Expanding tonal space - Revision history

Holger Stoltenberg: Newline removed

2026-01-14T20:01:11Z

Newline removed

← Older revision		Revision as of 20:01, 14 January 2026
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	\|<br>mode8 maj (8:10:12)<br>'''pure major''' for reference<br><br>\|\|[[File:TS_maj_lin_8_C_P-I_Fig-5.mp3\|80px]]		\|<br>mode8 maj (8:10:12)<br>'''pure major''' for reference<br><br>\|\|[[File:TS_maj_lin_8_C_P-I_Fig-5.mp3\|80px]]
	\|}		\|}
	~~<br>~~


	The model of tonal space is well suited for the design and comparison of chords. No matter what intervals we '''mark on any horizontal line''', the result will always be a chord made up of just intervals that share a common denominator. Therefore, any such chord is a local subset of the harmonic series at its proper position.		The model of tonal space is well suited for the design and comparison of chords. No matter what intervals we '''mark on any horizontal line''', the result will always be a chord made up of just intervals that share a common denominator. Therefore, any such chord is a local subset of the harmonic series at its proper position.

Holger Stoltenberg: Fig.5 has been amended with a few audio samples

2026-01-14T19:58:22Z

Fig.5 has been amended with a few audio samples

← Older revision		Revision as of 19:58, 14 January 2026
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	==General applicability==		==General applicability==
	In the model discussed so far a ''chord'' is composed of at least two stacked intervals with frequency ratios taken from the harmonic series in ascending order. The chord should be footed on the tonic of the particular mode. Skipped harmonics within a chord may remain mute. Fig.5 shows a comparison of four augmented chords that sound quite different:		In the model discussed so far a ''chord'' is composed of at least two stacked intervals with frequency ratios taken from the harmonic series in ascending order. The chord should be footed on the tonic of the particular mode. Skipped harmonics within a chord may remain mute. Fig.5 shows a comparison of four augmented chords that sound quite different. <br>
	[[File:Fig-5 tonal space 753i aug.png\|~~480px~~\|~~center]]~~		Listen to the following audio examples...
	~~<center><small>~~<u>Fig.5</u>: Selection of different augmented chords <~~/small~~><~~/center~~>		[[File:Fig-5 tonal space 753i aug.png\|thumb\|420px\|right\|<u>Fig.5</u>: Selection of different augmented chords]]
			:{\| class="wikitable"
			!Chord
			!Play
			\|-
			\|mode16 aug (16:20:25)<br>2 stacked pure 3rds\|\|[[File:TS_aug_L16_p25_C_P-I_Fig-5.mp3\|80px]]
			\|-
			\|mode14 aug (14:18:22)<br>sounds equal to (7:9:11)\|\|[[File:TS_aug_L14_C_P-I_Fig-5.mp3\|80px]]
			\|-
			\|mode11 aug (11:14:17)<br>no, this is not major...\|\|[[File:TS_aug_L11-14-17_C_P-I_Fig-5.mp3\|80px]]
			\|-
			\|mode10 aug (10:13:16)<br>extra wide 3rd\|\|[[File:TS_aug_L10_C_P-I_Fig-5.mp3\|80px]]
			\|-
			\|<br>mode8 maj (8:10:12)<br>'''pure major''' for reference<br><br>\|\|[[File:TS_maj_lin_8_C_P-I_Fig-5.mp3\|80px]]
			\|}
			<br>

Holger Stoltenberg: Links in section "Find out more about tonal space" updated

2025-04-06T12:54:21Z

Links in section "Find out more about tonal space" updated

← Older revision		Revision as of 12:54, 6 April 2025
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	====[[Expanding tonal space/planar extensions\|Part II: <span style="font-weight:normal">Planar extensions</span>]]====		====[[Expanding tonal space/planar extensions\|Part II: <span style="font-weight:normal">Planar extensions</span>]]====
	====[[Expanding tonal space/projections\|Part III: <span style="font-weight:normal">Projections</span>]]====		====[[Expanding tonal space/projections\|Part III: <span style="font-weight:normal">Projections</span>]]====
			====[[Expanding tonal space/third dimension\|Part IV: <span style="font-weight:normal">Third dimension</span>]]====

