# Talk:Delta-rational chord

## Trying to sort this stuff out / clean it up

I noticed some recent changes in this department, and as I tried to catch up on the topic, I found it to be a bit of a mess at the moment. Please correct me if I've gotten anything about this situation wrong:

chord type illustrative examples
actual chord deltas delta ratio set
frequency ratio are items all integers? delta signature * reduced delta signature (class) are items all the same? unique undirected ratios between the deltas are items all rational?
DR (delta-rational): at least one rational in delta ratio set FDR (fully delta-rational): all rationals in delta ratio set JI (just intonation) but not isodifferential ** :

chord contains only integers (and therefore delta ratio set contains only rationals), and delta ratio set does not ≠ {1}

4:5:7:8 yes, all +1+2+1 +1+2+1 no, not all { 1, 2 } yes
3:5:9:11 +2+4+2
3:4:7:9 +1+3+2 +1+3+2 { 3/2, 2, 3 }
isoharmonic

(JI and isodifferential): chord contains only integers (and therefore delta ratio set contains only rationals), and more specifically delta ratio set = {1}

class i 4:5:6 +1+1 +1+1 yes, all { 1 }
4:5:6:7 +1+1+1 +1+1+1
3:4:5:6
class ii 3:5:7:9 +2+2+2
5:7:9:11
class iii 1:4:7:10 +3+3+3
2:5:8:11
... ... ...
not JI, but isodifferential:

chord contains non-integers, and delta ratio set = {1}

ɸ:(ɸ+1):(ɸ+2):(ɸ+3) no, not all or none +1+1+1
1:ɸ:(2ɸ-1):(3ɸ-2) +(ɸ-1)+(ɸ-1)+(ɸ-1)
(incompletely) DR:

at least one rational in delta ratio set, but not all rationals (therefore the chord contains non-integers)

4:5:τ:7:9 +1+(τ-5)+(7-τ)+2 +1+(τ-5)+(7-τ)+2 (irrelevant for categorization) { (7-τ)/(τ-5), 7-τ, τ-5, 2/(τ-5), 2, 2/(7-τ) } no, but at least one
5:τ:8:(3+τ) +(τ-5)+(8-τ)+(τ-5) +1+(8-τ)/(τ-5)+1 { 1, (8-τ)/(τ-5) }
1:(1+a):(1+a+b):(1+a+2b):(1+3a+2b), with a/b irrational +a+b+b+2a +a+b+b+2a { a/b, 1, 2, 2a/b }
not DR: no rationals in delta ratio set

(therefore the chord contains non-integers)

4:5:τ:7 +1+(τ-5)+(7-τ) +1+(τ-5)+(7-τ) { (7-τ)/(τ-5), 7-τ, τ-5 } no, none
5:τ:7 +(τ-5)+(7-τ) +1+(7-τ)/(τ-5) { (7-τ)/(τ-5) }

* Sometimes when found formatted as a ratio, e.g. +1+2+1 as 1:2:1, these have been referred to as "isoratios", or at least I think that's what's happening; this term appears on a few pages, but is never defined.

1. If I've got that right, though, then the name is no good; these are neither ratios (and we do care about something similar that does consist of ratios, i.e. what I'm calling "The Set" here for now, but it needs a better name), nor are they necessarily "iso" (which means "the same", which does apply for isoharmonic and isodifferential chords).
2. By the way, on the Otonality and utonality page, it says, "All chords with isoratios that can be reduced to 1:1, 1:1:1, 1:1:1:1 etc., are otonal", but what does it mean for a chord to have isoratios, or to be reduced to 1:1 or 1:1:1 etc., or for isoratios to be reduced to that, if that's the way that sentence is meant to be parsed?
3. As for an actual name for what I've put "The Set" as a placeholder here, maybe just call these the "delta ratios" (which I think avoids implying that the ratios have integer elements and thus the chords are possibly delta-rational)? I've asked ChatGPT if there's any existing mathematical terminology we could leverage for ratios between deltas of ratios, but it can't think of anything.
4. Also, am I correct in assuming that The Set includes all ratios between all elements, in a similar way to how a 4:5:6:7 chord not only contains 4:5, 5:6, and 6:7, but also 4:6, 5:7, and 4:7? If so, then 4:5:(5+e):(7+e) is a DR, because its delta signature is +1+e+2 and therefore its The Set is { e/2, 2, e }, and since The Set contains a rational number, it's DR. However, if we only consider ratios between neighboring deltas, then its The Set is only { e/2, e }, which contains no rationals, and therefore it's not DR.

** Isodifferential chords have many alternative names: linear, equal-hertz, equal-beating, and proportional-beating chords. Some of these terms redirect to the DR page, others to the isoharmonic chord page, and some have no page. I recommend we consolidate the isoharmonic and DR pages and do a better job of breaking down these different classifications, such as per a table like this one. Then, in the Types of chords section of the Chord page, we should replace "isoharmonic" and "linear" with "DR".

Once this topic is cleaned up, I will also update the relevant footnote in D&D's Guide to use refined terminology (i.e. not isoratio or linear chord), and since the DR & RTT section of the DR page seems to give a good explanation for optimizing, at this point it'd be better to link directly there.

Sorry if I've made this more complicated than it needs to be. I'm just struggling to make sense of the materials that are currently sparse and/or spread across several pages at this time. (Note: I have revised the above table for accuracy and more completeness.)

--Cmloegcmluin (talk) 20:27, 18 February 2024 (UTC)

Okay, I know that no discussion has occurred here, but a bunch of discussion occurred on the #wiki channel of the Discord, so I went ahead and took my best shot at cleaning up and refactoring these materials. I ran out of time to deal with the DR and RTT section, where I'd like to provide much more detail on how to derive those results, since it took days of back-and-forth with Inthar for me to be able to make sense of it personally, and I'm concerned that by leaving off at the very simple case of triads, we're leaving a lot of important questions unanswered and important conceptual distinctions hidden.

Feedback welcome, or feel free to revise/undo as you see fit. --Cmloegcmluin (talk) 03:46, 21 February 2024 (UTC)