User:MasonGreen1/Naughty and nice harmonics - Revision history

Fredg999: Fredg999 moved page Naughty and nice harmonics to User:MasonGreen1/Naughty and nice harmonics without leaving a redirect: XW:NG

2025-08-21T20:56:10Z

Fredg999 moved page Naughty and nice harmonics to User:MasonGreen1/Naughty and nice harmonics without leaving a redirect: XW:NG

← Older revision	Revision as of 20:56, 21 August 2025
(No difference)

ArrowHead294 at 13:13, 10 April 2025

2025-04-10T13:13:12Z

← Older revision		Revision as of 13:13, 10 April 2025
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	A common assumption made by [[~~Just_intonation\|~~just intonation]] composers and theorists is that intervals that occur lower on the harmonic series are always more consonant than those involving higher harmonics.		A common assumption made by [[just intonation]] composers and theorists is that intervals that occur lower on the harmonic series are always more consonant than those involving higher harmonics.

	However, in practice, this may be an oversimplification. It might be better to view the decrease in consonance as a ''general'' trend, but not a constant one. In particular, some higher harmonics (particularly the 19th) seem to provide more consonant intervals than their position in the harmonic series might suggest, while lower harmonics (like the 11th) aren't so good. An analogy could be made with atomic nuclei; nuclei heavier than iron get less stable with increasing atomic number, but certain numbers of protons and neutrons are "magic" and yield more stable nuclei than the overall trend would suggest, while other numbers (such as 43 protons, corresponding with technetium) are unstable.		However, in practice, this may be an oversimplification. It might be better to view the decrease in consonance as a ''general'' trend, but not a constant one. In particular, some higher harmonics (particularly the 19th) seem to provide more consonant intervals than their position in the harmonic series might suggest, while lower harmonics (like the 11th) aren't so good. An analogy could be made with atomic nuclei; nuclei heavier than iron get less stable with increasing atomic number, but certain numbers of protons and neutrons are "magic" and yield more stable nuclei than the overall trend would suggest, while other numbers (such as 43 protons, corresponding with technetium) are unstable.
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	The concept of naughtiness and niceness is rather subjective, and some listeners and composers might disagree with my (Mason Green's) analysis and actually enjoy the sound of undecimal intervals. However, as a general rule, I would argue that		The concept of naughtiness and niceness is rather subjective, and some listeners and composers might disagree with my (Mason Green's) analysis and actually enjoy the sound of undecimal intervals. However, as a general rule, I would argue that

	~~<ul><li>~~Harmonics that can form strong triads or tetrads are "nice". The 16:19:24 triad sounds very good (it is close to the 12edo minor triad), 17 and 13 can also be used to build triads which sound reasonably good (14:17:21 and 10:13:15, respectively) but 11 cannot. Triads such as 8:11:12 have a lot of "tension" due to the small size of the 11:12.~~</li><li>~~Harmonics whose intervals are extremely close to one another are "naughty", since this results in higher harmonic entropy. 17 fails this test because 12:17 and 17:24 are extremely close to one another. 11 also runs into problems; 9:11 and 22:27 are very close to one another, as are 8:11 and 11:15. Meanwhile, 13 and 19 perform better in this regard; their intervals are spaced apart more evenly.~~</li></ul>~~		* Harmonics that can form strong triads or tetrads are "nice". The 16:19:24 triad sounds very good (it is close to the 12edo minor triad), 17 and 13 can also be used to build triads which sound reasonably good (14:17:21 and 10:13:15, respectively) but 11 cannot. Triads such as 8:11:12 have a lot of "tension" due to the small size of the 11:12.
			* Harmonics whose intervals are extremely close to one another are "naughty", since this results in higher harmonic entropy. 17 fails this test because 12:17 and 17:24 are extremely close to one another. 11 also runs into problems; 9:11 and 22:27 are very close to one another, as are 8:11 and 11:15. Meanwhile, 13 and 19 perform better in this regard; their intervals are spaced apart more evenly.

