinkers to reason about frequency ratios like 2:3, 4:5, etc than it is for us to reason about wavelengths. But to the ancient Greeks, who thought in lengths, it was the overtones which were related by the harmonic mean, and it the undertones that were related by the arithmetic mean. If one checks the work of Zarlino, Marchetto, Schlesinger, Partch, and probably many others, this assumption that "arithmetic" refers to subharmonic relationships is perpetuated. For example, here's a quote from p69 of Genesis of a Music, where Partch defines the "Arithmetical Proportion":

an ancient principle of scale structure by which a sounding body is divided into a number of exactly equal parts, and a scale constructed from the resulting relationships, or ratios; this does not produce equal scale steps. The Arithmetical Proportion has been seized upon by some theorists as the ancient substantiation of "minor tonality", since, starting with the one part that gives the highest tone, it results in a series of tones the exact reverse of a fundamental and its series of partials, down to an arbitrary low tone determined by the number of equal parts used (see page 174). The Arithmetical Proportion may be considered as a demonstration of Utonality ("minor tonality"). 

However, my research shows that since the internet era, we find about a 50/50 split between statements where people assume "arithmetic" refers to string length per the historical usage, and statements assuming frequency per modern intuition. Notably, however, when we break the data down by unique individuals, we find many many more individual people who assume "arithmetic" referring to frequency than we do those who assume it referring to length. In other words, we have a small but very vocal minority of people who continue to use "arithmetic" in the original sense — most prominently Margo Schulter — and a large but scattered majority of non-experts (such as Shaahin Mohajeri, Bill Wesley, Paul Hjelmstad, Joakim Bang Larsen) who come to the table thinking of frequency ratios like 8:9 and 9:10 and take the arithmetic means of those, i.e. casually assuming frequency.

If you want to view the research, it's available here: https://docs.google.com/spreadsheets/d/1wUhccKnyobJIQf3Ra1wQ7amYSMlV-VVmV5FFnyHLrnQ/edit?usp=sharing This link includes the raw data, links, and several charts you may find interesting.

My conclusion is that it is unwise to propagate a system which assumes "arithmetic" referring either to length or to frequency. This is why Paul has been explicit about which resource he is sequencing arithmetically, using "arithmetic frequency sequence" since at least as far back as 2017, and why when Billy and I got together with Paul to codify and complete his design on the wiki in 2021, we maintained this approach.

Shaahin's term "ADO" is fine as a one-off personal name he would use for his compositions, but it is not acceptable as a term we should propagate in the community. (By the way, until you came to the problem recently, my research shows that ADO had very little traction in the community; only two other people besides Shaahin himself — and also you and CompactStar, of course — have claimed to use it. To be fair, I only found three other people adopting the system designed by Paul, Billy, and I. So it's a total of 5 people versus 6 people. Which is hardly a statistically significant difference.) It encourages the sort of problematic writings I've called out recently from CompactStar, who is taking the existence of ADO (and especially your propagation of it) as a sign that they can treat "arithmetic" as evoking frequency (e.g. in Arithmetic MOS scale or Arithmetic interval chain), despite the fact that for educated microtonalists that word implies the exact opposite thing: length. And so I can not accept "IDO" either, since it has the same deal-breaker of a problem.

Shaahin's (and your) approach are comparable to if Paul, Billy, and I had decided to write up FS, PS, and LS — frequency sequences, pitch sequences, and length sequences. Maybe some people would assume we meant arithmetic sequences, but what some people assumed harmonic sequences, or geometric sequences? It's just not safe to be less than fully explicit in this domain.

Fixing IDO; make it IFDO

As stated previously, however, I do believe that there is some value in the generalization of this idea to any power mean; this allows your system to cover ground which our system does not cover. And so, I don't recommend we completely reject it yet.

