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)