	Overall, while individual opinions may vary, it might be best to consider the 11th harmonic unusually "naughty" or discordant relative to its position, while the 19 is unusually "nice", and the 13 and 17 are somewhere in between.		Overall, while individual opinions may vary, it might be best to consider the 11th harmonic unusually "naughty" or discordant relative to its position, while the 19 is unusually "nice", and the 13 and 17 are somewhere in between.

	==Applications==		== Applications ==
			It is possible to derive a generalized measure of [[harmonic entropy]] by weighting certain harmonics more than others. For instance, if we have designated the 11th harmonic as "naughty" (and there are good reasons to do so as given above) we can recalculate the harmonic entropy by simply ignoring all possible matches that contain a factor of 11. This causes 12edo to appear even ''stronger'' as a tuning system, since it avoids the 11th harmonic. It could be that over time, listeners have gradually been doing this mentally; since Western music does not use the 11th harmonic, our brains can automatically rule it out, thereby improving the ease with which other intervals are identified. Conversely, this also means that when someone used to 12edo actually ''does'' hear an undecimal interval, they will perceive it as something very unsettling or discordant.
	It is possible to derive a generalized measure of [[~~Harmonic_Entropy\|~~harmonic entropy]] by weighting certain harmonics more than others. For instance, if we have designated the 11th harmonic as "naughty" (and there are good reasons to do so as given above) we can recalculate the harmonic entropy by simply ignoring all possible matches that contain a factor of 11. This causes 12edo to appear even ''stronger'' as a tuning system, since it avoids the 11th harmonic. It could be that over time, listeners have gradually been doing this mentally; since Western music does not use the 11th harmonic, our brains can automatically rule it out, thereby improving the ease with which other intervals are identified. Conversely, this also means that when someone used to 12edo actually ''does'' hear an undecimal interval, they will perceive it as something very unsettling or discordant.

	Because the 11 is naughty, damping or omitting the 11th harmonic partial from the sound of synthesized tones or physical instruments may result in a more pleasant timbre. Similarly, the 19th harmonic could be amplified louder than it would otherwise be.		Because the 11 is naughty, damping or omitting the 11th harmonic partial from the sound of synthesized tones or physical instruments may result in a more pleasant timbre. Similarly, the 19th harmonic could be amplified louder than it would otherwise be.

	It might be a good idea to use a modified [[~~The_Riemann_Zeta_Function_and_Tuning~~\|Z function]] when analyzing higher edos, removing the component corresponding to the 11th harmonic (thus making it a no-elevens Z function) while increasing the weighting of the component corresponding to 19. Using such a function even more firmly establishes 12edo's position as supreme among small edos, since 12edo matches the 19th harmonic very closely while avoiding the 11th.		It might be a good idea to use a modified [[the Riemann zeta function and tuning\|Z function]] when analyzing higher edos, removing the component corresponding to the 11th harmonic (thus making it a no-elevens Z function) while increasing the weighting of the component corresponding to 19. Using such a function even more firmly establishes 12edo's position as supreme among small edos, since 12edo matches the 19th harmonic very closely while avoiding the 11th.

	Among higher edos, the one that has the most to gain is [[53edo\|53edo]], which closely matches the 13 and 19, while matching the 17 less well and completely avoiding the 11. Even using the ''un''modified Z function, 53edo is already a zeta peak, integral, ''and'' gap edo (and it shares that rare distinction with 12edo). Using the modified Z-function improves its standing even further.		Among higher edos, the one that has the most to gain is [[53edo\|53edo]], which closely matches the 13 and 19, while matching the 17 less well and completely avoiding the 11. Even using the ''un''modified Z function, 53edo is already a zeta peak, integral, ''and'' gap edo (and it shares that rare distinction with 12edo). Using the modified Z-function improves its standing even further.
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	Of the other two "triple zeta edos" in this range (19edo and 31edo), 19edo fares fairly well, too; while it does not match the "nice" 19th or the 17th, it does match the 13th and avoids the "naughty" 11th. 31edo, on the other hand, does not look nearly as good when using the modified Z-function, since it closely matches the 11th "naughty" harmonic while not matching the 13th, 17th, or "nice" 19th nearly as well.		Of the other two "triple zeta edos" in this range (19edo and 31edo), 19edo fares fairly well, too; while it does not match the "nice" 19th or the 17th, it does match the 13th and avoids the "naughty" 11th. 31edo, on the other hand, does not look nearly as good when using the modified Z-function, since it closely matches the 11th "naughty" harmonic while not matching the 13th, 17th, or "nice" 19th nearly as well.