However, in order for your system to not be forbiddingly confusing, it must be modified to be explicit about the resource divided or sequenced, like our system does. So, your system must be changed at least from ADO to AFDO, and from IDO to IFDO. To be clear, I think it's clear from my research that you must explicitly include "frequency" in the name, or you'll be doomed to conflict and confusion.

In fact, I even came across some concerns about how EDO does not explicitly specify the divided resource: https://discord.com/channels/332357996569034752/780300193110818826/861278258350784522 "I think it'd be a good idea to clarify that the equal divisions are of log(frequency)". This agrees with Paul/Bill/I's decision to treat EDO as short for EPDO, where pitch of course = log(freq).

Of course, EDO is not the same as EFDO (it is the same as EPDO), but you could use GFDO, for geometric frequency division of the octave.

Here's a series of tables that will show the relationship between the three musical resources — frequency, pitch, and length — and the three key mathematical means/progressions — arithmetic, geometric, and harmonic — and how our two systems approach the space differently. Here's a part of the system I designed with Paul and Billy:

	frequency	pitch	length
arithmetic (p = 1)	AFS	APS	ALS
geometric (p = 0)	(unused)	(unused)	(of low interest)
harmonic → "inverse-arithmetic" (p = -1)	(unused)	(of low interest)	(of low interest)

And then here's your system:

	frequency	pitch	length
arithmetic (p = 1)	ADO	(unused/unusable)	(unused/unusable)
geometric (p = 0)	(EDO)	(unused/unusable)	(of low interest)
harmonic → "inverse-arithmetic" (p = -1)	IDO	(of low interest)	(of low interest)

This isn't a very direct comparison, since your system so far only deals with octaves, and deals with divisions rather than sequences. My main point here is conveyed by the coloration pattern, which runs diagonally across this space. Cells with the same color describe the same type of tuning phenomenon, but in different ways. But if your system was made unproblematic by being made explicit that it deals with frequency, and then extended to sequences and generalized beyond octaves for a more direct comparison with our system, then this is what it would look like:

	frequency	pitch	length
arithmetic (p = 1)	AFS	(unused)	(unused)
geometric (p = 0)	GFS	(unused)	(of low interest)
harmonic → "inverse-arithmetic" (p = -1)	IFS	(of low interest)	(of low interest)

Note at this point that we both agree on AFS in the blue top-left corner, for arithmetic frequency sequence. So we could actually combine our systems. Here's how that would look like:

	frequency	pitch	length
arithmetic (p = 1)	AFS	APS	ALS
geometric (p = 0)	GFS	GPS	(of low interest)
harmonic → "inverse-arithmetic" (p = -1)	IFS	(of low interest)	(of low interest)

I also completed the chart by filling in the last remaining viable cell: GPS in the middle for geometric pitch sequence.

Looking at this completed chart, we can see that another term for an arithmetic pitch sequence (e.g. something like Carlos alpha, beta, or gamma), such as how a user of our system would think of things, would be a geometric frequency sequence in your system; APS = GFS. And similarly, ALS = GPS = HFS (the last of which is to say, that a subharmonic series could be understood as a harmonic sequence of frequencies).

And so now we need to take a look at what makes your system have potentially new value, because my criticisms I shared with Fredg999 above still hold, i.e. I think it's clear that the system I co-designed is not only the unambiguous system that theorists need here, but also the more accessible system for music makers. But your system has one advantage, which is that it can cover theoretical space that our system cannot. It's the space between the three rows of this table, and beyond it:

	frequency	pitch	length
super-arithmetic (p > 1)	pFS	pPS	pLS
arithmetic (p = 1)	AFS	APS	ALS
super-geometric (1 > p > 0)	pFS	pPS	pLS
geometric (p = 0)	GFS	GPS	(of low interest)
sub-geometric (0 > p > -1)	pFS	pPS	pLS
harmonic → "inverse-arithmetic" (p = -1)	IFS	(of low interest)	(of low interest)
sub-inverse-arithmetic (-1 > p)	pFS	pPS	pLS

By the way, these terms "super-arithmetic", "sub-geometric", etc. are not well-thought-out coinages. Just needed to fill out the table.

