User:Astaryuu/15edo Notes/Notes and chords - Revision history

Astaryuu: /* Inside you are two fifths (and two tritones) */

2026-08-27T19:57:54Z

Inside you are two fifths (and two tritones)

← Older revision		Revision as of 19:57, 27 August 2026
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	This means we end up calling 9\15 the major fifth, 8\15 the minor fifth and major tritone, 7\15 the minor tritone, and 2\15 the perfect second. (Note that in 12edo, the tritone is probably best described as neutral, like the neutral second described earlier.) After all, our "neutral second" isn't really neutral either. Usually neutral seconds are defined as 150¢ specifically. Thirds and fourths are still somewhat normal relative to 12edo, so they don't flip even though there are audible differences.		This means we end up calling 9\15 the major fifth, 8\15 the minor fifth and major tritone, 7\15 the minor tritone, and 2\15 the perfect second. (Note that in 12edo, the tritone is probably best described as neutral, like the neutral second described earlier.) After all, our "neutral second" isn't really neutral either. Usually neutral seconds are defined as 150¢ specifically. Thirds and fourths are still somewhat normal relative to 12edo, so they don't flip even though there are audible differences.

	Of course, chords are named after the third, so we can't just call them major and minor based on which fifth they use. Instead, we call chords built using the "major" fifth natural, while ones built with the "minor" wolf fifth are "wolf." For example, 0-5-9 is the major chord, while 0-5-8 is the wolfmajor chord. Since the wolf fifth is a tritone, wolf chords can be used for similar tritone-y stuff as the 12edo tritone - but note that neither this tritone nor the previous one are as dissonant as the 12edo tritone. Even edos perfectly hit the ~~1 to square root~~ of 2 ~~interval that~~ defines the tritone, while odd edos can't.		Of course, chords are named after the third, so we can't just call them major and minor based on which fifth they use. Instead, we call chords built using the "major" fifth natural, while ones built with the "minor" wolf fifth are "wolf." For example, 0-5-9 is the major chord, while 0-5-8 is the wolfmajor chord. Since the wolf fifth is a tritone, wolf chords can be used for similar tritone-y stuff as the 12edo tritone - but note that neither this tritone nor the previous one are as dissonant as the 12edo tritone. Even edos perfectly hit the midpoint of [[2/1]], which defines the tritone, while odd edos can't do that.

	==Sixths and sevenths==		==Sixths and sevenths==

Astaryuu: /* Inside you are two fifths (and two tritones) */

2026-08-27T19:54:59Z

Inside you are two fifths (and two tritones)

← Older revision		Revision as of 19:54, 27 August 2026
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	==Inside you are two fifths (and two tritones)==		==Inside you are two fifths (and two tritones)==
	15edo... doesn't really have a fifthy fifth. 12edo's fifth sounds the way it does because it's an extremely good estimation of a so-called "perfect fifth". The best fifth in 15edo is 9\15, but that sounds notably sharp. Again, you can probably get used to it, but it'll mean 12edo strategies might sound a bit off in it. That's not what it means to be "wolf", though.		15edo... doesn't really have a fifthy fifth. 12edo's fifth sounds the way it does because it's an extremely good estimation of a so-called "perfect fifth" (in technical terms, it's 1.9¢ flat of [[3/2]], the second-smallest numerator and denominator you can put in an interval non-trivially). The best fifth in 15edo is 9\15, but that sounds notably sharp. Again, you can probably get used to it, but it'll mean 12edo strategies might sound a bit off in it. That's not what it means to be "wolf", though.

	The actual wolf fifth, is the valentine semitone ''below'' that 720¢ interval. At 640¢, meet 8\15, the wolf fifth. You could think of this as four neutral seconds stacked on top of each other, but at that point the margin between the neutral second and 2\15 goes beyond the diesis. This is actually also a tritone. Since 15 is odd, there is no true position halfway between octaves - but 7\15 and 8\15 bracket that midpoint.		The actual wolf fifth, is the valentine semitone ''below'' that 720¢ interval. At 640¢, meet 8\15, the wolf fifth. You could think of this as four neutral seconds stacked on top of each other, but at that point the margin between the neutral second and 2\15 goes beyond the diesis. This is actually also a tritone. Since 15 is odd, there is no true position halfway between octaves - but 7\15 and 8\15 bracket that midpoint.