	Thus, if "naughtiness and niceness" of higher harmonics is taken into account, 53edo looks like one of the best candidates for extending 12edo. It sounds similar to 12edo in many respects (both tunings closely match the perfect fifth, and while 53edo is not a meantone tuning, it does closely approximate Pythagorean tuning, as does 12edo). 53edo does not contain 12edo as a subset, but [[~~60edo\|~~60edo]] (which is consistent as a no-elevens, no-seventeens 27 limit system) does, making that tuning a good option as well.		Thus, if "naughtiness and niceness" of higher harmonics is taken into account, 53edo looks like one of the best candidates for extending 12edo. It sounds similar to 12edo in many respects (both tunings closely match the perfect fifth, and while 53edo is not a meantone tuning, it does closely approximate Pythagorean tuning, as does 12edo). 53edo does not contain 12edo as a subset, but [[60edo]] (which is consistent as a no-elevens, no-seventeens 27 limit system) does, making that tuning a good option as well.

	==See also==		== See also ==
	* [[Shadow]]		* [[Shadow]]
	* [[Substitute harmonic]]s		* [[Substitute harmonic]]s

BudjarnLambeth: /* See also */

2024-12-16T02:35:44Z

BudjarnLambeth: Add see also

2024-12-16T02:35:28Z

Add see also

← Older revision		Revision as of 02:35, 16 December 2024
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	Thus, if "naughtiness and niceness" of higher harmonics is taken into account, 53edo looks like one of the best candidates for extending 12edo. It sounds similar to 12edo in many respects (both tunings closely match the perfect fifth, and while 53edo is not a meantone tuning, it does closely approximate Pythagorean tuning, as does 12edo). 53edo does not contain 12edo as a subset, but [[60edo\|60edo]] (which is consistent as a no-elevens, no-seventeens 27 limit system) does, making that tuning a good option as well.		Thus, if "naughtiness and niceness" of higher harmonics is taken into account, 53edo looks like one of the best candidates for extending 12edo. It sounds similar to 12edo in many respects (both tunings closely match the perfect fifth, and while 53edo is not a meantone tuning, it does closely approximate Pythagorean tuning, as does 12edo). 53edo does not contain 12edo as a subset, but [[60edo\|60edo]] (which is consistent as a no-elevens, no-seventeens 27 limit system) does, making that tuning a good option as well.

			==See also==
			* [[Shadow]]
			* [[Substitute harmonics]]

	[[Category:Just intonation]]		[[Category:Just intonation]]
	[[Category:Harmonic series]]		[[Category:Harmonic series]]

Fredg999 category edits: Removing from Category:Theory using Cat-a-lot

2023-04-23T21:45:45Z

Removing from Category:Theory using Cat-a-lot

← Older revision		Revision as of 21:45, 23 April 2023
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	[[Category:Just intonation]]		[[Category:Just intonation]]
	[[Category:Harmonic series]]		[[Category:Harmonic series]]
	~~[[Category:Theory]]~~

BudjarnLambeth: Categorised this uncategorised page

2023-04-23T20:19:30Z

Categorised this uncategorised page

← Older revision		Revision as of 20:19, 23 April 2023
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	Thus, if "naughtiness and niceness" of higher harmonics is taken into account, 53edo looks like one of the best candidates for extending 12edo. It sounds similar to 12edo in many respects (both tunings closely match the perfect fifth, and while 53edo is not a meantone tuning, it does closely approximate Pythagorean tuning, as does 12edo). 53edo does not contain 12edo as a subset, but [[60edo\|60edo]] (which is consistent as a no-elevens, no-seventeens 27 limit system) does, making that tuning a good option as well.		Thus, if "naughtiness and niceness" of higher harmonics is taken into account, 53edo looks like one of the best candidates for extending 12edo. It sounds similar to 12edo in many respects (both tunings closely match the perfect fifth, and while 53edo is not a meantone tuning, it does closely approximate Pythagorean tuning, as does 12edo). 53edo does not contain 12edo as a subset, but [[60edo\|60edo]] (which is consistent as a no-elevens, no-seventeens 27 limit system) does, making that tuning a good option as well.
			[[Category:Just intonation]]
			[[Category:Harmonic series]]
			[[Category:Theory]]