And also, I note that as far as I know, there's no particular reason to be interested in these spaces. That's why I used the term "potential" above. It's the only reason I don't ask you to completely remove IDO (or IFDO) from the wiki entirely.

Finally, I note that CompactStar's RMS-based tuning proposed on Discord which got this discussion kicked off would be a 2FDO.

Is it even safe for Paul, Billy, and I to use "arithmetic" at all?

I think there's one big remaining question here, and that's whether — given how problematic the word "arithmetic" is — is it okay that Paul's/Billy's/my system uses it at all? That is, while Margo is someone who consistently assumes arithmetic to refer to string length, she is also open to applying the term arithmetic to frequency or pitch when it is made explicit. But will everyone like her who assumes arithmetic defaults to length be open to it applying to other resources?

My answer is: I'm not sure, but I think so. In any case, we can't allow ancient historical usage to block effective modern usage. So I contend that the page I created for Arithmetic tunings is still acceptable, on the grounds that it uses "arithmetic" in the way it actually means: referring to arithmetic means or sequences/progressions as applied to musical scales or tunings. That said, I should certainly at least add some historical notes along the lines of the stuff I reported here from my research.

As a thought experiment, though, what if we decided that "arithmetic" was irredeemably tainted with confusion, and dismissed it entirely? What would the page "arithmetic tunings" be instead, then? Well, almost certainly it would be "equal-step tunings". However, there is already a page with that name, where it doesn't quite mean the same thing. The existing Equal-step tuning page assumes pitch as the default resource, whereas I would be here wanting it to be general to frequency and length too. This would begin to necessitate a major overhaul, along the lines of insisting that the existing equal-step tuning page be changed to equal-cents or something like that. There is a precedent in "equal-hertz" which is an old, outdated term for "isoharmonic", and we could then have "equal-wavelength" to complete the set. But anyway, I don't think such an overhaul is reasonable, nor am I interested in dealing with such a thing anyway.

Conclusion

So I'll try to come to a concluding statement now. I don't particularly care if you do choose to extend your system from divisions to sequences. Or if you generalize it beyond octaves. But I insist that you leave Shaahin's work alone and do not try to extend it. It was never designed to be coherent and comprehensive in the way folks like you and me want systems to be. If you do still believe it is of theoretical interest to think of and explore tunings described by various power progressions of frequencies, then by all means do so, but please recognize the historical danger involved using the word "arithmetic" unqualified, and I strongly recommend that you and CompactStar take all the recent explosion of stuff you created with ADOs and IDOs and change them to AFDOs and IFDOs, please. This is a big opportunity to get the community educated on the relationship between these three progressions and these three musical resources. I can't describe how disheartening it was for me during my week of research to see the same confusion playing out over and over and over and over and over again when people would use "arithmetic" without being explicit about with regards to what. I don't want to leave future impressionable newcomers like CompactStar vulnerable to conflating columns of the tables in this post with their rows.

The rest of the stuff in this post is just supporting information.

Paul's explanations

Here are some samples of Paul's explanations over the past 20 years or so, about the confusion inherent in the situation:

2002

https://yahootuninggroupsultimatebackup.github.io/harmonic_entropy/topicId_581#584

the arithmetic progression of the frequencies in the harmonic series? that would make sense, but unfortunately you have to be careful around tuning folks -- 'arithmetic', as opposed to 'harmonic', divisions of intervals historically refer to the arithmetic division of a string, which result in a *subharmonic*, rather than a *harmonic*, series.