Astaryuu: /* The semitone */

2026-08-27T19:51:21Z

The semitone

← Older revision		Revision as of 19:51, 27 August 2026
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	==The semitone==		==The semitone==
	In 12edo, an "edostep" is 100 cents exactly - this is the definition of a cent, in fact. However, 15edo's edostep has ~~edosteps~~ of 80¢. This is initially noticeably distinct from the semitone, which in standard harmony is usually defined as 1\12 exactly. Note the notation: two numbers on either side of a back slash, in the form X\Y, mean X steps of Y, so 1\12 is one step in 12edo.		In 12edo, an "edostep" is 100 cents exactly - this is the definition of a cent, in fact. However, 15edo's edostep has a size of 80¢. This is initially noticeably distinct from the semitone, which in standard harmony is usually defined as 1\12 exactly. Note the notation: two numbers on either side of a back slash, in the form X\Y, mean X steps of Y, so 1\12 is one step in 12edo.

	While it's not quite the 100¢ "standard" semitone size, 1\15 is similar enough to the 12edo semitone that one can get used to it as the new semitone, especially if used in the right context. It's also a close approximation of the [[valentine]] semitone, which is why 15edo's chromatic scale is also called "valentine".		While it's not quite the 100¢ "standard" semitone size, 1\15 is similar enough to the 12edo semitone that one can get used to it as the new semitone, especially if used in the right context. It's also a close approximation of the [[valentine]] semitone, which is why 15edo's chromatic scale is also called "valentine".

Astaryuu: /* Sixths and sevenths */

2026-02-08T05:39:23Z

Sixths and sevenths

← Older revision		Revision as of 05:39, 8 February 2026
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	12\15 is in a weird case; I tend to call it the "wolf seventh" though, as it has the same interval with the wolf fifth (8\15) that the wolf fifth has with the minor third (4\15). However, it is "technically" a harmonic seventh, and forms harmonic seventh chords.		12\15 is in a weird case; I tend to call it the "wolf seventh" though, as it has the same interval with the wolf fifth (8\15) that the wolf fifth has with the minor third (4\15). However, it is "technically" a harmonic seventh, and forms harmonic seventh chords.

			==Putting it all together==
			The following is a table of chords that I have created in messing around with 15edo. They are all justified on 0, so not every scale is going to have these. A whole separate page can go into what chords naturally occur in each mode of a given scale.

			{\| class="wikitable sortable"
			\|-
			! Contents
			! Name
			! Other names
			! Notes
			\|-
			\| 0 2 8
			\| wolfsuspended-2
			\|
			\|
			\|-
			\| 0 3 8
			\| wolfminor
			\|
			\| I offset wolf-fifth chords by one compared to natural-fifth chords.
			\|-
			\| 0 4 8
			\| wolfmajor
			\| diminished
			\| In a minor piece, this will sound diminished.
			\|-
			\| 0 5 8
			\| wolfsupermajor
			\| wolfsuspended-4
			\|
			\|-
			\| 0 3 9
			\| subminor
			\| suspended-2
			\| With a proper 9\15 interval, 3\15 can appear either to be a wolf third or a major second.
			\|-
			\| 0 4 9
			\| minor
			\| \|\|
			\|-
			\| 0 5 9
			\| major
			\| \|\|
			\|-
			\| 0 6 9
			\| suspended-4
			\| \|\|
			\|-
			\| 0 5 10
			\| augmented
			\| \|\|
			\|-
			\| 0 4 8 12
			\| wolfmajor 7th
			\| diminished 7th
			\| See also the original 0-4-8 chord.
			\|-
			\| 0 4 9 13
			\| minor 7th
			\| \|\|
			\|-
			\| 0 5 9 13
			\| dominant 7th
			\| \|\|
			\|-
			\| 0 5 9 14
			\| major 7th
			\| \|\|
			\|-
			\| 0 5 10 13
			\| augmented 7th
			\| \|\|
			\|-
			\| 0 5 10 14
			\| augmented major 7th
			\| \|\|
			\|}

Astaryuu: /* Sixths and sevenths */

2026-02-06T18:13:55Z

Sixths and sevenths

← Older revision		Revision as of 18:13, 6 February 2026
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	13\15 and 14\15 are seventh-like, with 13\15 as a minor 7th and 14\15 as a major 7th. This also means that if it exists, 13\15 will likely be your guy for seventh chords, as dominant sevenths are the default.		13\15 and 14\15 are seventh-like, with 13\15 as a minor 7th and 14\15 as a major 7th. This also means that if it exists, 13\15 will likely be your guy for seventh chords, as dominant sevenths are the default.

