Delta-rational chord - Revision history

Inthar: /* Examples */

2026-05-19T01:26:31Z

Examples

← Older revision		Revision as of 01:26, 19 May 2026
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	=== Examples ===		=== Examples ===
	# The chord 0c-400c-724.7c is a +1+1 chord (approximately 3.8473:4.8473:5.8473) and so is isodifferential (hence fully DR). It is close to the 15edo triad (0-5-9)\15 (0c-400c-720c).		# The chord 0c-400c-724.7c is a +1+1 chord (approximately 3.8473:4.8473:5.8473) and so is isodifferential (hence fully DR). It is close to the 15edo triad (0-5-9)\15 (0c-400c-720c).
	## The Blackwood temperament is best tuned with a sharp 5/4 because of the sharpness of its 3/2. 0c-397.2c-720c (the Blackwood major chord in pure-octave DR-optimized tuning) is within 5 cents of the above chord.		#* The Blackwood temperament is best tuned with a sharp 5/4 because of the sharpness of its 3/2. 0c-397.2c-720c (the Blackwood major chord in pure-octave DR-optimized tuning) is within 5 cents of the above chord.
	# The chord 0c-281c-734.7c-923.6c is a +1+2+1 chord (approximately 5.675:6.675:8.675:9.675), and so is fully DR (but not isodifferential). It is close to the 13edo tetrad (0-3-8-10)\13 (0c-276.9c-738.5c-923.1c).		# The chord 0c-281c-734.7c-923.6c is a +1+2+1 chord (approximately 5.675:6.675:8.675:9.675), and so is fully DR (but not isodifferential). It is close to the 13edo tetrad (0-3-8-10)\13 (0c-276.9c-738.5c-923.1c).
	# The chord 0c-258.3c-771.7c-944.7c is a +1+?+1 chord (approximately 6.214:7.214:9.704:10.704), and thus a ''partially'' (not fully) DR tetrad. It is close to the 14edo tetrad (0-3-9-11)\14 (0c-257.1c-771.4c-942.9c).		# The chord 0c-258.3c-771.7c-944.7c is a +1+?+1 chord (approximately 6.214:7.214:9.704:10.704), and thus a ''partially'' (not fully) DR tetrad. It is close to the 14edo tetrad (0-3-9-11)\14 (0c-257.1c-771.4c-942.9c).

Inthar: /* Examples */

2026-05-19T01:26:10Z

Examples

← Older revision		Revision as of 01:26, 19 May 2026
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	=== Examples ===		=== Examples ===
	# The chord 0c-400c-724.7c is a +1+1 chord (approximately 3.8473:4.8473:5.8473) and so is isodifferential (hence fully DR). It is close to the 15edo triad (0-5-9)\15 (0c-400c-720c).		# The chord 0c-400c-724.7c is a +1+1 chord (approximately 3.8473:4.8473:5.8473) and so is isodifferential (hence fully DR). It is close to the 15edo triad (0-5-9)\15 (0c-400c-720c).
	## The Blackwood temperament is best tuned with a sharp 5/4 because of the sharpness of its 3/2. 0c-397.2c-720c (the ~~blackwood~~ major chord in DR-optimized tuning) is within 5 cents of the above chord.		## The Blackwood temperament is best tuned with a sharp 5/4 because of the sharpness of its 3/2. 0c-397.2c-720c (the Blackwood major chord in pure-octave DR-optimized tuning) is within 5 cents of the above chord.
	# The chord 0c-281c-734.7c-923.6c is a +1+2+1 chord (approximately 5.675:6.675:8.675:9.675), and so is fully DR (but not isodifferential). It is close to the 13edo tetrad (0-3-8-10)\13 (0c-276.9c-738.5c-923.1c).		# The chord 0c-281c-734.7c-923.6c is a +1+2+1 chord (approximately 5.675:6.675:8.675:9.675), and so is fully DR (but not isodifferential). It is close to the 13edo tetrad (0-3-8-10)\13 (0c-276.9c-738.5c-923.1c).
	# The chord 0c-258.3c-771.7c-944.7c is a +1+?+1 chord (approximately 6.214:7.214:9.704:10.704), and thus a ''partially'' (not fully) DR tetrad. It is close to the 14edo tetrad (0-3-9-11)\14 (0c-257.1c-771.4c-942.9c).		# The chord 0c-258.3c-771.7c-944.7c is a +1+?+1 chord (approximately 6.214:7.214:9.704:10.704), and thus a ''partially'' (not fully) DR tetrad. It is close to the 14edo tetrad (0-3-9-11)\14 (0c-257.1c-771.4c-942.9c).

