Harmony of 23edo - Revision history

FloraC: More cleanup

2025-03-23T09:28:10Z

More cleanup

FloraC: +links; fix prescriptive tone

2025-03-23T09:20:08Z

+links; fix prescriptive tone

← Older revision		Revision as of 09:20, 23 March 2025
Line 1:		Line 1:
	If you take a look at the intervals of [[23edo]], you~~'ll~~ find that this system does not contain good representations of the ~~harmonics~~ 3, 5, 7, 11, or 13, which appear as central in most just intonation systems. Rather than ~~trivialize 23edo by calling~~ it "atonal" or "inharmonic", ~~you should~~ consider the higher-limit harmonies that could serve as useful sonorities, perhaps even ~~'consonances'~~, in the context of careful composition. 23edo contains intervals which approach very well the harmonics 9, 17, 21, 23, 33, 35, 55, 79 ~~&~~ 117. Let's compare the ~~cents~~ values to see how close 23edo intervals come to these harmonics (and other intervals):		If you take a look at the intervals of [[23edo]], you will find that this system does not contain good representations of the [[harmonic]]s [[3/1\|3]], [[5/1\|5]], [[7/1\|7]], [[11/1\|11]], or [[13/1\|13]], which appear as central in most [[just intonation]] systems. Rather than it ''atonal'' or ''inharmonic'', we could consider the higher-limit harmonies that could serve as useful sonorities, perhaps even [[consonance]]s, in the context of careful composition. 23edo contains intervals which approach very well the harmonics [[9/1\|9]], [[17/1\|17]], [[21/1\|21]], [[23/1\|23]], [[33/1\|33]], [[35/1\|35]], [[55/1\|55]], [[79/1\|79]] and [[117/1\|117]]. Let us compare the [[cent]]s values to see how close 23edo intervals come to these harmonics (and other intervals):

	{\| class="wikitable"		{\| class="wikitable"

Secretcoffee at 10:19, 19 July 2023

2023-07-19T10:19:12Z

← Older revision		Revision as of 10:19, 19 July 2023
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	\| \| '''none'''		\| \| '''none'''
	\|}		\|}
	You'll see that intervals of 23edo come within 5 cents of 9/8; 3 cents of 23/16 and 31/16; 2 cents of 33/32, 21/16, 35/32, & 55/32; & 1 cent of 17/16, 79/64, & 117/64. Due to the notable accuracy of 17/16, it also makes sense to treat the interval of 4 steps as 289/256 instead of 9/8, which has a distinct sound in spite of being only 6 cents sharper than 9/8. (<'''And let's also note the excellent representations of 14/11 and its inverse, 11/7!!! In fact they might be considered good enough that a chain of 23 such intervals would be a reasonable way to acoustically tune this temperament''' -- AKJ) Of course, it also has perfect unisons & octaves, by definition. This means we could potentially build a very strange (& slightly mistuned) harmonic chord which, reduced to within one octave, we could write as frequency ratios 64:66:68:70:72:79:84:92:110:117. I find this cluster a little hard to listen to, whether tuned to JI or 23edo, so I'd like to consider smaller chords, triads & tetrads, as a starting point.		You'll see that intervals of 23edo come within 5 cents of 9/8; 3 cents of 23/16 and 31/16; 2 cents of 33/32, 21/16, 35/32, & 55/32; & 1 cent of 17/16, 79/64, & 117/64. Due to the notable accuracy of 17/16, it also makes sense to treat the interval of 4 steps as 289/256 instead of 9/8, which has a distinct sound in spite of being only 6 cents sharper than 9/8. (<'''And let's also note the excellent representations of 14/11 and its inverse, 11/7!!! In fact they might be considered good enough that a chain of 23 such intervals would be a reasonable way to acoustically tune this temperament''' -- AKJ) Of course, it also has perfect unisons & octaves, by definition. This means we could potentially build a very strange (& slightly mistuned) harmonic chord which, reduced to within one octave, we could write as frequency ratios 64:66:68:70:72:79:84:92:110:117:124. I find this cluster a little hard to listen to, whether tuned to JI or 23edo, so I'd like to consider smaller chords, triads & tetrads, as a starting point.