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

Show changes

Wikispaces>MasonGreen1: Imported revision 567517253 - Original comment:

2015-11-23T14:54:32Z

**Imported revision 567517253 - Original comment: **

← Older revision		Revision as of 14:54, 23 November 2015
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	<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:MasonGreen1\|MasonGreen1]] and made on <tt>2015-11-23 14:53:24 UTC</tt>.<br>		: This revision was by author [[User:MasonGreen1\|MasonGreen1]] and made on <tt>2015-11-23 14:54:32 UTC</tt>.<br>
	: The original revision id was <tt>~~567517025~~</tt>.<br>		: The original revision id was <tt>567517253</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>
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	It might be a good idea to use a modified [[The Riemann Zeta Function and Tuning\|Z function]] when analyzing higher edos, removing the component corresponding to the 11th harmonic (thus making it a no-elevens Z function) while increasing the weighting of the component corresponding to 19. Using such a function even more firmly establishes 12edo's position as supreme among small edos, since 12edo matches the 19th harmonic very closely while avoiding the 11th.		It might be a good idea to use a modified [[The Riemann Zeta Function and Tuning\|Z function]] when analyzing higher edos, removing the component corresponding to the 11th harmonic (thus making it a no-elevens Z function) while increasing the weighting of the component corresponding to 19. Using such a function even more firmly establishes 12edo's position as supreme among small edos, since 12edo matches the 19th harmonic very closely while avoiding the 11th.

	Among higher edos, the one that has the most to gain is [[53edo]], which closely matches the 13 and 19, while matching the 17 less well and completely avoiding the 11.		Among higher edos, the one that has the most to gain is [[53edo]], which closely matches the 13 and 19, while matching the 17 less well and completely avoiding the 11. Even using the //un//modified Z function, 53edo is already a zeta peak, integral, //and// gap edo (and it shares that rare distinction with 12edo). Using the modified Z-function improves its standing even further.

	~~Even using~~ the ~~//un//modified Z function, 53edo is already a~~ zeta ~~peak, integral, //and// gap edo~~ (and ~~it shares that rare distinction with 12edo~~)~~. Using the modified Z-function improves its standing even further.~~		Of the other two "triple zeta edos" in this range (19edo and 31edo), 19edo fares fairly well, too; while it does not match the "nice" 19th or the 17th, it does match the 13th and avoids the "naughty" 11th. 31edo, on the other hand, does not look nearly as good when using the modified Z-function, since it closely matches the 11th "naughty" harmonic while not matching the 13th, 17th, or "nice" 19th nearly as well.

	19edo fares fairly well, too; while it does not match the "nice" 19th or the 17th, it does match the 13th and avoids the "naughty" 11th. 31edo, on the other hand, does not look nearly as good when using the modified Z-function, since it closely matches the 11th "naughty" harmonic while not matching the 13th, 17th, or "nice" 19th nearly as well.