Vector at 01:09, 19 May 2026

2026-05-19T01:09:26Z

← Older revision		Revision as of 01:09, 19 May 2026
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	=== Examples ===		=== Examples ===
	# The chord 0c-400c-724.7c is a +1+1 chord (approximately 3.8473:4.8473:5.8473) and so is isodifferential (hence fully DR). It is close to the 15edo triad (0-5-9)\15 (0c-400c-720c).		# The chord 0c-400c-724.7c is a +1+1 chord (approximately 3.8473:4.8473:5.8473) and so is isodifferential (hence fully DR). It is close to the 15edo triad (0-5-9)\15 (0c-400c-720c).
			## The Blackwood temperament is best tuned with a sharp 5/4 because of the sharpness of its 3/2. 0c-397.2c-720c (the blackwood major chord in DR-optimized tuning) is within 5 cents of the above chord.
	# The chord 0c-281c-734.7c-923.6c is a +1+2+1 chord (approximately 5.675:6.675:8.675:9.675), and so is fully DR (but not isodifferential). It is close to the 13edo tetrad (0-3-8-10)\13 (0c-276.9c-738.5c-923.1c).		# The chord 0c-281c-734.7c-923.6c is a +1+2+1 chord (approximately 5.675:6.675:8.675:9.675), and so is fully DR (but not isodifferential). It is close to the 13edo tetrad (0-3-8-10)\13 (0c-276.9c-738.5c-923.1c).
	# The chord 0c-258.3c-771.7c-944.7c is a +1+?+1 chord (approximately 6.214:7.214:9.704:10.704), and thus a ''partially'' (not fully) DR tetrad. It is close to the 14edo tetrad (0-3-9-11)\14 (0c-257.1c-771.4c-942.9c).		# The chord 0c-258.3c-771.7c-944.7c is a +1+?+1 chord (approximately 6.214:7.214:9.704:10.704), and thus a ''partially'' (not fully) DR tetrad. It is close to the 14edo tetrad (0-3-9-11)\14 (0c-257.1c-771.4c-942.9c).

Inthar: bold is only for article titles or terms

2026-05-18T21:02:41Z

bold is only for article titles or terms

← Older revision		Revision as of 21:02, 18 May 2026
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	Inversions and revoicings of DR chords may not be DR, unlike the case with JI chords where inversions and revoicings of JI chords stay JI.		Inversions and revoicings of DR chords may not be DR, unlike the case with JI chords where inversions and revoicings of JI chords stay JI.

	Delta-rational chords provide a ~~'''~~non-JI-based approach to concordance~~'''~~, since chords that are delta-rational with simple (i.e. low-number) delta signatures tend to be perceived as more concordant than other chords, even when the ratios between the notes themselves are not rational.		Delta-rational chords provide a non-JI-based approach to concordance, since chords that are delta-rational with simple (i.e. low-number) delta signatures tend to be perceived as more concordant than other chords, even when the ratios between the notes themselves are not rational.

	== Definitions ==		== Definitions ==

Kili: bold

2026-05-13T16:53:59Z

bold

← Older revision		Revision as of 16:53, 13 May 2026
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	Inversions and revoicings of DR chords may not be DR, unlike the case with JI chords where inversions and revoicings of JI chords stay JI.		Inversions and revoicings of DR chords may not be DR, unlike the case with JI chords where inversions and revoicings of JI chords stay JI.

	Delta-rational chords provide a non-JI-based approach to concordance, since chords that are delta-rational with simple (i.e. low-number) delta signatures tend to be perceived as more concordant than other chords, even when the ratios between the notes themselves are not rational.		Delta-rational chords provide a '''non-JI-based approach to concordance''', since chords that are delta-rational with simple (i.e. low-number) delta signatures tend to be perceived as more concordant than other chords, even when the ratios between the notes themselves are not rational.

