Chord complexity - Revision history

ArrowHead294 at 22:46, 20 January 2026

2026-01-20T22:46:32Z

ArrowHead294 at 22:28, 20 January 2026

2026-01-20T22:28:06Z

← Older revision		Revision as of 22:28, 20 January 2026
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	One possible way forward is to imagine that the incoming JI chord as a set of upper harmonics of some fundamental frequency—the GCD of the notes of the chord—and we want to quantify how strongly the chord matches that virtual fundamental. We can make some very basic assumptions:		One possible way forward is to imagine that the incoming JI chord as a set of upper harmonics of some fundamental frequency—the GCD of the notes of the chord—and we want to quantify how strongly the chord matches that virtual fundamental. We can make some very basic assumptions:

	1. Given some fundamental frequency ''f'', an ''N''-note chord built from very high harmonics of ''f'' will be a weaker match than an N-note chord built from ~~lower harmonics of~~ ''f''. In other words, 4:5:6 matches "1" better than 5:6:7. This is just a restatement of our definition of the simple complexity above.		1. Given some fundamental frequency ''f'', an ''N''-note chord built from very high harmonics of ''f'' will be a weaker match than an N-note chord built from ''f''{{`s}} lower harmonics. In other words, 4:5:6 matches "1" better than 5:6:7. This is just a restatement of our definition of the simple complexity above.

	2. Given some fundamental frequency ''f'' and a chord built from ''f''{{`s}} harmonics, adding ''another'' note from ''f''{{`s}} harmonics always ''increases'' the strength of the match to ''f''. In other words, 4:5:6:7 matches "1" better than 4:5:6.		2. Given some fundamental frequency ''f'' and a chord built from ''f''{{`s}} harmonics, adding ''another'' note from ''f''{{`s}} harmonics always ''increases'' the strength of the match to ''f''. In other words, 4:5:6:7 matches "1" better than 4:5:6.
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	The second proposition is the interesting one. It means that the chord 1:2 evokes "1" less than 1:2:3, which is less than 1:2:3:4, and so on, so that the chord 1:2:3:4:... evokes the frequency "1" most strongly.		The second proposition is the interesting one. It means that the chord 1:2 evokes "1" less than 1:2:3, which is less than 1:2:3:4, and so on, so that the chord 1:2:3:4:... evokes the frequency "1" most strongly.

	Strictly speaking, this phenomenon—the reinforcement of virtual pitch—is most evident if the notes of the chord are played with sine waves, with volume decreasing as you get higher into the harmonic series. In that situation, the chord 1:2:3:4:5:6:7:... is basically something like a sawtooth wave. It isn't quite so apparent that if you instead have all harmonics at equal volume, the resulting "delta comb" should really be viewed as more "consonant" than a sine wave in an absolute sense. This is even more true if, instead of sine waves, all of the notes are being played with some arbitrary harmonic timbre! Still, though, we still view the basic spirit of this as a "good-enough" rule of thumb which is simple enough to be worth modeling. (As we will see, we depart from strict adherence to this criterion anyway.)		Strictly speaking, this phenomenon—the reinforcement of virtual pitch—is most evident if the notes of the chord are played with sine waves, with volume decreasing as you get higher into the harmonic series. In that situation, the chord 1:2:3:4:5:6:7:... is basically something like a sawtooth wave. It isn't quite so apparent that if you instead have all harmonics at equal volume, the resulting "delta comb" should really be viewed as more "consonant" than a sine wave in an absolute sense. This is even more true if, instead of sine waves, all of the notes are being played with some arbitrary harmonic timbre! Still, though, we still view the basic spirit of this as a "good enough" rule of thumb which is simple enough to be worth modeling. (As we will see, we depart from strict adherence to this criterion anyway.)

	== Dirichlet complexity ==		== Dirichlet complexity ==

ArrowHead294 at 20:37, 20 January 2026

2026-01-20T20:37:40Z

Show changes

ArrowHead294: Update

2026-01-15T21:41:10Z

Update

← Older revision		Revision as of 21:41, 15 January 2026
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	1. Given some fundamental frequency ''f'', an ''N''-note chord built from very high harmonics of ''f'' will be a weaker match than an N-note chord built from lower harmonics of ''f''. In other words, 4:5:6 matches "1" better than 5:6:7. This is just a restatement of our definition of the simple complexity above.		1. Given some fundamental frequency ''f'', an ''N''-note chord built from very high harmonics of ''f'' will be a weaker match than an N-note chord built from lower harmonics of ''f''. In other words, 4:5:6 matches "1" better than 5:6:7. This is just a restatement of our definition of the simple complexity above.