	I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, & 23, excluding (at least for now) 33, 35, 55, 79, & 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.		I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, & 23, excluding (at least for now) 33, 35, 55, 79, & 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.

Secretcoffee at 10:18, 19 July 2023

2023-07-19T10:18:06Z

← Older revision		Revision as of 10:18, 19 July 2023
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	\| \| '''none'''		\| \| '''none'''
	\|}		\|}
	You'll see that intervals of 23edo come within 5 cents of 9/8; 3 cents of 23/16 and 31/16; 2 cents of 33/32, 21/16, 35/32, & 55/32; & 1 cent of 17/16, 79/64, & 117/64. Due to the notable accuracy of 17/16, it also makes sense to treat the interval of 4 steps as 289/256 instead of 9/8, which has a distinct sound in spite of being only 6 cents sharper than 9/8. (<'''And let's also note the excellent representations of 14/11 and its inverse, 11/7!!! In fact they might be considered good ~~enought~~ that a chain of 23 such intervals would be a reasonable way to acoustically tune this temperament''' -- AKJ) Of course, it also has perfect unisons & octaves, by definition. This means we could potentially build a very strange (& slightly mistuned) harmonic chord which, reduced to within one octave, we could write as frequency ratios 64:66:68:70:72:79:84:92:110:117. I find this cluster a little hard to listen to, whether tuned to JI or 23edo, so I'd like to consider smaller chords, triads & tetrads, as a starting point.		You'll see that intervals of 23edo come within 5 cents of 9/8; 3 cents of 23/16 and 31/16; 2 cents of 33/32, 21/16, 35/32, & 55/32; & 1 cent of 17/16, 79/64, & 117/64. Due to the notable accuracy of 17/16, it also makes sense to treat the interval of 4 steps as 289/256 instead of 9/8, which has a distinct sound in spite of being only 6 cents sharper than 9/8. (<'''And let's also note the excellent representations of 14/11 and its inverse, 11/7!!! In fact they might be considered good enough that a chain of 23 such intervals would be a reasonable way to acoustically tune this temperament''' -- AKJ) Of course, it also has perfect unisons & octaves, by definition. This means we could potentially build a very strange (& slightly mistuned) harmonic chord which, reduced to within one octave, we could write as frequency ratios 64:66:68:70:72:79:84:92:110:117. I find this cluster a little hard to listen to, whether tuned to JI or 23edo, so I'd like to consider smaller chords, triads & tetrads, as a starting point.

	I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, & 23, excluding (at least for now) 33, 35, 55, 79, & 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.		I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, & 23, excluding (at least for now) 33, 35, 55, 79, & 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.

Secretcoffee at 10:16, 19 July 2023

2023-07-19T10:16:23Z

← Older revision		Revision as of 10:16, 19 July 2023
Line 164:		Line 164:
	\| \| +0.607		\| \| +0.607
	\|-		\|-
	\| \| 22		\| \| '''22'''
	\| \| 9 (1v)		\| \| '''9 (1v)'''
	\| \| 1147.826		\| \| '''1147.826'''
	\| \| 64/33		\| \| '''31/16'''
	\| \| ~~1146~~.~~727~~		\| \| '''1145.036'''
	\| \| +1.~~099~~		\| \| '''+2.791'''
	\|-		\|-
	\| \| '''23·· (or 0)'''		\| \| '''23·· (or 0)'''
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	\| \| '''none'''		\| \| '''none'''
	\|}		\|}
	You'll see that intervals of 23edo come within 5 cents of 9/8; 3 cents of 23/16; 2 cents of 33/32, 21/16, 35/32, & 55/32; & 1 cent of 17/16, 79/64, & 117/64. (<'''And let's also note the excellent representations of 14/11 and its inverse, 11/7!!! In fact they might be considered good enought that a chain of 23 such intervals would be a reasonable way to acoustically tune this temperament''' -- AKJ) Of course, it also has perfect unisons & octaves, by definition. This means we could potentially build a very strange (& slightly mistuned) harmonic chord which, reduced to within one octave, we could write as frequency ratios 64:66:68:70:72:79:84:92:110:117. I find this cluster a little hard to listen to, whether tuned to JI or 23edo, so I'd like to consider smaller chords, triads & tetrads, as a starting point.		You'll see that intervals of 23edo come within 5 cents of 9/8; 3 cents of 23/16 and 31/16; 2 cents of 33/32, 21/16, 35/32, & 55/32; & 1 cent of 17/16, 79/64, & 117/64. Due to the notable accuracy of 17/16, it also makes sense to treat the interval of 4 steps as 289/256 instead of 9/8, which has a distinct sound in spite of being only 6 cents sharper than 9/8. (<'''And let's also note the excellent representations of 14/11 and its inverse, 11/7!!! In fact they might be considered good enought that a chain of 23 such intervals would be a reasonable way to acoustically tune this temperament''' -- AKJ) Of course, it also has perfect unisons & octaves, by definition. This means we could potentially build a very strange (& slightly mistuned) harmonic chord which, reduced to within one octave, we could write as frequency ratios 64:66:68:70:72:79:84:92:110:117. I find this cluster a little hard to listen to, whether tuned to JI or 23edo, so I'd like to consider smaller chords, triads & tetrads, as a starting point.