	== Definitions ==		== Definitions ==

Inthar at 05:56, 8 April 2026

2026-04-08T05:56:57Z

← Older revision		Revision as of 05:56, 8 April 2026
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	Inversions and revoicings of DR chords may not be DR, unlike the case with JI chords where inversions and revoicings of JI chords stay JI.		Inversions and revoicings of DR chords may not be DR, unlike the case with JI chords where inversions and revoicings of JI chords stay JI.

	Delta-rational chords provide a non-JI-based approach to concordance, since chords that are delta-rational with simple (i. e. low-number) delta signatures tend to be perceived as more concordant than other chords, even when the ratios between the notes themselves are not rational.		Delta-rational chords provide a non-JI-based approach to concordance, since chords that are delta-rational with simple (i.e. low-number) delta signatures tend to be perceived as more concordant than other chords, even when the ratios between the notes themselves are not rational.

	== Definitions ==		== Definitions ==

Inthar: /* Acoustics */

2026-04-08T04:39:15Z

Acoustics

← Older revision		Revision as of 04:39, 8 April 2026
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	{{adv\|The reason that we experience the DR effect even with non-JI chords may be that detecting the period of the temporal envelope is one of two mechanisms used by the human auditory system to detect the [[virtual fundamental]] of a set of harmonics:}}		{{adv\|The reason that we experience the DR effect even with non-JI chords may be that detecting the period of the temporal envelope is one of two mechanisms used by the human auditory system to detect the [[virtual fundamental]] of a set of harmonics:}}
	# {{adv\|Lower (called ''resolved'') harmonics are matched to a harmonic series template. Evidence from auditory cortex recordings suggests this mechanism involves harmonic template neurons.<ref>Feng, L., & Wang, X. (2017). Harmonic template neurons in primate auditory cortex underlying complex sound processing. Proceedings of the National Academy of Sciences, 114(5), E840-E848.</ref>}}		# {{adv\|Lower (called ''resolved'') harmonics are matched to a harmonic series template. Evidence from auditory cortex recordings taken from primates suggests this mechanism involves harmonic template neurons.<ref>Feng, L., & Wang, X. (2017). Harmonic template neurons in primate auditory cortex underlying complex sound processing. Proceedings of the National Academy of Sciences, 114(5), E840-E848.</ref>}}
	# {{adv\|Higher (called ''unresolved'') harmonics are thought to be processed to determine the period of the temporal envelope. Certain psychoacoustic studies suggest that the harmonic series template ("LCJI") mechanism gradually stops working starting around harmonic 10 (a transition zone, not a hard cutoff) and we start using the temporal envelope periodicity ("DR") mechanism.<ref>Bernstein, J. G., & Oxenham, A. J. (2003). Pitch discrimination of diotic and dichotic tone complexes: Harmonic resolvability or harmonic number?. The Journal of the Acoustical Society of America, 113(6), 3323-3334.</ref><ref>Moore, B. C., & Gockel, H. E. (2011). Resolvability of components in complex tones and implications for theories of pitch perception. Hearing research, 276(1-2), 88-97.</ref>}}		# {{adv\|Higher (called ''unresolved'') harmonics are thought to be processed to determine the period of the temporal envelope. Certain psychoacoustic studies suggest that the harmonic series template ("LCJI") mechanism gradually stops working starting around harmonic 10 (a transition zone, not a hard cutoff) and we start using the temporal envelope periodicity ("DR") mechanism.<ref>Bernstein, J. G., & Oxenham, A. J. (2003). Pitch discrimination of diotic and dichotic tone complexes: Harmonic resolvability or harmonic number?. The Journal of the Acoustical Society of America, 113(6), 3323-3334.</ref><ref>Moore, B. C., & Gockel, H. E. (2011). Resolvability of components in complex tones and implications for theories of pitch perception. Hearing research, 276(1-2), 88-97.</ref>}}

