Talk:IFDO: Difference between revisions

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:::::::::: 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. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk: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:

# 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.

# 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 ''sub''harmonic 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:

{| class="wikitable"

|+

!

!frequency

!pitch

!length

|-

!arithmetic (''p'' = 1)

|style="background-color: #9fc5e8;" |AFS

|style="background-color: #ea9999;" |APS

|style="background-color: #ffd966;" |ALS

|-

!geometric (''p'' = 0)

|style="background-color: #ea9999;" |(unused)

|style="background-color: #ffd966;" |(unused)

|style="background-color: #666666;" |(of low interest)

|-

!harmonic → "inverse-arithmetic" (''p'' = -1)

|style="background-color: #ffd966;" |(unused)

|style="background-color: #666666;" |(of low interest)

|}

And then here's your system:

{| class="wikitable"

|+

!

!frequency

!pitch

!length

|-

!arithmetic (''p'' = 1)

|style="background-color: #9fc5e8;" |ADO

|style="background-color: #ea9999;" |(unused/unusable)

|style="background-color: #ffd966;" |(unused/unusable)

|-

!geometric (''p'' = 0)

|style="background-color: #ea9999;" |(EDO)

|style="background-color: #ffd966;" |(unused/unusable)

|style="background-color: #666666;" |(of low interest)

|-

!harmonic → "inverse-arithmetic" (''p'' = -1)

|style="background-color: #ffd966;" |IDO

|style="background-color: #666666;" |(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:

{| class="wikitable"

|+

!

!frequency

!pitch

!length

|-

!arithmetic (''p'' = 1)

|style="background-color: #9fc5e8;" |AFS

|style="background-color: #ea9999;" |(unused)

|style="background-color: #ffd966;" |(unused)

|-

!geometric (''p'' = 0)

|style="background-color: #ea9999;"|GFS

|style="background-color: #ffd966;" |(unused)

|style="background-color: #666666;" |(of low interest)

|-

!harmonic → "inverse-arithmetic" (''p'' = -1)

|style="background-color: #ffd966;" |IFS

|style="background-color: #666666;" |(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:

{| class="wikitable"

|+

!

!frequency

!pitch

!length

|-

!arithmetic (''p'' = 1)

|style="background-color: #9fc5e8;" |AFS

|style="background-color: #ea9999;" |APS

|style="background-color: #ffd966;" |ALS

|-

!geometric (''p'' = 0)

|style="background-color: #ea9999;" |GFS

|style="background-color: #ffd966;" |GPS

|style="background-color: #666666;" |(of low interest)

|-

!harmonic → "inverse-arithmetic" (''p'' = -1)

|style="background-color: #ffd966;" |IFS

|style="background-color: #666666;" |(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:

{| class="wikitable"

|+

!

!frequency

!pitch

!length

|-

!super-arithmetic (''p'' > 1)

|''p''FS

|''p''PS

|''p''LS

|-

!arithmetic (''p'' = 1)

|style="background-color: #9fc5e8;" |AFS

|style="background-color: #ea9999;" |APS

|style="background-color: #ffd966;" |ALS

|-

!super-geometric (1 > ''p'' > 0)

|''p''FS

|''p''PS

|''p''LS

|-

!geometric (''p'' = 0)

|style="background-color: #ea9999;" |GFS

|style="background-color: #ffd966;" |GPS

|style="background-color: #666666;" |(of low interest)

|-

!sub-geometric (0 > ''p'' > -1)

|''p''FS

|''p''PS

|''p''LS

|-

!harmonic → "inverse-arithmetic" (''p'' = -1)

|style="background-color: #ffd966;" |IFS

|style="background-color: #666666;" |(of low interest)

|-

!sub-inverse-arithmetic (-1 > ''p'')

|''p''FS

|''p''PS

|''p''LS

|}

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/

--[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 22:52, 8 April 2023 (UTC)