	I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, & 23, excluding (at least for now) 33, 35, 55, 79, & 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.		I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, & 23, excluding (at least for now) 33, 35, 55, 79, & 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.

Secretcoffee at 10:05, 19 July 2023

2023-07-19T10:05:59Z

← Older revision		Revision as of 10:05, 19 July 2023
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	\| \| [[Armodue]] note		\| \| [[Armodue]] note
	\| \| [[Cent]]s sizes		\| \| [[Cent]]s sizes
	\| \| ~~Nearest harmonic~~		\| \| Just interval
	\| \| Cents		\| \| Cents
	\| \| "Error"		\| \| Error
	\|-		\|-
	\| \| 0		\| \| 0

Fredg999: Misc. edits, categories

2022-01-22T06:06:51Z

Misc. edits, categories

← Older revision		Revision as of 06:06, 22 January 2022
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	If you take a look at the intervals of [[~~23edo\|~~23edo]], you'll find that this system does not contain good representations of the harmonics 3, 5, 7, 11, or 13, which appear as central in most just intonation systems. Rather than trivialize 23edo by calling it "atonal" or "inharmonic", you should consider the higher-limit harmonies that could serve as useful sonorities, perhaps even 'consonances', in the context of careful composition. ~~[[23edo\|~~23edo]] contains intervals which approach very well the harmonics 9, 17, 21, 23, 33, 35, 55, 79 & 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics (and other intervals):		If you take a look at the intervals of [[23edo]], you'll find that this system does not contain good representations of the harmonics 3, 5, 7, 11, or 13, which appear as central in most just intonation systems. Rather than trivialize 23edo by calling it "atonal" or "inharmonic", you should consider the higher-limit harmonies that could serve as useful sonorities, perhaps even 'consonances', in the context of careful composition. 23edo contains intervals which approach very well the harmonics 9, 17, 21, 23, 33, 35, 55, 79 & 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics (and other intervals):

	{\| class="wikitable"		{\| class="wikitable"
	\|-		\|-
	\| \| Degrees		\| \| Degrees
	\| \| Armodue note		\| \| [[Armodue]] note
	\| \| ~~Cents~~ sizes		\| \| [[Cent]]s sizes
	\| ~~\| [[nearest_harmonic~~\|Nearest ~~Harmonic]]~~		\| \| Nearest harmonic
	\| \| Cents		\| \| Cents
	\| \| "Error"		\| \| "Error"
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	Thus we produce ten triads, five tetrads, & one pentad, 16 chords, which, with their inversions (given), doubles to 32 chords. I've written then in a closed position (within one octave), & I recommend trying different voicings. Moving chord tones up & down by octaves, you can unmuddy a muddy chord.		Thus we produce ten triads, five tetrads, & one pentad, 16 chords, which, with their inversions (given), doubles to 32 chords. I've written then in a closed position (within one octave), & I recommend trying different voicings. Moving chord tones up & down by octaves, you can unmuddy a muddy chord.

	==Triads==		== Triads ==
	'''16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21)'''.		'''16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21)'''.