	12\15 is in a weird case; I tend to call it the "wolf seventh" though, as it has the same interval with the wolf fifth (8\15) that the wolf fifth has with the minor third (4\15).		12\15 is in a weird case; I tend to call it the "wolf seventh" though, as it has the same interval with the wolf fifth (8\15) that the wolf fifth has with the minor third (4\15). However, it is "technically" a harmonic seventh, and forms harmonic seventh chords.

	~~Technically, in chords, any seventh can serve as the new note on top of a three-note chord~~. However, ~~I think~~ it ~~best to, if possible, stick to the wolf~~ seventh ~~for wolf chords~~, and ~~the minor~~ seventh ~~for non-wolf~~ chords.

Astaryuu: /* Inside you are two fifths (and two tritones) */

2026-02-06T18:08:34Z

Inside you are two fifths (and two tritones)

← Older revision		Revision as of 18:08, 6 February 2026
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	Of course, chords are named after the third, so we can't just call them major and minor based on which fifth they use. Instead, we call chords built using the "major" fifth natural, while ones built with the "minor" wolf fifth are "wolf." For example, 0-5-9 is the major chord, while 0-5-8 is the wolfmajor chord. Since the wolf fifth is a tritone, wolf chords can be used for similar tritone-y stuff as the 12edo tritone - but note that neither this tritone nor the previous one are as dissonant as the 12edo tritone. Even edos perfectly hit the 1 to square root of 2 interval that defines the tritone, while odd edos can't.		Of course, chords are named after the third, so we can't just call them major and minor based on which fifth they use. Instead, we call chords built using the "major" fifth natural, while ones built with the "minor" wolf fifth are "wolf." For example, 0-5-9 is the major chord, while 0-5-8 is the wolfmajor chord. Since the wolf fifth is a tritone, wolf chords can be used for similar tritone-y stuff as the 12edo tritone - but note that neither this tritone nor the previous one are as dissonant as the 12edo tritone. Even edos perfectly hit the 1 to square root of 2 interval that defines the tritone, while odd edos can't.

			==Sixths and sevenths==
			In this system we have two sixths: 10\15 and 11\15. The former makes the augmented chord, 0-5-10, while if it exists in the scale I'm using, I prefer 11\15 for those spicy 6th chords you may have heard about.

			13\15 and 14\15 are seventh-like, with 13\15 as a minor 7th and 14\15 as a major 7th. This also means that if it exists, 13\15 will likely be your guy for seventh chords, as dominant sevenths are the default.

			12\15 is in a weird case; I tend to call it the "wolf seventh" though, as it has the same interval with the wolf fifth (8\15) that the wolf fifth has with the minor third (4\15).

			Technically, in chords, any seventh can serve as the new note on top of a three-note chord. However, I think it best to, if possible, stick to the wolf seventh for wolf chords, and the minor seventh for non-wolf chords.

Astaryuu: /* Inside you are two fifths (and two tritones) */

2025-08-19T09:21:38Z

Inside you are two fifths (and two tritones)

← Older revision		Revision as of 09:21, 19 August 2025
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	It might honestly be better to treat the "wolf/perfect" fifths as "minor/major" - they are a semitone apart, like the major and minor second. Technically, this would make the neutral second a "perfect second" and the tritones "minor/major" too. All of these sound incredibly cursed to the 12edo brain, but that's kinda the point of xenharmony, so I'm just gonna do it.		It might honestly be better to treat the "wolf/perfect" fifths as "minor/major" - they are a semitone apart, like the major and minor second. Technically, this would make the neutral second a "perfect second" and the tritones "minor/major" too. All of these sound incredibly cursed to the 12edo brain, but that's kinda the point of xenharmony, so I'm just gonna do it.