	2. Given some fundamental frequency ''f'' and a chord built from ''f''{{-'}}s harmonics, adding ''another'' note from ''f''{{-'}}s harmonics always ''increases'' the strength of the match to ''f''. In other words, 4:5:6:7 matches "1" better than 4:5:6.		2. Given some fundamental frequency ''f'' and a chord built from ''f''{{`s}} harmonics, adding ''another'' note from ''f''{{`s}} harmonics always ''increases'' the strength of the match to ''f''. In other words, 4:5:6:7 matches "1" better than 4:5:6.

	The second proposition is the interesting one. It means that the chord 1:2 evokes "1" less than 1:2:3, which is less than 1:2:3:4, and so on, so that the chord 1:2:3:4:... evokes the frequency "1" most strongly.		The second proposition is the interesting one. It means that the chord 1:2 evokes "1" less than 1:2:3, which is less than 1:2:3:4, and so on, so that the chord 1:2:3:4:... evokes the frequency "1" most strongly.

Overthink: replaced deprecated template with nowiki

2025-12-02T22:56:27Z

replaced deprecated template with nowiki

← Older revision		Revision as of 22:56, 2 December 2025
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	When we are only looking at dyads made from harmonic sounds, many of the psychoacoustic qualities associated with consonance above simplify to the same basic metric, which is that they are strongest for dyads that are close to simple (numerically small) frequency ratios. In general, for some ratio ''n''/''d'', these qualities tend to decrease as ''n'' and ''d'' increase, unless ''n''/''d'' is a complex (numerically large) ratio that happens to also be very close to a simple ratio. In that situation, the perception of the complex ratio per se starts to be eclipsed by the perception of it as a slightly-detuned version of the nearby simpler ratio.		When we are only looking at dyads made from harmonic sounds, many of the psychoacoustic qualities associated with consonance above simplify to the same basic metric, which is that they are strongest for dyads that are close to simple (numerically small) frequency ratios. In general, for some ratio ''n''/''d'', these qualities tend to decrease as ''n'' and ''d'' increase, unless ''n''/''d'' is a complex (numerically large) ratio that happens to also be very close to a simple ratio. In that situation, the perception of the complex ratio per se starts to be eclipsed by the perception of it as a slightly-detuned version of the nearby simpler ratio.

	If we don't care about modeling the latter effect, and only care about modeling the complexity of a ratio directly, then for ''n''/''d'', any function of ''n'' and ''d'' that is monotonically increasing in either variable will do. The [[Height\|height]] functions on this Wiki are some simple examples of this. The two most commonly used are the [[Benedetti height]]/[[Tenney height]] of ''n''*''d'' and {{nowrap\|log(''n'' ~~{{asterisk}}~~ ''d'')}}, and the [[Weil height]] of {{nowrap\|max(''n'', ''d'')}} or {{nowrap\|log(max(''n'', ''d''))}}, which have the useful property that their logarithmic versions are norms on the space of [[monzos]] (in particular, the first is a type of L1 norm).		If we don't care about modeling the latter effect, and only care about modeling the complexity of a ratio directly, then for ''n''/''d'', any function of ''n'' and ''d'' that is monotonically increasing in either variable will do. The [[Height\|height]] functions on this Wiki are some simple examples of this. The two most commonly used are the [[Benedetti height]]/[[Tenney height]] of ''n''''d'' and {{nowrap\|log(''n''''d'')}}, and the [[Weil height]] of {{nowrap\|max(''n'', ''d'')}} or {{nowrap\|log(max(''n'', ''d''))}}, which have the useful property that their logarithmic versions are norms on the space of [[monzos]] (in particular, the first is a type of L1 norm).

	Note that the Benedetti height and Tenney height are basically the same thing; it is fairly common when talking about height functions to equivocate between the logarithmic and non-logarithmic versions of the same function, as they rank rationals the same either way. We will sometimes equivocate between the two names, but in general the name "Benedetti height" has been given to the non-logarithmic version and the name "Tenney height" to the logarithmic version.		Note that the Benedetti height and Tenney height are basically the same thing; it is fairly common when talking about height functions to equivocate between the logarithmic and non-logarithmic versions of the same function, as they rank rationals the same either way. We will sometimes equivocate between the two names, but in general the name "Benedetti height" has been given to the non-logarithmic version and the name "Tenney height" to the logarithmic version.