Line 193:		Line 193:
	18/17 (98.955, error: +5.393)		18/17 (98.955, error: +5.393)

	===16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21).===		=== 16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21). ===
	17/16 (104.955, error -0.607)		17/16 (104.955, error -0.607)

Line 200:		Line 200:
	21/17 (365.825, error: -0.608)		21/17 (365.825, error: -0.608)

	===16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21).===		=== 16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21). ===
	17/16 (104.955, error -0.607)		17/16 (104.955, error -0.607)

Line 207:		Line 207:
	23/17 (523.319, error: -1.578)		23/17 (523.319, error: -1.578)

	===16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19).===		=== 16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19). ===
	18/16 = 9/8 (203.910, error +4.786)		18/16 = 9/8 (203.910, error +4.786)

Line 214:		Line 214:
	21/18 = 7/6 (266.871, error: -6.001)		21/18 = 7/6 (266.871, error: -6.001)

	===16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19).===		=== 16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19). ===
	18/16 = 9/8 (203.910, error +4.786)		18/16 = 9/8 (203.910, error +4.786)

Line 221:		Line 221:
	23/18 (424.364, error: -6.973)		23/18 (424.364, error: -6.973)

	===16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14).===		=== 16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14). ===
	21/16 (470.781, error -1.216)		21/16 (470.781, error -1.216)

Line 228:		Line 228:
	23/21 (157.493, error: -0.971)		23/21 (157.493, error: -0.971)

	===17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21).===		=== 17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21). ===
	18/17 (98.955, error: +5.393)		18/17 (98.955, error: +5.393)

Line 235:		Line 235:
	21/18 = 7/6 (266.871, error: -6.001)		21/18 = 7/6 (266.871, error: -6.001)

	===17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21).===		=== 17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21). ===
	18/17 (98.955, error: +5.393)		18/17 (98.955, error: +5.393)

Line 242:		Line 242:
	23/18 (424.364, error: -6.973)		23/18 (424.364, error: -6.973)

	===17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16).===		=== 17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16). ===
	21/17 (365.825, error: -0.608)		21/17 (365.825, error: -0.608)

Line 249:		Line 249:
	23/21 (157.493, error: -0.971)		23/21 (157.493, error: -0.971)

	===18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18).===		=== 18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18). ===
	21/18 = 7/6 (266.871, error: -6.001)		21/18 = 7/6 (266.871, error: -6.001)

Line 256:		Line 256:
	23/21 (157.493, error: -0.971)		23/21 (157.493, error: -0.971)

	==Tetrads==		== Tetrads ==
	'''16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21)'''.		'''16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21)'''.

Line 271:		Line 271:
	21/18 = 7/6 (266.871, error: -6.001)		21/18 = 7/6 (266.871, error: -6.001)

	===16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21).===		=== 16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21). ===
	17/16 (104.955, error -0.607)		17/16 (104.955, error -0.607)

Line 284:		Line 284:
	23/18 (424.364, error: -6.973)		23/18 (424.364, error: -6.973)

	===16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21).===		=== 16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21). ===
	17/16 (104.955, error -0.607)		17/16 (104.955, error -0.607)

Line 297:		Line 297:
	23/21 (157.493, error: -0.971)		23/21 (157.493, error: -0.971)

	===16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19).===		=== 16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19). ===
	18/16 = 9/8 (203.910, error +4.786)		18/16 = 9/8 (203.910, error +4.786)

Line 310:		Line 310:
	23/21 (157.493, error: -0.971)		23/21 (157.493, error: -0.971)

	===17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21).===		=== 17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21). ===
	18/17 (98.955, error: +5.393)		18/17 (98.955, error: +5.393)

Line 323:		Line 323:
	23/21 (157.493, error: -0.971)		23/21 (157.493, error: -0.971)

	==~~Pentad~~==		== Pentads ==
	'''16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21)'''.		'''16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21)'''.