@@ Line 68: / Line 68: @@
 :::::::::: 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. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk: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:
+# 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.
+# 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 ''sub''harmonic 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:
+{| class="wikitable"
+|+
+!
+!frequency
+!pitch
+!length
+|-
+!arithmetic (''p'' = 1)
+|style="background-color: #9fc5e8;" |AFS
+|style="background-color: #ea9999;" |APS
+|style="background-color: #ffd966;" |ALS
+|-
+!geometric (''p'' = 0)
+|style="background-color: #ea9999;" |(unused)
+|style="background-color: #ffd966;" |(unused)
+|style="background-color: #666666;" |(of low interest)
+|-
+!harmonic → "inverse-arithmetic" (''p'' = -1)
+|style="background-color: #ffd966;" |(unused)
+|style="background-color: #666666;" |(of low interest)
+|style="background-color: #666666;" |(of low interest)
+|}
+And then here's your system:
+{| class="wikitable"
+|+
+!
+!frequency
+!pitch
+!length
+|-
+!arithmetic (''p'' = 1)
+|style="background-color: #9fc5e8;" |ADO
+|style="background-color: #ea9999;" |(unused/unusable)
+|style="background-color: #ffd966;" |(unused/unusable)
+|-
+!geometric (''p'' = 0)
+|style="background-color: #ea9999;" |(EDO)
+|style="background-color: #ffd966;" |(unused/unusable)
+|style="background-color: #666666;" |(of low interest)
+|-
+!harmonic → "inverse-arithmetic" (''p'' = -1)
+|style="background-color: #ffd966;" |IDO
+|style="background-color: #666666;" |(of low interest)
+|style="background-color: #666666;" |(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:
+{| class="wikitable"
+|+
+!
+!frequency
+!pitch
+!length
+|-
+!arithmetic (''p'' = 1)
+|style="background-color: #9fc5e8;" |AFS
+|style="background-color: #ea9999;" |(unused)
+|style="background-color: #ffd966;" |(unused)
+|-
+!geometric (''p'' = 0)
+|style="background-color: #ea9999;"|GFS
+|style="background-color: #ffd966;" |(unused)
+|style="background-color: #666666;" |(of low interest)
+|-
+!harmonic → "inverse-arithmetic" (''p'' = -1)
+|style="background-color: #ffd966;" |IFS
+|style="background-color: #666666;" |(of low interest)
+|style="background-color: #666666;" |(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:
+{| class="wikitable"
+|+
+!
+!frequency
+!pitch
+!length
+|-
+!arithmetic (''p'' = 1)
+|style="background-color: #9fc5e8;" |AFS
+|style="background-color: #ea9999;" |APS
+|style="background-color: #ffd966;" |ALS
+|-
+!geometric (''p'' = 0)
+|style="background-color: #ea9999;" |GFS
+|style="background-color: #ffd966;" |GPS
+|style="background-color: #666666;" |(of low interest)
+|-
+!harmonic → "inverse-arithmetic" (''p'' = -1)
+|style="background-color: #ffd966;" |IFS
+|style="background-color: #666666;" |(of low interest)
+|style="background-color: #666666;" |(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:
+{| class="wikitable"
+|+
+!
+!frequency
+!pitch
+!length
+|-
+!super-arithmetic (''p'' > 1)
+|''p''FS
+|''p''PS
+|''p''LS
+|-
+!arithmetic (''p'' = 1)
+|style="background-color: #9fc5e8;" |AFS
+|style="background-color: #ea9999;" |APS
+|style="background-color: #ffd966;" |ALS
+|-
+!super-geometric (1 > ''p'' > 0)
+|''p''FS
+|''p''PS
+|''p''LS
+|-
+!geometric (''p'' = 0)
+|style="background-color: #ea9999;" |GFS
+|style="background-color: #ffd966;" |GPS
+|style="background-color: #666666;" |(of low interest)
+|-
+!sub-geometric (0 > ''p'' > -1)
+|''p''FS
+|''p''PS
+|''p''LS
+|-
+!harmonic → "inverse-arithmetic" (''p'' = -1)
+|style="background-color: #ffd966;" |IFS
+|style="background-color: #666666;" |(of low interest)
+|style="background-color: #666666;" |(of low interest)
+|-
+!sub-inverse-arithmetic (-1 > ''p'')
+|''p''FS
+|''p''PS
+|''p''LS
+|}
+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:
+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.
+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).
+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 . . .
+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.
+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.
+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/
+--[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 22:52, 8 April 2023 (UTC)