ArrowHead294: /* N-adic simple complexity/otonalness */

2025-01-20T14:43:49Z

N-adic simple complexity/otonalness

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	= ''N''-adic simple complexity/otonalness =		= ''N''-adic simple complexity/otonalness =
	Suppose that {{nowrap\|''x''<sub>1</sub> : ''x''<sub>2</sub> : ''x''<sub>3</sub> : ... : ''x''<sub>''N''</sub>}} is some ''N''-note JI chord. Then some function {{nowrap\|''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''N''</sub>}} can be said to measure the '''simple complexity''' or '''otonalness''' of our N-note chord if it is monotonically increasing in each variable separately. We will sometimes use the notation {{nowrap\|'''X''' {{=}} ''x''<sub>1</sub> : ''x''<sub>2</sub> : ''x''<sub>3</sub> : ... : ''x''<sub>''N''</sub>}}, so that our simple complexity function can be written as ''f''('''X''').		Suppose that {{nowrap\|''x''<sub>1</sub> : ''x''<sub>2</sub> : ''x''<sub>3</sub> : ... : ''x''<sub>''N''</sub>}} is some ''N''-note JI chord. Then some function {{nowrap\|''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''N''</sub>)}} can be said to measure the '''simple complexity''' or '''otonalness''' of our N-note chord if it is monotonically increasing in each variable separately. We will sometimes use the notation {{nowrap\|'''X''' {{=}} ''x''<sub>1</sub> : ''x''<sub>2</sub> : ''x''<sub>3</sub> : ... : ''x''<sub>''N''</sub>}}, so that our simple complexity function can be written as ''f''('''X''').

	We may also look at functions ''f'' which are defined for chords with a variable number of notes. This can also be thought of as a family of functions {{nowrap\|''f''<sub>1</sub>, ''f''<sub>2</sub>, ''f''<sub>3</sub>, ...}}, each giving the simple complexity for chords of size 1, 2, 3, etc.		We may also look at functions ''f'' which are defined for chords with a variable number of notes. This can also be thought of as a family of functions {{nowrap\|''f''<sub>1</sub>, ''f''<sub>2</sub>, ''f''<sub>3</sub>, ...}}, each giving the simple complexity for chords of size 1, 2, 3, etc.

ArrowHead294 at 16:36, 15 January 2025

2025-01-15T16:36:24Z

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	$$		$$

	Now, we note that {{nowrap\|log(''b''/''a'')}} can basically be thought of as a function of the span of the dyad. The span in cents would be <math>\text{cents}(b/a) = 1200\log_2(b/a) = 1200\log(b/a)/\log 2</math>, so we have <math>\log(b/a) = \text{cents}(b/a) \log(2)/1200</math>.<ref group="note">In fact, this can also be thought of as a representation of the span in terms of a different unit: rather than cents, we are using "nepers", where one "neper" is equal to {{nowrap\|1200 log<sub>2</sub>''e'' {{=}} 1731.234{{c}}}}, rather than the typical units of cents or octaves—perfectly legitimate, if not a bit strange, and used rather frequently in the writings of the late [[Martin Gough]].</ref> Thus, the above expression is a monotonic function purely in terms of the span. Putting it all together, we have		Now, we note that {{nowrap\|log(''b''/''a'')}} can basically be thought of as a function of the span of the dyad. The span in cents would be <math>\text{cents}(b/a) = 1200\log_2(b/a) = 1200\log(b/a)/\log 2</math>, so we have <math>\log(b/a) = \text{cents}(b/a) \log(2)/1200</math>.<ref group="note">In fact, this can also be thought of as a representation of the span in terms of a different unit: rather than cents, we are using "nepers", where one "neper" is equal to {{nowrap\|1200 log<sub>2</sub>''e'' {{=}} 1731.234{{cent}}}}, rather than the typical units of cents or octaves—perfectly legitimate, if not a bit strange, and used rather frequently in the writings of the late [[Martin Gough]].</ref> Thus, the above expression is a monotonic function purely in terms of the span. Putting it all together, we have

	$$		$$

ArrowHead294 at 23:53, 25 December 2024

2024-12-25T23:53:12Z

← Older revision		Revision as of 23:53, 25 December 2024
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	1. Given some fundamental frequency ''f'', an ''N''-note chord built from very high harmonics of ''f'' will be a weaker match than an N-note chord built from lower harmonics of ''f''. In other words, 4:5:6 matches "1" better than 5:6:7. This is just a restatement of our definition of the simple complexity above.		1. Given some fundamental frequency ''f'', an ''N''-note chord built from very high harmonics of ''f'' will be a weaker match than an N-note chord built from lower harmonics of ''f''. In other words, 4:5:6 matches "1" better than 5:6:7. This is just a restatement of our definition of the simple complexity above.