	Thus, if "naughtiness and niceness" of higher harmonics is taken into account, 53edo looks like one of the best candidates for extending 12edo. It sounds similar to 12edo in many respects (both tunings closely match the perfect fifth, and while 53edo is not a meantone tuning, it does closely approximate Pythagorean tuning, as does 12edo). 53edo does not contain 12edo as a subset, but [[60edo]] (which is consistent as a no-elevens, no-seventeens 27 limit system) does, making that tuning a good option as well.</pre></div>		Thus, if "naughtiness and niceness" of higher harmonics is taken into account, 53edo looks like one of the best candidates for extending 12edo. It sounds similar to 12edo in many respects (both tunings closely match the perfect fifth, and while 53edo is not a meantone tuning, it does closely approximate Pythagorean tuning, as does 12edo). 53edo does not contain 12edo as a subset, but [[60edo]] (which is consistent as a no-elevens, no-seventeens 27 limit system) does, making that tuning a good option as well.</pre></div>
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	It might be a good idea to use a modified <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning">Z function</a> when analyzing higher edos, removing the component corresponding to the 11th harmonic (thus making it a no-elevens Z function) while increasing the weighting of the component corresponding to 19. Using such a function even more firmly establishes 12edo's position as supreme among small edos, since 12edo matches the 19th harmonic very closely while avoiding the 11th.<br />		It might be a good idea to use a modified <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning">Z function</a> when analyzing higher edos, removing the component corresponding to the 11th harmonic (thus making it a no-elevens Z function) while increasing the weighting of the component corresponding to 19. Using such a function even more firmly establishes 12edo's position as supreme among small edos, since 12edo matches the 19th harmonic very closely while avoiding the 11th.<br />
	<br />		<br />
	Among higher edos, the one that has the most to gain is <a class="wiki_link" href="/53edo">53edo</a>, which closely matches the 13 and 19, while matching the 17 less well and completely avoiding the 11.~~<br />~~		Among higher edos, the one that has the most to gain is <a class="wiki_link" href="/53edo">53edo</a>, which closely matches the 13 and 19, while matching the 17 less well and completely avoiding the 11. Even using the <em>un</em>modified Z function, 53edo is already a zeta peak, integral, <em>and</em> gap edo (and it shares that rare distinction with 12edo). Using the modified Z-function improves its standing even further.<br />
	~~<br />~~
	Even using the <em>un</em>modified Z function, 53edo is already a zeta peak, integral, <em>and</em> gap edo (and it shares that rare distinction with 12edo). Using the modified Z-function improves its standing even further.<br />
	<br />		<br />
	19edo fares fairly well, too; while it does not match the &quot;nice&quot; 19th or the 17th, it does match the 13th and avoids the &quot;naughty&quot; 11th. 31edo, on the other hand, does not look nearly as good when using the modified Z-function, since it closely matches the 11th &quot;naughty&quot; harmonic while not matching the 13th, 17th, or &quot;nice&quot; 19th nearly as well.<br />		Of the other two &quot;triple zeta edos&quot; in this range (19edo and 31edo), 19edo fares fairly well, too; while it does not match the &quot;nice&quot; 19th or the 17th, it does match the 13th and avoids the &quot;naughty&quot; 11th. 31edo, on the other hand, does not look nearly as good when using the modified Z-function, since it closely matches the 11th &quot;naughty&quot; harmonic while not matching the 13th, 17th, or &quot;nice&quot; 19th nearly as well.<br />
	<br />		<br />
	Thus, if &quot;naughtiness and niceness&quot; of higher harmonics is taken into account, 53edo looks like one of the best candidates for extending 12edo. It sounds similar to 12edo in many respects (both tunings closely match the perfect fifth, and while 53edo is not a meantone tuning, it does closely approximate Pythagorean tuning, as does 12edo). 53edo does not contain 12edo as a subset, but <a class="wiki_link" href="/60edo">60edo</a> (which is consistent as a no-elevens, no-seventeens 27 limit system) does, making that tuning a good option as well.</body></html></pre></div>		Thus, if &quot;naughtiness and niceness&quot; of higher harmonics is taken into account, 53edo looks like one of the best candidates for extending 12edo. It sounds similar to 12edo in many respects (both tunings closely match the perfect fifth, and while 53edo is not a meantone tuning, it does closely approximate Pythagorean tuning, as does 12edo). 53edo does not contain 12edo as a subset, but <a class="wiki_link" href="/60edo">60edo</a> (which is consistent as a no-elevens, no-seventeens 27 limit system) does, making that tuning a good option as well.</body></html></pre></div>

Wikispaces>MasonGreen1: Imported revision 567517025 - Original comment:

2015-11-23T14:53:24Z

**Imported revision 567517025 - Original comment: **

Wikispaces>MasonGreen1: Imported revision 567516825 - Original comment:

2015-11-23T14:52:25Z

**Imported revision 567516825 - Original comment: **

← Older revision		Revision as of 14:53, 23 November 2015
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	<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:MasonGreen1\|MasonGreen1]] and made on <tt>2015-11-23 14:52:25 UTC</tt>.<br>		: This revision was by author [[User:MasonGreen1\|MasonGreen1]] and made on <tt>2015-11-23 14:53:24 UTC</tt>.<br>
	: The original revision id was <tt>~~567516825~~</tt>.<br>		: The original revision id was <tt>567517025</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>
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	Because the 11 is naughty, damping or omitting the 11th harmonic partial from the sound of synthesized tones or physical instruments may result in a more pleasant timbre. Similarly, the 19th harmonic could be amplified louder than it would otherwise be.		Because the 11 is naughty, damping or omitting the 11th harmonic partial from the sound of synthesized tones or physical instruments may result in a more pleasant timbre. Similarly, the 19th harmonic could be amplified louder than it would otherwise be.

	~~Similarly, it~~ might be a good idea to use a modified [[The Riemann Zeta Function and Tuning\|Z function]] when analyzing higher edos, removing the component corresponding to the 11th harmonic (thus making it a no-elevens Z function) while increasing the weighting of the component corresponding to 19. Using such a function even more firmly establishes 12edo's position as supreme among small edos, since 12edo matches the 19th harmonic very closely while avoiding the 11th.		It might be a good idea to use a modified [[The Riemann Zeta Function and Tuning\|Z function]] when analyzing higher edos, removing the component corresponding to the 11th harmonic (thus making it a no-elevens Z function) while increasing the weighting of the component corresponding to 19. Using such a function even more firmly establishes 12edo's position as supreme among small edos, since 12edo matches the 19th harmonic very closely while avoiding the 11th.

	Among higher edos, the one that has the most to gain is [[53edo]], which closely matches the 13 and 19, while matching the 17 less well and completely avoiding the 11.		Among higher edos, the one that has the most to gain is [[53edo]], which closely matches the 13 and 19, while matching the 17 less well and completely avoiding the 11.
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	Because the 11 is naughty, damping or omitting the 11th harmonic partial from the sound of synthesized tones or physical instruments may result in a more pleasant timbre. Similarly, the 19th harmonic could be amplified louder than it would otherwise be.<br />		Because the 11 is naughty, damping or omitting the 11th harmonic partial from the sound of synthesized tones or physical instruments may result in a more pleasant timbre. Similarly, the 19th harmonic could be amplified louder than it would otherwise be.<br />
	<br />		<br />
	~~Similarly, it~~ might be a good idea to use a modified <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning">Z function</a> when analyzing higher edos, removing the component corresponding to the 11th harmonic (thus making it a no-elevens Z function) while increasing the weighting of the component corresponding to 19. Using such a function even more firmly establishes 12edo's position as supreme among small edos, since 12edo matches the 19th harmonic very closely while avoiding the 11th.<br />		It might be a good idea to use a modified <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning">Z function</a> when analyzing higher edos, removing the component corresponding to the 11th harmonic (thus making it a no-elevens Z function) while increasing the weighting of the component corresponding to 19. Using such a function even more firmly establishes 12edo's position as supreme among small edos, since 12edo matches the 19th harmonic very closely while avoiding the 11th.<br />
	<br />		<br />
	Among higher edos, the one that has the most to gain is <a class="wiki_link" href="/53edo">53edo</a>, which closely matches the 13 and 19, while matching the 17 less well and completely avoiding the 11.<br />		Among higher edos, the one that has the most to gain is <a class="wiki_link" href="/53edo">53edo</a>, which closely matches the 13 and 19, while matching the 17 less well and completely avoiding the 11.<br />