Inthar: use new Cite ext

2026-04-08T04:31:16Z

use new Cite ext

← Older revision		Revision as of 04:31, 8 April 2026
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	{{adv\|The reason that we experience the DR effect even with non-JI chords may be that detecting the period of the temporal envelope is one of two mechanisms used by the human auditory system to detect the [[virtual fundamental]] of a set of harmonics:}}		{{adv\|The reason that we experience the DR effect even with non-JI chords may be that detecting the period of the temporal envelope is one of two mechanisms used by the human auditory system to detect the [[virtual fundamental]] of a set of harmonics:}}
	# {{adv\|Lower (called ''resolved'') harmonics are matched to a harmonic series template. Evidence from auditory cortex recordings suggests this mechanism involves harmonic template neurons.<~~sup~~>~~[1]~~</~~sup~~>}}		# {{adv\|Lower (called ''resolved'') harmonics are matched to a harmonic series template. Evidence from auditory cortex recordings suggests this mechanism involves harmonic template neurons.<ref>Feng, L., & Wang, X. (2017). Harmonic template neurons in primate auditory cortex underlying complex sound processing. Proceedings of the National Academy of Sciences, 114(5), E840-E848.</ref>}}
	# {{adv\|Higher (called ''unresolved'') harmonics are thought to be processed to determine the period of the temporal envelope. Certain psychoacoustic studies suggest that the harmonic series template ("LCJI") mechanism gradually stops working starting around harmonic 10 (a transition zone, not a hard cutoff) and we start using the temporal envelope periodicity ("DR") mechanism.<~~sup~~>[2~~][3]~~</~~sup~~>}}		# {{adv\|Higher (called ''unresolved'') harmonics are thought to be processed to determine the period of the temporal envelope. Certain psychoacoustic studies suggest that the harmonic series template ("LCJI") mechanism gradually stops working starting around harmonic 10 (a transition zone, not a hard cutoff) and we start using the temporal envelope periodicity ("DR") mechanism.<ref>Bernstein, J. G., & Oxenham, A. J. (2003). Pitch discrimination of diotic and dichotic tone complexes: Harmonic resolvability or harmonic number?. The Journal of the Acoustical Society of America, 113(6), 3323-3334.</ref><ref>Moore, B. C., & Gockel, H. E. (2011). Resolvability of components in complex tones and implications for theories of pitch perception. Hearing research, 276(1-2), 88-97.</ref>}}

	== See also ==		== See also ==
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	== References ==		== References ==
	<~~sup>[1]</sup> Feng, L., & Wang, X. (2017). Harmonic template neurons in primate auditory cortex underlying complex sound processing. Proceedings of the National Academy of Sciences, 114(5), E840-E848.~~		<references/>

	<sup>[2]</sup> Bernstein, J. G., & Oxenham, A. J. (2003). Pitch discrimination of diotic and dichotic tone complexes: Harmonic resolvability or harmonic number?. The Journal of the Acoustical Society of America, 113(6), 3323-3334.

	~~<sup>[3]<~~/~~sup~~> ~~Moore, B. C., & Gockel, H. E. (2011). Resolvability of components in complex tones and implications for theories of pitch perception. Hearing research, 276(1-2), 88-97.~~

	== External links ==		== External links ==
	* [https://turbofishcrow.github.io/delta/ Inthar's DR chord explorer]		* [https://turbofishcrow.github.io/delta/ Inthar's DR chord explorer]

	{{cat\|Atypical ratios		{{cat\|Atypical ratios
	Delta-rational harmony\|*		Delta-rational harmony\|*
	}}		}}

Inthar: /* Definitions */

2026-04-01T16:01:34Z

Definitions

← Older revision		Revision as of 16:01, 1 April 2026
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	Deltas that are ''free'', i.e. not required to be related to any other deltas are indicated with +?. For example, saying that a tetrad is "+1+?+1" means the first two notes and the last two notes have equal frequency difference (thus the ratio between the differences is 1/1), but the middle two notes are not in any simple relationship with the two outer intervals.		Deltas that are ''free'', i.e. not required to be related to any other deltas are indicated with +?. For example, saying that a tetrad is "+1+?+1" means the first two notes and the last two notes have equal frequency difference (thus the ratio between the differences is 1/1), but the middle two notes are not in any simple relationship with the two outer intervals.

	{{adv\|If you have some sets of deltas related to each other but not to other sets of increments, you could write the related sets with variables a, b, ... or use one fewer letter by writing one set with positive integers without variables: a delta signature +a +b +a +b can also be written +1 +c +1 +c where c = b/a.}}		{{adv\|If you have some sets of deltas related to each other but not to other sets of increments, you could write the related sets with variables a, b, ... or use one fewer letter by writing one set with positive integers without variables: a delta signature +a+b+a+b can also be written +1+c+1+c where c = b/a.}}

	=== Examples ===		=== Examples ===

Inthar: /* Definitions */

2026-04-01T16:01:13Z

Definitions

← Older revision		Revision as of 16:01, 1 April 2026
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	Thus all isodifferential chords (including isoharmonic JI chords) are fully delta-rational, and all fully delta-rational chords (including all JI chords) are delta-rational.		Thus all isodifferential chords (including isoharmonic JI chords) are fully delta-rational, and all fully delta-rational chords (including all JI chords) are delta-rational.