Line 345:		Line 345:

	23/21 (157.493, error: -0.971)		23/21 (157.493, error: -0.971)

	[[Category:23edo]]		[[Category:23edo]]
	[[Category:~~harmony~~]]		[[Category:Harmony]]
	[[Category:~~interval~~]]		[[Category:Pentad]]
	[[Category:~~pentad~~]]		[[Category:Tetrad]]
	[[Category:~~tetrad]]~~		[[Category:Triad]]
	~~[[Category:theory]]~~
	~~[[Category:triad~~]]

Keenan Pepper category edits: rm Category:edo

2020-12-04T23:21:14Z

rm Category:edo

← Older revision		Revision as of 23:21, 4 December 2020
Line 346:		Line 346:
	23/21 (157.493, error: -0.971)		23/21 (157.493, error: -0.971)
	[[Category:23edo]]		[[Category:23edo]]
	~~[[Category:edo]]~~
	[[Category:harmony]]		[[Category:harmony]]
	[[Category:interval]]		[[Category:interval]]

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

Show changes

Wikispaces>keenanpepper: Imported revision 294270202 - Original comment:

2012-01-22T14:51:51Z

**Imported revision 294270202 - Original comment: **

← Older revision		Revision as of 14:51, 22 January 2012
Line 1:		Line 1:
	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
	: This revision was by author [[User:~~Osmiorisbendi~~\|~~Osmiorisbendi~~]] and made on <tt>2012-01-~~21 17~~:18:32 UTC</tt>.<br>		: This revision was by author [[User:keenanpepper\|keenanpepper]] and made on <tt>2012-01-22 14:51:51 UTC</tt>.<br>
	: The original revision id was <tt>~~294146236~~</tt>.<br>		: The original revision id was <tt>294270202</tt>.<br>
	: The revision comment was: <tt></tt><br>		: The revision comment was: <tt></tt><br>
	The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>		The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
Line 36:		Line 36:
	I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, & 23, excluding (at least for now) 33, 35, 55, 79, & 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.		I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, & 23, excluding (at least for now) 33, 35, 55, 79, & 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.

	Thus we produce ten triads, five tetrads, & one ~~quintad~~, 16 chords, which, with their inversions (given), doubles to 32 chords. I've written then in a closed position (within one octave), & I recommend trying different voicings. Moving chord tones up & down by octaves, you can unmuddy a muddy chord.		Thus we produce ten triads, five tetrads, & one pentad, 16 chords, which, with their inversions (given), doubles to 32 chords. I've written then in a closed position (within one octave), & I recommend trying different voicings. Moving chord tones up & down by octaves, you can unmuddy a muddy chord.

	==~~[[Triad]]s~~==		==Triads==
	16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21).		16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21).
	17/16 (104.955, error -0.607)		17/16 (104.955, error -0.607)
Line 90:		Line 90:


	==~~[[Tetrad]]s~~==		==Tetrads==
	16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21).		16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21).
	17/16 (104.955, error -0.607)		17/16 (104.955, error -0.607)
Line 132:		Line 132:


	==~~[[Quintad]]~~==		==Pentad==
	16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21).		16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21).
	17/16 (104.955, error -0.607)		17/16 (104.955, error -0.607)
Line 505:		Line 505:
	I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, &amp; 23, excluding (at least for now) 33, 35, 55, 79, &amp; 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.<br />		I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, &amp; 23, excluding (at least for now) 33, 35, 55, 79, &amp; 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.<br />
	<br />		<br />
	Thus we produce ten triads, five tetrads, &amp; one ~~quintad~~, 16 chords, which, with their inversions (given), doubles to 32 chords. I've written then in a closed position (within one octave), &amp; I recommend trying different voicings. Moving chord tones up &amp; down by octaves, you can unmuddy a muddy chord.<br />		Thus we produce ten triads, five tetrads, &amp; one pentad, 16 chords, which, with their inversions (given), doubles to 32 chords. I've written then in a closed position (within one octave), &amp; I recommend trying different voicings. Moving chord tones up &amp; down by octaves, you can unmuddy a muddy chord.<br />
	<br />		<br />
	<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Triads"></a><!-- ws:end:WikiTextHeadingRule:0 -->~~<a class="wiki_link" href="/Triad">Triad</a>s~~</h2>		<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Triads"></a><!-- ws:end:WikiTextHeadingRule:0 -->Triads</h2>
	<strong>16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21)</strong>.<br />		<strong>16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21)</strong>.<br />
	17/16 (104.955, error -0.607)<br />		17/16 (104.955, error -0.607)<br />
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	<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="x-Tetrads"></a><!-- ws:end:WikiTextHeadingRule:20 -->~~<a class="wiki_link" href="/Tetrad">Tetrad</a>s~~</h2>		<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="x-Tetrads"></a><!-- ws:end:WikiTextHeadingRule:20 -->Tetrads</h2>
	<strong>16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21)</strong>.<br />		<strong>16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21)</strong>.<br />
	17/16 (104.955, error -0.607)<br />		17/16 (104.955, error -0.607)<br />
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	<!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="x-~~Quintad~~"></a><!-- ws:end:WikiTextHeadingRule:30 -->~~<a class="wiki_link" href="/Quintad">Quintad</a>~~</h2>		<!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="x-Pentad"></a><!-- ws:end:WikiTextHeadingRule:30 -->Pentad</h2>
	<strong>16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21)</strong>.<br />		<strong>16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21)</strong>.<br />
	17/16 (104.955, error -0.607)<br />		17/16 (104.955, error -0.607)<br />