	2. Given some fundamental frequency ''f'' and a chord built from ''f''{{'}}s harmonics, adding ''another'' note from ''f''{{'}}s harmonics always ''increases'' the strength of the match to ''f''. In other words, 4:5:6:7 matches "1" better than 4:5:6.		2. Given some fundamental frequency ''f'' and a chord built from ''f''{{-'}}s harmonics, adding ''another'' note from ''f''{{-'}}s harmonics always ''increases'' the strength of the match to ''f''. In other words, 4:5:6:7 matches "1" better than 4:5:6.

	The second proposition is the interesting one. It means that the chord 1:2 evokes "1" less than 1:2:3, which is less than 1:2:3:4, and so on, so that the chord 1:2:3:4:... evokes the frequency "1" most strongly.		The second proposition is the interesting one. It means that the chord 1:2 evokes "1" less than 1:2:3, which is less than 1:2:3:4, and so on, so that the chord 1:2:3:4:... evokes the frequency "1" most strongly.

ArrowHead294: Get bold text to work within «pre» elements

2024-11-18T16:43:59Z

Get bold text to work within «pre» elements

← Older revision		Revision as of 16:43, 18 November 2024
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	Given all of this, it may be useful to see some examples. Let's look at the set of all chords of size 1-5 notes, with coefficients of at most 5, just to see what we get. Higher values on the list are "stronger" or "less complex". This is with the Benedetti Height with {{nowrap\|''s'' {{=}} 1}}:		Given all of this, it may be useful to see some examples. Let's look at the set of all chords of size 1-5 notes, with coefficients of at most 5, just to see what we get. Higher values on the list are "stronger" or "less complex". This is with the Benedetti Height with {{nowrap\|''s'' {{=}} 1}}:

	<pre>		<pre<includeonly />>
	'''Benedetti Height, s=1'''		'''Benedetti Height, s=1'''
	1:2:3:4:5 - 0.52103		1:2:3:4:5 - 0.52103
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	If we do this instead with the Weil height, we get something which looks slightly tidier at first glance, but which on further inspection has some strange features. Let's take a look:		If we do this instead with the Weil height, we get something which looks slightly tidier at first glance, but which on further inspection has some strange features. Let's take a look:

	<pre>		<pre<includeonly />>
	'''Weil Height, s=1'''		'''Weil Height, s=1'''
	1:2:3:4:5 - 1		1:2:3:4:5 - 1
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	We can always interpolate between the two as well; if we go with the Tenney-Weil height for {{nowrap\|''k'' {{=}} 0.5}}, and again {{nowrap\|''s'' {{=}} 1}}, we get something intermediate:		We can always interpolate between the two as well; if we go with the Tenney-Weil height for {{nowrap\|''k'' {{=}} 0.5}}, and again {{nowrap\|''s'' {{=}} 1}}, we get something intermediate:

	<pre>		<pre<includeonly />>
	'''Benedetti-Weil Height, k=0.5, s=1'''		'''Benedetti-Weil Height, k=0.5, s=1'''
	1:2:3:4:5 - 0.72183		1:2:3:4:5 - 0.72183
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	Let's focus on just the Tenney height, and look at how the value of ''s'' changes things. If we instead set {{nowrap\|''s'' {{=}} 0.5}}, we get the following:		Let's focus on just the Tenney height, and look at how the value of ''s'' changes things. If we instead set {{nowrap\|''s'' {{=}} 0.5}}, we get the following:

	<pre>		<pre<includeonly />>
	'''Benedetti height, s=0.5'''		'''Benedetti height, s=0.5'''
	1:2:3:4:5 - 0.10421		1:2:3:4:5 - 0.10421
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	Likewise, if we make s larger and move it towards infinity, the Tenney height tends toward just the geometric mean, without being divided by any normalizing term. Then we get this:		Likewise, if we make s larger and move it towards infinity, the Tenney height tends toward just the geometric mean, without being divided by any normalizing term. Then we get this:

	<pre>		<pre<includeonly />>
	'''Benedetti height, s=Inf'''		'''Benedetti height, s=Inf'''
	1 - 1		1 - 1
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	The viewpoint of this author is that the most sensible results are when we have {{nowrap\|''s'' {{=}} 1}}, with Tenney height preferred to Weil height. The all-around best seems to be the Tenney-Weil height with k somewhere near 0.5, although having {{nowrap\|''k'' {{=}} 0}} is probably good enough and slightly simpler to work with, particularly since it reduces to an L1 norm for dyads. We will do an extended listing of both, this time with at-most tetrads and with coefficients up to 7. First the Tenney height:		The viewpoint of this author is that the most sensible results are when we have {{nowrap\|''s'' {{=}} 1}}, with Tenney height preferred to Weil height. The all-around best seems to be the Tenney-Weil height with k somewhere near 0.5, although having {{nowrap\|''k'' {{=}} 0}} is probably good enough and slightly simpler to work with, particularly since it reduces to an L1 norm for dyads. We will do an extended listing of both, this time with at-most tetrads and with coefficients up to 7. First the Tenney height:

	<pre>		<pre<includeonly />>
	'''Benedetti height, s=1, tetrads with max-coefficient=7'''		'''Benedetti height, s=1, tetrads with max-coefficient=7'''
	1:2:3:4 - 0.55334		1:2:3:4 - 0.55334
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	The results seem reasonably sensible to me, although with a few little caveats here and there—we have 1:7 ranked above 4:5:6:7, partly because there is no notion of octave-equivalence involved, and partly because Tenney height may not be prioritizing small-span intervals quite enough. But this is at least ballpark-sensible. We can tweak it slightly by looking at the Tenney-Weil norm with {{nowrap\|''k'' {{=}} 0.5}} and {{nowrap\|''s'' {{=}} 1}}:		The results seem reasonably sensible to me, although with a few little caveats here and there—we have 1:7 ranked above 4:5:6:7, partly because there is no notion of octave-equivalence involved, and partly because Tenney height may not be prioritizing small-span intervals quite enough. But this is at least ballpark-sensible. We can tweak it slightly by looking at the Tenney-Weil norm with {{nowrap\|''k'' {{=}} 0.5}} and {{nowrap\|''s'' {{=}} 1}}:

	<pre>		<pre<includeonly />>
	'''Benedetti-Weil height, s=1, k=0.5, tetrads with max-coefficient=7'''		'''Benedetti-Weil height, s=1, k=0.5, tetrads with max-coefficient=7'''
	1:2:3:4 - 0.74387		1:2:3:4 - 0.74387

Mike Battaglia at 22:56, 4 November 2024

2024-11-04T22:56:10Z

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	Much of tuning theory involves looking at intervals. However, when intervals are combined into chords, they can sometimes form synergies that aren't immediately apparent if one is only focused on dyads. Fortunately, many of the metrics we use to evaluate intervals generalize very easily to larger chords, and we will look at some in this article.		Much of tuning theory involves looking at intervals. However, when intervals are combined into chords, they can sometimes form synergies that aren't immediately apparent if one is only focused on dyads. Fortunately, many of the metrics we use to evaluate intervals generalize very easily to larger chords, and we will look at some in this article.

	The terms dyad, triad, tetrad, etc are used here to refer to any collection of 2, 3, or 4 pitches, without any notion of octave equivalence and with doubling allowed. Thus we will refer to, for instance, 4:5:6:8 as a tetrad even though the "4" and the "8" are the same pitch class. We will ~~also~~ sometimes refer to the special dyad 1:1, triad 1:1:1, tetrad 1:1:1:1, and so on, even though only one pitch is used.		The terms dyad, triad, tetrad, etc are used here to refer to any collection of 2, 3, or 4 pitches, without any notion of octave equivalence and with doubling allowed. Thus we will refer to, for instance, 4:5:6:8 as a tetrad even though the "4" and the "8" are the same pitch class. We will even sometimes refer to the special dyad 1:1, triad 1:1:1, tetrad 1:1:1:1, and so on, when mathematically convenient, and treat them as such even though only one pitch is used.