2002

https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_39746#39926

in the old days, before the physical harmonic series was known (a natural phenomenon in which any tone tends to be accompanied by tones with integer multiples of its FREQUENCY), string lengths on monochords were used to compare pitches, and many theorists demanded simple proportions in those string lengths. string length is approximately inversely proportional to frequency. if these string-length proportions were ARITHMETIC (that is, in proportions belonging to the series 1, 2, 3, 4, 5 . . .), the result was what is today known as SUBHARMONIC. if these proportions were HARMONIC (that is, in proportions belonging to the series 1/1, 1/2, 1/3, 1/4, 1/5 . . .), the result was what is today still known as HARMONIC (though typically expressed today as a set of ARITHMETIC proportions in FREQUENCY).

2003

https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_8130#8160

The reason for this is historical. We say whole number multiples today because everyone since Fourier talks about frequency measurements. In the old days, it was string length measurements (or still today, period or wavelength) where the numbers are *inversely proportional* to the frequency numbers. So the harmonic series in the old days *was* 1/1, 1/2, 1/3, 1/4, etc, and yet it's the same harmonic series that today goes 1, 2, 3, 4 . . .

2019

https://www.facebook.com/groups/untwelve/posts/2447957548592730/?comment_id=2453839764671175

[Arithmetic Division of the Octave] could mean several different things. Prior to the 17th century, it would typically have meant a Utonal series. More recently, it would typically mean an Otonal series. One can reckon pitches in different ways and what is "arithmetic" one way is not arithmetic in the other ways.

2019

https://www.facebook.com/groups/xenharmonicmath/posts/1536904806449736/?comment_id=1536991326441084

Janne Karimäki:

The name of such scales is ADO, meaning arithmetic division of the octave. You have 5-ADO, 10-ADO, and so on there.

Paul Erlich:

Careful because "arithmetic" typically means Utonal while this scale is "harmonic" or Otonal. I've only ever seen one person use the ADO moniker and it's caused a lot of confusion.

Janne Karimäki:

It makes perfect sense to me.

Paul Erlich:

The Arithmetic Proportion or Arithmetic Division usually refers us subharmonic aka Utonal series, not harmonic series. That's because the terminology originates in antiquity, prior to the scientific revolution, and string lengths were still used to reckon pitches.

Janne Karimäki:

There is also the name EDL, equal division of length, which directly applies to strings.

Paul Erlich:

True but without context of that "opposite", the preponderance of conflicting terminology out there tends to confuse a few people every time.

2020

https://www.facebook.com/groups/497105067092502/posts/1752376808231982/?comment_id=1752536931549303&reply_comment_id=1753252624811067

Paul Erlich:

Steve, that may have been true prior to the 17th century, when the nature of pitch as rapid vibration was not yet known, and string lengths were used to reckon and describe scales, but today musical divisions are reckoned according to frequency, so 4:5:6 becomes the arithmetic mean division and 10:12:15 becomes the harmonic mean division, as Joakim has said. What is confusing is that the etymology comes from prior to the 17th century, so what we call the "harmonic series" today is named after how it looks with respect to string divisions, even though the harmonic series is an arithmetic progression in modern (frequency-based) terms.

Margo Schulter:

Paul, I agree that how one approaches this depends on what sources one follows; and to me, the 17th century is rather ultramodern, so that I follow the more traditional terminology. But it's a traditional view that the harmonic division (as defined by string lengths) is indeed the most harmonious for vertical sonorities. Thus _Cpmpendium de musica_ (c. 1350?) argues that the fifth is best placed below the fourth in a three-voice sonority dividing the octave because there is a natural series of numbers 2-3-4 (with the 3:2 fifth preceding the 4:3 fourth). This source also gives what might be called the quaternario or series 1-2-3-4, in which 2-3-4 is concluded (with the 1-2-3 sonority, an octave plus an upper fifth forming a 12th above the lowest voice, also a common 14th-century close). Zarlino likewise uses the senario, 1-2-3-4-5-6, to demonstrate that the division of the fifth where the major third precedes the minor third is more harmonious and "joyful." So the conclusions of these sources fit the later concept of the harmonic series.