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	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:MasonGreen1\|MasonGreen1]] and made on <tt>2015-11-23 14:51:05 UTC</tt>.<br>		: This revision was by author [[User:MasonGreen1\|MasonGreen1]] and made on <tt>2015-11-23 14:52:25 UTC</tt>.<br>
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	* Harmonics that can form strong triads or tetrads are "nice". The 16:19:24 triad sounds very good (it is close to the 12edo minor triad), 17 and 13 can also be used to build triads which sound reasonably good (14:17:21 and 10:13:15, respectively) but 11 cannot. Triads such as 8:11:12 have a lot of "tension" due to the small size of the 11:12.		* Harmonics that can form strong triads or tetrads are "nice". The 16:19:24 triad sounds very good (it is close to the 12edo minor triad), 17 and 13 can also be used to build triads which sound reasonably good (14:17:21 and 10:13:15, respectively) but 11 cannot. Triads such as 8:11:12 have a lot of "tension" due to the small size of the 11:12.
	* Harmonics whose intervals are extremely close to one another are "naughty", since this results in higher harmonic entropy. 17 fails this test because 12:17 and 17:24 are extremely close to one another. 11 also runs into problems; 9:11 and 22:27 are very close to one another, as are 8:11 and 11:15. Meanwhile, 13 and 19 perform better in this regard.		* Harmonics whose intervals are extremely close to one another are "naughty", since this results in higher harmonic entropy. 17 fails this test because 12:17 and 17:24 are extremely close to one another. 11 also runs into problems; 9:11 and 22:27 are very close to one another, as are 8:11 and 11:15. Meanwhile, 13 and 19 perform better in this regard; their intervals are spaced apart more evenly.

	Overall, while individual opinions may vary, it might be best to consider the 11th harmonic unusually "naughty" or discordant relative to its position, while the 19 is unusually "nice", and the 13 and 17 are somewhere in between.		Overall, while individual opinions may vary, it might be best to consider the 11th harmonic unusually "naughty" or discordant relative to its position, while the 19 is unusually "nice", and the 13 and 17 are somewhere in between.
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	The concept of naughtiness and niceness is rather subjective, and some listeners and composers might disagree with my (Mason Green's) analysis and actually enjoy the sound of undecimal intervals. However, as a general rule, I would argue that<br />		The concept of naughtiness and niceness is rather subjective, and some listeners and composers might disagree with my (Mason Green's) analysis and actually enjoy the sound of undecimal intervals. However, as a general rule, I would argue that<br />
	<br />		<br />
	<ul><li>Harmonics that can form strong triads or tetrads are &quot;nice&quot;. The 16:19:24 triad sounds very good (it is close to the 12edo minor triad), 17 and 13 can also be used to build triads which sound reasonably good (14:17:21 and 10:13:15, respectively) but 11 cannot. Triads such as 8:11:12 have a lot of &quot;tension&quot; due to the small size of the 11:12.</li><li>Harmonics whose intervals are extremely close to one another are &quot;naughty&quot;, since this results in higher harmonic entropy. 17 fails this test because 12:17 and 17:24 are extremely close to one another. 11 also runs into problems; 9:11 and 22:27 are very close to one another, as are 8:11 and 11:15. Meanwhile, 13 and 19 perform better in this regard.</li></ul><br />		<ul><li>Harmonics that can form strong triads or tetrads are &quot;nice&quot;. The 16:19:24 triad sounds very good (it is close to the 12edo minor triad), 17 and 13 can also be used to build triads which sound reasonably good (14:17:21 and 10:13:15, respectively) but 11 cannot. Triads such as 8:11:12 have a lot of &quot;tension&quot; due to the small size of the 11:12.</li><li>Harmonics whose intervals are extremely close to one another are &quot;naughty&quot;, since this results in higher harmonic entropy. 17 fails this test because 12:17 and 17:24 are extremely close to one another. 11 also runs into problems; 9:11 and 22:27 are very close to one another, as are 8:11 and 11:15. Meanwhile, 13 and 19 perform better in this regard; their intervals are spaced apart more evenly.</li></ul><br />
	Overall, while individual opinions may vary, it might be best to consider the 11th harmonic unusually &quot;naughty&quot; or discordant relative to its position, while the 19 is unusually &quot;nice&quot;, and the 13 and 17 are somewhere in between.<br />		Overall, while individual opinions may vary, it might be best to consider the 11th harmonic unusually &quot;naughty&quot; or discordant relative to its position, while the 19 is unusually &quot;nice&quot;, and the 13 and 17 are somewhere in between.<br />
	<br />		<br />