	This means we end up calling 9\15 the major fifth, 8\15 the minor fifth and major tritone, 7\15 the minor tritone, and 2\15 the perfect second. After all, our "neutral second" isn't really neutral either. Usually neutral seconds are defined as 150¢ specifically. Thirds and fourths are still somewhat normal relative to 12edo, so they don't flip even though there are audible differences.		This means we end up calling 9\15 the major fifth, 8\15 the minor fifth and major tritone, 7\15 the minor tritone, and 2\15 the perfect second. (Note that in 12edo, the tritone is probably best described as neutral, like the neutral second described earlier.) After all, our "neutral second" isn't really neutral either. Usually neutral seconds are defined as 150¢ specifically. Thirds and fourths are still somewhat normal relative to 12edo, so they don't flip even though there are audible differences.

	Of course, chords are named after the third, so we can't just call them major and minor based on which fifth they use. Instead, we call chords built using the "major" fifth natural, while ones built with the "minor" wolf fifth are "wolf." For example, 0-5-9 is the major chord, while 0-5-8 is the wolfmajor chord. Since the wolf fifth is a tritone, wolf chords can be used for similar tritone-y stuff as the 12edo tritone - but note that neither this tritone nor the previous one are as dissonant as the 12edo tritone. Even edos perfectly hit the 1 to square root of 2 interval that defines the tritone, while odd edos can't.		Of course, chords are named after the third, so we can't just call them major and minor based on which fifth they use. Instead, we call chords built using the "major" fifth natural, while ones built with the "minor" wolf fifth are "wolf." For example, 0-5-9 is the major chord, while 0-5-8 is the wolfmajor chord. Since the wolf fifth is a tritone, wolf chords can be used for similar tritone-y stuff as the 12edo tritone - but note that neither this tritone nor the previous one are as dissonant as the 12edo tritone. Even edos perfectly hit the 1 to square root of 2 interval that defines the tritone, while odd edos can't.

Astaryuu: /* Inside you are two fifths (and two tritones) */

2025-08-19T09:20:24Z

Inside you are two fifths (and two tritones)

← Older revision		Revision as of 09:20, 19 August 2025
Line 31:		Line 31:
	This means we end up calling 9\15 the major fifth, 8\15 the minor fifth and major tritone, 7\15 the minor tritone, and 2\15 the perfect second. After all, our "neutral second" isn't really neutral either. Usually neutral seconds are defined as 150¢ specifically. Thirds and fourths are still somewhat normal relative to 12edo, so they don't flip even though there are audible differences.		This means we end up calling 9\15 the major fifth, 8\15 the minor fifth and major tritone, 7\15 the minor tritone, and 2\15 the perfect second. After all, our "neutral second" isn't really neutral either. Usually neutral seconds are defined as 150¢ specifically. Thirds and fourths are still somewhat normal relative to 12edo, so they don't flip even though there are audible differences.

	Of course, chords are named after the third, so we can't just call them major and minor based on which fifth they use. Instead, we call chords built using the "major" fifth natural, while ones built with the "minor" wolf fifth are "wolf." For example, 0-5-9 is the major chord, while 0-5-8 is the wolfmajor chord. Since the wolf fifth is a tritone, wolf chords can be used for similar tritone-y stuff as the 12edo tritone - but note that neither this tritone nor the previous one are as dissonant as the 12edo tritone.		Of course, chords are named after the third, so we can't just call them major and minor based on which fifth they use. Instead, we call chords built using the "major" fifth natural, while ones built with the "minor" wolf fifth are "wolf." For example, 0-5-9 is the major chord, while 0-5-8 is the wolfmajor chord. Since the wolf fifth is a tritone, wolf chords can be used for similar tritone-y stuff as the 12edo tritone - but note that neither this tritone nor the previous one are as dissonant as the 12edo tritone. Even edos perfectly hit the 1 to square root of 2 interval that defines the tritone, while odd edos can't.

Astaryuu: /* Inside you are two fifths (and two tritones) */

2025-08-19T09:17:49Z

Inside you are two fifths (and two tritones)

← Older revision		Revision as of 09:17, 19 August 2025
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	The actual wolf fifth, is the valentine semitone ''below'' that 720¢ interval. At 640¢, meet 8\15, the wolf fifth. You could think of this as four neutral seconds stacked on top of each other, but at that point the margin between the neutral second and 2\15 goes beyond the diesis. This is actually also a tritone. Since 15 is odd, there is no true position halfway between octaves - but 7\15 and 8\15 bracket that midpoint.		The actual wolf fifth, is the valentine semitone ''below'' that 720¢ interval. At 640¢, meet 8\15, the wolf fifth. You could think of this as four neutral seconds stacked on top of each other, but at that point the margin between the neutral second and 2\15 goes beyond the diesis. This is actually also a tritone. Since 15 is odd, there is no true position halfway between octaves - but 7\15 and 8\15 bracket that midpoint.