As a bonus, here's another great statement I found from Tony Durham, in 2021:

https://www.facebook.com/groups/xenharmonicmath/posts/2100727053400839/?comment_id=2101967246610153&reply_comment_id=2103106736496204

One reason why the nomenclature is confused: it depends whether you are thinking in terms of frequency, wavelength or subjective pitch. Harry Partch built acoustic instruments. He thought in wavelengths, so it was natural for him to describe Utonal sequences as arithmetic. But somebody who works in electronics, and thinks in frequencies, is more likely to describe the overtone sequence as arithmetic. Each is right in its way. I've tried to summarise this in a table. Feedback welcome: I didn't spend long on this and it is probably far from perfect.

More background information on Shaahin's and our ideas

As I've pointed out earlier, Shaahin's work is clearly not striving toward a coherent system that fits into existing practice as smoothly as possible (as Paul, Billy, and I did). Or at least, if this was his intention, he failed quite badly at it.

I note that for him it was not simply ADO and EDL. He also had ARDO, ARD, AID, and ADL. I have given up on trying to understand the relationship between all of these things. But if you'd like to try to make sense of it, a lot of the links he's shared on Facebook and Yahoo Groups! are dead now, but I found them on the Internet Archive's Wayback Machine, so you can explore yourself:

https://web.archive.org/web/20210612173344/http://www.96edo.com/Sometuningmodels.html

https://web.archive.org/web/20210211003137/https://sites.google.com/site/240edo/arithmeticdivisionsoflenght

https://web.archive.org/web/20210211002126/https://sites.google.com/site/240edo/equaldivisionsoflength(edl)

We found it particularly frustrating that he acknowledged his A(R)DO as a otonal system, but did not leverage that terminology, instead, unnecessarily coining something new.

I haven't gone back and re-read these recently, but I recognize these as the two main discussions where Paul, Billy, and I developed our system, if these might be of interest here:

https://www.facebook.com/groups/xenharmonic2/posts/4023222704364668

https://www.facebook.com/groups/497105067092502/permalink/1980938532042474/

--Cmloegcmluin (talk) 22:52, 8 April 2023 (UTC)

Additional thoughts

Okay, sorry for how lengthy my side of this thread is getting. But I had some further thoughts overnight. I'll try to keep it brief.

The page I created for Arithmetic tunings is like the first row of the tables I shared above. Like this:

	frequency	pitch	length
arithmetic (p = 1)	AFS, OS, EFD, OD	APS, AS, E(P)D	ALS, US, ELD, UD

While your system is like the first column. If you created a central page for them, it'd be called Frequency sequences. Again, we overlap on the top-left cell, for AFS. Like this:

	frequency
super-arithmetic (p > 1)	pFS, pFD
arithmetic (p = 1)	AFS, AFD
super-geometric (1 > p > 0)	pFS, pFD
geometric (p = 0)	GFS
sub-geometric (0 > p > -1)	pFS, pFD
harmonic → "inverse-arithmetic" (p = -1)	IFS, IFD
sub-inverse-arithmetic (-1 > p)	pFS, pFD

There's no reason to worry about the middle of the table, i.e. where I had "GPS" for "geometric pitch sequence". Because of the diagonal equivalence pattern, it's only necessary to think in terms of possible arithmetic sequences (our system), or possible frequency sequences (your system).

These two systems have reason to coexist. For some people, our system may make more sense. For others, yours may make more sense. Though I repeat, that I think our system is more natural for actual musicians, while yours may be more natural for mathematicians or engineers.

A vs E

But what my system calls an "EFD", for "equal frequency division", yours calls an "AFD", for "arithmetic frequency division". Per Talk:Arithmetic tuning, Paul, Mike, and I find "arithmetic division" incorrect and prefer "equal division".