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	\| \| +0.607		\| \| +0.607
	\|-		\|-
	\| \| 22		\| \| '''22'''
	\| \| 9 (1v)		\| \| '''9 (1v)'''
	\| \| 1147.826		\| \| '''1147.826'''
	\| \| 64/33		\| \| '''31/16'''
	\| \| ~~1146~~.~~727~~		\| \| '''1145.036'''
	\| \| +1.~~099~~		\| \| '''+2.791'''
	\|-		\|-
	\| \| '''23·· (or 0)'''		\| \| '''23·· (or 0)'''
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	\| \| '''none'''		\| \| '''none'''
	\|}		\|}
	You'll see that intervals of 23edo come within 5 cents of 9/8; 3 cents of 23/16; 2 cents of 33/32, 21/16, 35/32, & 55/32; & 1 cent of 17/16, 79/64, & 117/64. (<'''And let's also note the excellent representations of 14/11 and its inverse, 11/7!!! In fact they might be considered good enought that a chain of 23 such intervals would be a reasonable way to acoustically tune this temperament''' -- AKJ) Of course, it also has perfect unisons & octaves, by definition. This means we could potentially build a very strange (& slightly mistuned) harmonic chord which, reduced to within one octave, we could write as frequency ratios 64:66:68:70:72:79:84:92:110:117. I find this cluster a little hard to listen to, whether tuned to JI or 23edo, so I'd like to consider smaller chords, triads & tetrads, as a starting point.		You'll see that intervals of 23edo come within 5 cents of 9/8; 3 cents of 23/16 and 31/16; 2 cents of 33/32, 21/16, 35/32, & 55/32; & 1 cent of 17/16, 79/64, & 117/64. Due to the notable accuracy of 17/16, it also makes sense to treat the interval of 4 steps as 289/256 instead of 9/8, which has a distinct sound in spite of being only 6 cents sharper than 9/8. (<'''And let's also note the excellent representations of 14/11 and its inverse, 11/7!!! In fact they might be considered good enought that a chain of 23 such intervals would be a reasonable way to acoustically tune this temperament''' -- AKJ) Of course, it also has perfect unisons & octaves, by definition. This means we could potentially build a very strange (& slightly mistuned) harmonic chord which, reduced to within one octave, we could write as frequency ratios 64:66:68:70:72:79:84:92:110:117. I find this cluster a little hard to listen to, whether tuned to JI or 23edo, so I'd like to consider smaller chords, triads & tetrads, as a starting point.

	I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, & 23, excluding (at least for now) 33, 35, 55, 79, & 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.		I'd also like to set an arbitrary limit on how high up the harmonic series we will go. I'll set my limit at the 23rd harmonic. I'll consider harmonics 1, 9, 17, 21, & 23, excluding (at least for now) 33, 35, 55, 79, & 117. Those sonorities could no doubt prove useful to a thoughful composer, but for this study, I'll leave them out.

← Older revision		Revision as of 10:05, 19 July 2023
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	\| \| [[Armodue]] note		\| \| [[Armodue]] note
	\| \| [[Cent]]s sizes		\| \| [[Cent]]s sizes
	\| \| ~~Nearest harmonic~~		\| \| Just interval
	\| \| Cents		\| \| Cents
	\| \| "Error"		\| \| Error
	\|-		\|-
	\| \| 0		\| \| 0