	= Basics =		= Basics =

← Older revision		Revision as of 22:46, 20 January 2026
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	When we are only looking at dyads made from harmonic sounds, many of the psychoacoustic qualities associated with consonance above simplify to the same basic metric, which is that they are strongest for dyads that are close to simple (numerically small) frequency ratios. In general, for some ratio ''n''/''d'', these qualities tend to decrease as ''n'' and ''d'' increase, unless ''n''/''d'' is a complex (numerically large) ratio that happens to also be very close to a simple ratio. In that situation, the perception of the complex ratio per se starts to be eclipsed by the perception of it as a slightly-detuned version of the nearby simpler ratio.		When we are only looking at dyads made from harmonic sounds, many of the psychoacoustic qualities associated with consonance above simplify to the same basic metric, which is that they are strongest for dyads that are close to simple (numerically small) frequency ratios. In general, for some ratio ''n''/''d'', these qualities tend to decrease as ''n'' and ''d'' increase, unless ''n''/''d'' is a complex (numerically large) ratio that happens to also be very close to a simple ratio. In that situation, the perception of the complex ratio per se starts to be eclipsed by the perception of it as a slightly-detuned version of the nearby simpler ratio.

	If we don't care about ~~modeling~~ the latter effect, and only care about ~~modeling~~ the complexity of a ratio directly, then for ''n''/''d'', any function of ''n'' and ''d'' that is monotonically increasing in either variable will do. The [[Height\|height]] functions on this Wiki are some simple examples of this. The two most commonly used are the [[Benedetti height]]/[[Tenney height]] of ''n''''d'' and {{nowrap\|log(''n''''d'')}}, and the [[Weil height]] of {{nowrap\|max(''n'', ''d'')}} or {{nowrap\|log(max(''n'', ''d''))}}, which have the useful property that their logarithmic versions are norms on the space of [[monzos]] (in particular, the first is a type of L1 norm).		If we don't care about modelling the latter effect, and only care about modelling the complexity of a ratio directly, then for ''n''/''d'', any function of ''n'' and ''d'' that is monotonically increasing in either variable will do. The [[Height\|height]] functions on this Wiki are some simple examples of this. The two most commonly used are the [[Benedetti height]]/[[Tenney height]] of ''n''''d'' and {{nowrap\|log(''n''''d'')}}, and the [[Weil height]] of {{nowrap\|max(''n'', ''d'')}} or {{nowrap\|log(max(''n'', ''d''))}}, which have the useful property that their logarithmic versions are norms on the space of [[monzos]] (in particular, the first is a type of L1 norm).

	Note that the Benedetti height and Tenney height are basically the same thing; it is fairly common when talking about height functions to equivocate between the logarithmic and non-logarithmic versions of the same function, as they rank rationals the same either way. We will sometimes equivocate between the two names, but in general the name "Benedetti height" has been given to the non-logarithmic version and the name "Tenney height" to the logarithmic version.		Note that the Benedetti height and Tenney height are basically the same thing; it is fairly common when talking about height functions to equivocate between the logarithmic and non-logarithmic versions of the same function, as they rank rationals the same either way. We will sometimes equivocate between the two names, but in general the name "Benedetti height" has been given to the non-logarithmic version and the name "Tenney height" to the logarithmic version.

	If we do care about ~~modeling~~ the aforementioned detuning effect, then [[Harmonic entropy]] is one way to model this, which has a free parameter determining how "tolerant" the listener's auditory system is to perceiving slightly detuned versions of simple ratios as slightly-off versions of themselves, rather than perceiving them as other, more complex ratios. Tenney and Weil height can also be used to seed the Harmonic entropy calculation to begin with, so that they can be thought of as "primitives" from which increasingly sophisticated models can be built.		If we do care about modelling the aforementioned detuning effect, then [[Harmonic entropy]] is one way to model this, which has a free parameter determining how "tolerant" the listener's auditory system is to perceiving slightly detuned versions of simple ratios as slightly-off versions of themselves, rather than perceiving them as other, more complex ratios. Tenney and Weil height can also be used to seed the Harmonic entropy calculation to begin with, so that they can be thought of as "primitives" from which increasingly sophisticated models can be built.