But it's not a simple fix for your system, because you're not really speaking about dividing arithmetically, which doesn't make grammatical sense, and that's our concern; you're talking about dividing with respect to the arithmetic mean/progression. Before I understood the relationship between means and progressions, I had suggested "arithmetic-mean-based division", but you noted that your system is as much about progressions as it is about means, and I see that now.

But I do have an observation which might help with this. Since the main new value for your system is how it can fill in between and beyond the values of p = 1, 0, and -1, it's really about power means. And I don't find that "power progressions" are written about much, so I believe that "mean" should take precedence over "progression" here. In other words, I suggest that your system might even better be thought of as the "frequency power mean sequence" system, and indeed AFD would stand for "arithmetic-mean-based frequency division".

Nonetheless, perhaps "frequency power progression tunings" or something like that, with AFD standing for "arithmetic-progression-based frequency division". Not sure. Just suggestions.

--Cmloegcmluin (talk) 18:15, 9 April 2023 (UTC)

I have not been following your discussion with Flora, but from what I've gathered, you believe that your system using the terms EFDO and ELDO is more intuitive for musicians due to naming resources frequency and length, whereas my/Flora's system using ADO and IDO is more generalizable. However, there is an easy way to have both of these advantages–use a spectrum of resources rather than means, where the "n-resource" is expressed in terms of frequency as fⁿ. The terms EFDO, ELDO, and E(P)DO can be maintained for n = 1, n = -1, and n → 0 respectively. For any other p, the term "EnDO" (equal n-resource division of the octave) can be used, so the tuning I described on Discord can be called an E2DO, and the tuning halfway between EDO and EFDO can be called an E0.5DO.

CompactStar (talk) 01:00, 10 April 2023 (UTC)

Hi CompactStar. Yes, you've gathered correctly; that is a big part of what I'm saying: that my team's system is more accessible, while yours is capable of generalizing to unexplored theoretical territory. (The other biggest part is that due to historical and contemporary usage patterns, "ADO" is unacceptable in a tuning name system.) Now, I do like how you're thinking here, attempting to synthesize the strengths of our two systems. I had not really thought of trying something like this. But unfortunately, I don't think I would recommend we go with this particular suggestion of yours. I wish I liked it... I really don't want to come across as inflexible or overly negative here. My problem is this: there's just no direct association of these resources — frequency, pitch, and length — with those numbers — 1, 0, and -1. In other words, in order to understand why a E(0.5)DO referred to a tuning between an EFDO and an EDO, one would have to understand several things:

1. arithmetic/geometric/harmonic progressions in the first place,
2. and then understand the blue/red/yellow "diagonal" relationship between frequency/pitch/length and arithmetic/geometric/harmonic progressions (that I showed off in the tables above),
3. and then understand the relationship between such progressions and the associated means,
4. and then understand how those famous means with special names (arithmetic, geometric, harmonic) generalize as power means (where p = 1, 0, -1, respectively).

So if we're going to discuss 0.5 tunings like this, I think we might as well use a more immediate and clear approach to it, as in the ... 2FDO, AFDO, (0.5)FDO, GFDO, (-0.5)FDO, IFDO, (-2)FDO ... continuum. If people are using this system already (i.e. your system, as revised according to my suggestions), then they should already know point #1, and possibly #3 too; it's really point #4 that's the interesting new thing. The key thing is they would never need to understand point #2. And this likens back to why I think our systems can coexist; because someone who thinks primarily in terms of frequency ratios like 7/6, 5/4, 3/1 etc. as well as in terms of these power means, well, they may potentially never have to learn anything about the "diagonal" relationship with pitch and string length (and vice versa, someone like me, for whom thinking about frequency, pitch, and length came naturally, would never have to learn about geometric and harmonic means, which indeed I had gotten away with my whole life without understanding, that is, of course, up until this whole issue came up earlier this year!) --Cmloegcmluin (talk) 03:20, 10 April 2023 (UTC)