	It might honestly be better to treat the "wolf/perfect" fifths as "minor/major" - they are a semitone apart, like the major and minor second. Technically, this would make the neutral second a "perfect second" and the tritones "minor/major" too. All of these sound incredibly cursed to the 12edo brain, but that's kinda the point of xenharmony, so I'm just gonna do it~~. That also resolves our issue from before with how four truly neutral seconds stacked on top of each other is noticeably flat of 8\15 - we can just call those "perfect seconds" instead~~.		It might honestly be better to treat the "wolf/perfect" fifths as "minor/major" - they are a semitone apart, like the major and minor second. Technically, this would make the neutral second a "perfect second" and the tritones "minor/major" too. All of these sound incredibly cursed to the 12edo brain, but that's kinda the point of xenharmony, so I'm just gonna do it.

	~~Generally~~, chords built using the "~~perfect~~" fifth ~~are~~ natural, while ones built with the "wolf" fifth are "wolf." For example, 0-5-9 is the major chord, while 0-5-8 is the wolfmajor chord. Since the wolf fifth is a tritone, wolf chords can be used for similar tritone-y stuff as the 12edo tritone - but note that neither this tritone nor the previous one are as dissonant as the 12edo tritone.		This means we end up calling 9\15 the major fifth, 8\15 the minor fifth and major tritone, 7\15 the minor tritone, and 2\15 the perfect second. After all, our "neutral second" isn't really neutral either. Usually neutral seconds are defined as 150¢ specifically. Thirds and fourths are still somewhat normal relative to 12edo, so they don't flip even though there are audible differences.

			Of course, chords are named after the third, so we can't just call them major and minor based on which fifth they use. Instead, we call chords built using the "major" fifth natural, while ones built with the "minor" wolf fifth are "wolf." For example, 0-5-9 is the major chord, while 0-5-8 is the wolfmajor chord. Since the wolf fifth is a tritone, wolf chords can be used for similar tritone-y stuff as the 12edo tritone - but note that neither this tritone nor the previous one are as dissonant as the 12edo tritone.

Astaryuu: /* Inside you are two fifths (and two tritones) */

2025-07-10T10:11:41Z

Inside you are two fifths (and two tritones)

← Older revision		Revision as of 10:11, 10 July 2025
Line 23:		Line 23:

	==Inside you are two fifths (and two tritones)==		==Inside you are two fifths (and two tritones)==
	15edo... doesn't really have a fifthy fifth. 12edo's fifth sounds the way it does because it's an extremely good estimation of a so-called "perfect fifth". The best fifth is 9\15, but that sounds notably sharp. Again, you can probably get used to it, but it'll mean 12edo strategies might sound a bit off in it. That's not what it means to be "wolf", though.		15edo... doesn't really have a fifthy fifth. 12edo's fifth sounds the way it does because it's an extremely good estimation of a so-called "perfect fifth". The best fifth in 15edo is 9\15, but that sounds notably sharp. Again, you can probably get used to it, but it'll mean 12edo strategies might sound a bit off in it. That's not what it means to be "wolf", though.

	The actual wolf fifth, is the valentine semitone ''below'' that 720¢ interval. At 640¢, meet 8\15, the wolf fifth. You could think of this as four neutral seconds stacked on top of each other, but at that point the margin between the neutral second and 2\15 goes beyond the diesis. This is actually also a tritone. Since 15 is odd, there is no true position halfway between octaves - but 7\15 and 8\15 bracket that midpoint.		The actual wolf fifth, is the valentine semitone ''below'' that 720¢ interval. At 640¢, meet 8\15, the wolf fifth. You could think of this as four neutral seconds stacked on top of each other, but at that point the margin between the neutral second and 2\15 goes beyond the diesis. This is actually also a tritone. Since 15 is odd, there is no true position halfway between octaves - but 7\15 and 8\15 bracket that midpoint.