Talk:Delta-rational chord - Revision history

ArrowHead294: Update links and clear up cluttered table formatting

2024-11-02T18:37:19Z

Update links and clear up cluttered table formatting

Cmloegcmluin: /* Trying to sort this stuff out / clean it up */

2024-02-21T03:46:56Z

Trying to sort this stuff out / clean it up

← Older revision		Revision as of 03:46, 21 February 2024
Line 126:		Line 126:

	--[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 20:27, 18 February 2024 (UTC)		--[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk: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. --[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 03:46, 21 February 2024 (UTC)

Cmloegcmluin: /* Trying to sort this stuff out / clean it up */ revise table for accuracy and more completeness

2024-02-19T20:10:32Z

Trying to sort this stuff out / clean it up: revise table for accuracy and more completeness

Show changes

Cmloegcmluin: fix another typo

2024-02-19T17:22:17Z

fix another typo

← Older revision		Revision as of 17:22, 19 February 2024
Line 85:		Line 85:
	this term appears on a few pages, but is never defined.		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", which <i>does</i> apply for isoharmonic and isodifferential chords).		# 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 <i>does</i> apply for isoharmonic and isodifferential chords).
	# 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?		# 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?
	# 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-<i>rational</i>)? 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.		# 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-<i>rational</i>)? 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.
	# 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 an integer, it's DR. However, if we only consider ratios between neighboring deltas, then its The Set is only { e/2, e }, which contains no integers, and therefore it's not DR.		# 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 an integer, it's DR. However, if we only consider ratios between neighboring deltas, then its The Set is only { e/2, e }, which contains no integers, and therefore it's not DR.

Cmloegcmluin: /* Trying to sort this stuff out / clean it up */ fix typo

2024-02-19T16:39:29Z

Trying to sort this stuff out / clean it up: fix typo

← Older revision		Revision as of 16:39, 19 February 2024
Line 23:		Line 23:
	! colspan="2" rowspan="1" \|JI (just intonation) but <i>not</i> isodifferential ** :		! colspan="2" rowspan="1" \|JI (just intonation) but <i>not</i> isodifferential ** :
	chord contains only integers (and therefore The Set contains only integers), and The Set does not ≠ {1}		chord contains only integers (and therefore The Set contains only integers), and The Set does not ≠ {1}
	\|4:5:7:9		\|4:5:7:8
	\| rowspan="5" \|yes, all		\| rowspan="5" \|yes, all
	\| +1+2+1		\| +1+2+1

Cmloegcmluin: /* Trying to sort this stuff out / clean it up */

2024-02-18T20:27:49Z

Trying to sort this stuff out / clean it up

← Older revision		Revision as of 20:27, 18 February 2024
Line 83:		Line 83:

	<nowiki>*</nowiki> 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;		<nowiki>*</nowiki> 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?		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", which <i>does</i> apply for isoharmonic and isodifferential chords).
			# 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?
			# 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-<i>rational</i>)? 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.
			# 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 an integer, it's DR. However, if we only consider ratios between neighboring deltas, then its The Set is only { e/2, e }, which contains no integers, and therefore it's not DR.

	<nowiki>**</nowiki> 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 [[Chord#Types_of_chords\|Types of chords section of the Chord]] page, we should replace "isoharmonic" and "linear" with "DR".		<nowiki>**</nowiki> 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 [[Chord#Types_of_chords\|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 [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#cite_note-7\|relevant footnote in D&D's Guide]] to use refined terminology (i.e. not isoratio or linear chord), and since the [[Delta-rational_chord#DR_and_RTT\|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.		Once this topic is cleaned up, I will also update the [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#cite_note-7\|relevant footnote in D&D's Guide]] to use refined terminology (i.e. not isoratio or linear chord), and since the [[Delta-rational_chord#DR_and_RTT\|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.

			--[[User:Cmloegcmluin\|Cmloegcmluin]] ([[User talk:Cmloegcmluin\|talk]]) 20:27, 18 February 2024 (UTC)

Cmloegcmluin: /* Trying to sort this stuff out / clean it up */ new section

2024-02-18T10:43:06Z

Trying to sort this stuff out / clean it up: new section

New page

== 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:

{| class="wikitable"
! colspan="4" rowspan="3" |chord type
! colspan="7" rowspan="1" |illustrative examples
|-
! colspan="2" |actual chord
! colspan="3" rowspan="1" |deltas
! colspan="2" |"The Set"
|-
!frequency ratio
!are items all integers?
!delta signature *
!are items all integers?
!are items all the same?
!unique [[Undirected_value|undirected]] ratios between the deltas
!are items all integers?
|-
! colspan="1" rowspan="7" |DR (delta-rational): at least one integer in The Set
! colspan="1" rowspan="6" |FDR (fully delta-rational): all integers in The Set
! colspan="2" rowspan="1" |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
| rowspan="5" |yes, all
| +1+2+1
| colspan="1" rowspan="5" |yes, all
|no, not all
|{ 1, 2 }
| rowspan="6" |yes
|-
! colspan="1" rowspan="4" |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
| colspan="1" rowspan="5" |yes, all
| colspan="1" rowspan="5" |{ 1 }
|-
!class ii
|3:5:7:9
| +2+2+2
|-
!class iii
|1:4:7:10
| +3+3+3
|-
!...
|...
|...
|-
! colspan="2" rowspan="1" |not JI, but isodifferential:
chord contains non-integers, and The Set = {1}
|1:ɸ:(2ɸ-1):(3ɸ-2)
| rowspan="4" |no, not all
| +(ɸ-1)+(ɸ-1)+(ɸ-1)
|no, none
|-
! colspan="3" rowspan="1" |(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
| colspan="1" |no, not all
|{ (7-τ)/(τ-5), 7-τ, τ-5, 2/(τ-5), 2, 2/(7-τ) }
|no, but at least one
|-
! colspan="4" rowspan="2" |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
| rowspan="2" |no, none
|{ (7-τ)/(τ-5), 7-τ, τ-5 }
| rowspan="2" |no, none
|-
|5:τ:7
| +(τ-5)+(7-τ)
|no, none
|{ (7-τ)/(τ-5) }
|}

<nowiki>*</nowiki> 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?

<nowiki>**</nowiki> 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 [[Chord#Types_of_chords|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 [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_fundamentals#cite_note-7|relevant footnote in D&D's Guide]] to use refined terminology (i.e. not isoratio or linear chord), and since the [[Delta-rational_chord#DR_and_RTT|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.