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 | "The Set" | ||||||||
| frequency ratio | are items all integers? | delta signature * | are items all integers? | are items all the same? | unique undirected ratios between the deltas | are items all integers? | ||||
| DR (delta-rational): at least one integer in The Set | FDR (fully delta-rational): all integers in The Set | JI (just intonation) but not isodifferential ** :
chord contains only integers (and therefore The Set contains only integers), and The Set does not ≠ {1} |
4:5:7:9 | yes, all | +1+2+1 | yes, all | no, not all | { 1, 2 } | yes | |
| isoharmonic
(JI and isodifferential): chord contains only integers (and therefore The Set contains only integers), and more specifically The Set = {1} |
class i | 4:5:6:7 | +1+1+1 | yes, all | { 1 } | |||||
| class ii | 3:5:7:9 | +2+2+2 | ||||||||
| class iii | 1:4:7:10 | +3+3+3 | ||||||||
| ... | ... | ... | ||||||||
| not JI, but isodifferential:
chord contains non-integers, and The Set = {1} |
1:ɸ:(2ɸ-1):(3ɸ-2) | no, not all | +(ɸ-1)+(ɸ-1)+(ɸ-1) | no, none | ||||||
| (incompletely) DR:
at least one integer in The Set, but not all integers (therefore the chord contains non-integers) |
4:5:τ:7:9 | +1+(τ-5)+(7-τ)+2 | no, at least two are, but not all | no, not all | { (7-τ)/(τ-5), 7-τ, τ-5, 2/(τ-5), 2, 2/(7-τ) } | no, but at least one | ||||
| not DR: no integers in The Set
(therefore the chord contains non-integers) |
4:5:τ:7 | +1+(τ-5)+(7-τ) | no, only one | no, none | { (7-τ)/(τ-5), 7-τ, τ-5 } | no, none | ||||
| 5:τ:7 | +(τ-5)+(7-τ) | no, none | { (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. 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"). 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?
** 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.