	Deltas that are ''free'', i.e. not required to be related to any other deltas are indicated with +?. For example, saying that a tetrad is "+1 +? +1" means the first two notes and the last two notes have equal frequency difference (thus the ratio between the differences is 1/1), but the middle two notes are not in any simple relationship with the two outer intervals.		Deltas that are ''free'', i.e. not required to be related to any other deltas are indicated with +?. For example, saying that a tetrad is "+1+?+1" means the first two notes and the last two notes have equal frequency difference (thus the ratio between the differences is 1/1), but the middle two notes are not in any simple relationship with the two outer intervals.

	{{adv\|If you have some sets of deltas related to each other but not to other sets of increments, you could write the related sets with variables a, b, ... or use one fewer letter by writing one set with positive integers without variables: a delta signature +a +b +a +b can also be written +1 +c +1 +c where c = b/a.}}		{{adv\|If you have some sets of deltas related to each other but not to other sets of increments, you could write the related sets with variables a, b, ... or use one fewer letter by writing one set with positive integers without variables: a delta signature +a +b +a +b can also be written +1 +c +1 +c where c = b/a.}}

← Older revision		Revision as of 04:39, 8 April 2026
Line 34:		Line 34:

	{{adv\|The reason that we experience the DR effect even with non-JI chords may be that detecting the period of the temporal envelope is one of two mechanisms used by the human auditory system to detect the [[virtual fundamental]] of a set of harmonics:}}		{{adv\|The reason that we experience the DR effect even with non-JI chords may be that detecting the period of the temporal envelope is one of two mechanisms used by the human auditory system to detect the [[virtual fundamental]] of a set of harmonics:}}
	# {{adv\|Lower (called ''resolved'') harmonics are matched to a harmonic series template. Evidence from auditory cortex recordings suggests this mechanism involves harmonic template neurons.<ref>Feng, L., & Wang, X. (2017). Harmonic template neurons in primate auditory cortex underlying complex sound processing. Proceedings of the National Academy of Sciences, 114(5), E840-E848.</ref>}}		# {{adv\|Lower (called ''resolved'') harmonics are matched to a harmonic series template. Evidence from auditory cortex recordings taken from primates suggests this mechanism involves harmonic template neurons.<ref>Feng, L., & Wang, X. (2017). Harmonic template neurons in primate auditory cortex underlying complex sound processing. Proceedings of the National Academy of Sciences, 114(5), E840-E848.</ref>}}
	# {{adv\|Higher (called ''unresolved'') harmonics are thought to be processed to determine the period of the temporal envelope. Certain psychoacoustic studies suggest that the harmonic series template ("LCJI") mechanism gradually stops working starting around harmonic 10 (a transition zone, not a hard cutoff) and we start using the temporal envelope periodicity ("DR") mechanism.<ref>Bernstein, J. G., & Oxenham, A. J. (2003). Pitch discrimination of diotic and dichotic tone complexes: Harmonic resolvability or harmonic number?. The Journal of the Acoustical Society of America, 113(6), 3323-3334.</ref><ref>Moore, B. C., & Gockel, H. E. (2011). Resolvability of components in complex tones and implications for theories of pitch perception. Hearing research, 276(1-2), 88-97.</ref>}}		# {{adv\|Higher (called ''unresolved'') harmonics are thought to be processed to determine the period of the temporal envelope. Certain psychoacoustic studies suggest that the harmonic series template ("LCJI") mechanism gradually stops working starting around harmonic 10 (a transition zone, not a hard cutoff) and we start using the temporal envelope periodicity ("DR") mechanism.<ref>Bernstein, J. G., & Oxenham, A. J. (2003). Pitch discrimination of diotic and dichotic tone complexes: Harmonic resolvability or harmonic number?. The Journal of the Acoustical Society of America, 113(6), 3323-3334.</ref><ref>Moore, B. C., & Gockel, H. E. (2011). Resolvability of components in complex tones and implications for theories of pitch perception. Hearing research, 276(1-2), 88-97.</ref>}}