	== Some Caveats in Expanding to Chords of Arbitrary Size ==		== Some Caveats in Expanding to Chords of Arbitrary Size ==
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	Now, of course, we have cheated somewhat—note that we have extended the chord in such a way that the differences between each frequency ratio are 2, making this an isoharmonic chord, which are known to strongly exhibit "periodicity buzz." Still, though, this general principle seems to hold to some degree, even if some of the notes are moved around by 1 here and there to form a non-isoharmonic chord, and it works well enough as a basic guiding principle to be viewed as significant, at least in the view of this author.		Now, of course, we have cheated somewhat—note that we have extended the chord in such a way that the differences between each frequency ratio are 2, making this an isoharmonic chord, which are known to strongly exhibit "periodicity buzz." Still, though, this general principle seems to hold to some degree, even if some of the notes are moved around by 1 here and there to form a non-isoharmonic chord, and it works well enough as a basic guiding principle to be viewed as significant, at least in the view of this author.

	So we would at least like some kind of reasonable starting point in ~~modeling~~ this phenomenon, so that we can compare chords of different sizes.		So we would at least like some kind of reasonable starting point in modelling this phenomenon, so that we can compare chords of different sizes.

	=== A simplified, but useful criterion ===		=== A simplified, but useful criterion ===
	One possible way forward is to imagine that the incoming JI chord as a set of upper harmonics of some fundamental frequency—the GCD of the notes of the chord—and we want to quantify how strongly the chord matches that virtual fundamental. We can make some very basic assumptions:		One possible way forward is to imagine that the incoming JI chord as a set of upper harmonics of some fundamental frequency—the GCD of the notes of the chord—and we want to quantify how strongly the chord matches that virtual fundamental. We can make some very basic assumptions:

	1. Given some fundamental frequency ''f''~~, an ''N''-note chord built from very high harmonics of ''f'' will be a weaker match than an N-note chord built from ''f''{{`s}} lower harmonics. In other words, 4:5:6 matches "1" better than 5:6~~:~~7. This is just a restatement of our definition of the simple complexity above.~~		Given some fundamental frequency ''f'':

	2. ~~Given some fundamental frequency~~ ''f'' ~~and~~ a chord built from ''f''{{`s}} harmonics, ~~adding~~ ''another'' note from ''f''{{`s}} harmonics always ''increases'' the strength of the match to ''f''. In other words, 4:5:6:7 matches "1" better than 4:5:6.		1. An ''N''-note chord built from very high harmonics of ''f'' will be a weaker match than an ''N''-note chord built from ''f''{{`s}} lower harmonics. In other words, 4:5:6 matches "1" better than 5:6:7. This is just a restatement of our definition of simple complexity above.

			2. Adding ''another'' note from ''f''{{`s}} harmonics to a chord built from ''f''{{`s}} harmonics always ''increases'' the strength of the match to ''f''. In other words, 4:5:6:7 matches "1" better than 4:5:6.

	The second proposition is the interesting one. It means that the chord 1:2 evokes "1" less than 1:2:3, which is less than 1:2:3:4, and so on, so that the chord 1:2:3:4:... evokes the frequency "1" most strongly.		The second proposition is the interesting one. It means that the chord 1:2 evokes "1" less than 1:2:3, which is less than 1:2:3:4, and so on, so that the chord 1:2:3:4:... evokes the frequency "1" most strongly.

	Strictly speaking, this phenomenon—the reinforcement of virtual pitch—is most evident if the notes of the chord are played with sine waves, with volume decreasing as you get higher into the harmonic series. In that situation, the chord 1:2:3:4:5:6:7:... is basically something like a sawtooth wave. It isn't quite so apparent that if you instead have all harmonics at equal volume, the resulting "delta comb" should really be viewed as more "consonant" than a sine wave in an absolute sense. This is even more true if, instead of sine waves, all of the notes are being played with some arbitrary harmonic timbre! Still, though, we still view the basic spirit of this as a "good enough" rule of thumb which is simple enough to be worth ~~modeling~~. (As we will see, we depart from strict adherence to this criterion anyway.)		Strictly speaking, this phenomenon—the reinforcement of virtual pitch—is most evident if the notes of the chord are played with sine waves, with volume decreasing as you get higher into the harmonic series. In that situation, the chord 1:2:3:4:5:6:7:... is basically something like a sawtooth wave. It isn't quite so apparent that if you instead have all harmonics at equal volume, the resulting "delta comb" should really be viewed as more "consonant" than a sine wave in an absolute sense. This is even more true if, instead of sine waves, all of the notes are being played with some arbitrary harmonic timbre! Still, though, we still view the basic spirit of this as a "good enough" rule of thumb which is simple enough to be worth modelling. (As we will see, we depart from strict adherence to this criterion anyway.)

	== Dirichlet complexity ==		== Dirichlet complexity ==