	==See also…==		==See also…==

Holger Stoltenberg: Section "Number of distinct intervals" and Table 1 inserted

2025-02-25T11:28:01Z

Section "Number of distinct intervals" and Table 1 inserted

@@ Line 94: / Line 94: @@
 ::::<math>
 r_{12edo}=integer\left (\frac{386c+50c}{100.0}  \right )\cdot 100=400</math> ¢
 ==A variety of projections of the model==

Holger Stoltenberg: Links to Part III added

2025-02-20T16:59:04Z

Links to Part III added

← Older revision		Revision as of 16:59, 20 February 2025
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	Nevertheless, this representation is the basis for a variety of ''realtime'' ''operations on chords'' that a future musical instrument can apply. In addition, the Cartesian projection of tonal space (with a straight horizontal line for each mode of an overtone scale) can be easily handled in a programming environment.		Nevertheless, this representation is the basis for a variety of ''realtime'' ''operations on chords'' that a future musical instrument can apply. In addition, the Cartesian projection of tonal space (with a straight horizontal line for each mode of an overtone scale) can be easily handled in a programming environment.
			(See [[Expanding tonal space/projections\|Part III]] for more on Cartesian and polar projections of tonal space.)

	==Polar projection of tonal space==		==Polar projection of tonal space==
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	== Find out more about tonal space… ==		== Find out more about tonal space… ==
	====[[Expanding tonal space/planar extensions\|Part II: <span style="font-weight:normal">Planar extensions</span>]]====		====[[Expanding tonal space/planar extensions\|Part II: <span style="font-weight:normal">Planar extensions</span>]]====
			====[[Expanding tonal space/projections\|Part III: <span style="font-weight:normal">Projections</span>]]====

	==See also…==		==See also…==

Holger Stoltenberg at 11:24, 18 February 2025

2025-02-18T11:24:42Z

← Older revision		Revision as of 11:24, 18 February 2025
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	==General applicability==		==General applicability==
	In the model discussed so far a ''chord'' is composed of at least two stacked intervals with frequency ratios taken from the harmonic series in ascending order. The chord should be footed on the tonic of the particular mode. Skipped harmonics within a chord may remain mute. Fig.5 shows a comparison of four augmented chords that sound quite different:		In the model discussed so far a ''chord'' is composed of at least two stacked intervals with frequency ratios taken from the harmonic series in ascending order. The chord should be footed on the tonic of the particular mode. Skipped harmonics within a chord may remain mute. Fig.5 shows a comparison of four augmented chords that sound quite different:
			[[File:Fig-5 tonal space 753i aug.png\|480px\|center]]
			<center><small><u>Fig.5</u>: Selection of different augmented chords </small></center>

Holger Stoltenberg at 11:23, 18 February 2025

2025-02-18T11:23:22Z

← Older revision		Revision as of 11:23, 18 February 2025
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	==General applicability==		==General applicability==
	In the model discussed so far a ''chord'' is composed of at least two stacked intervals with frequency ratios taken from the harmonic series in ascending order. The chord should be footed on the tonic of the particular mode. Skipped harmonics within a chord may remain mute. Fig.5 shows a comparison of four augmented chords that sound quite different:		In the model discussed so far a ''chord'' is composed of at least two stacked intervals with frequency ratios taken from the harmonic series in ascending order. The chord should be footed on the tonic of the particular mode. Skipped harmonics within a chord may remain mute. Fig.5 shows a comparison of four augmented chords that sound quite different:
	~~[[File:Fig-5 tonal space 753i aug.png\|480px\|center]]~~
	~~<center><small><u>Fig.5</u>: Selection of different augmented chords </small></center>~~
	The model of tonal space is well suited for the design and comparison of chords. No matter what intervals we '''mark on any horizontal line''', the result will always be a chord made up of just intervals that share a common denominator. Therefore, any such chord is a local subset of the harmonic series at its proper position.		The model of tonal space is well suited for the design and comparison of chords. No matter what intervals we '''mark on any horizontal line''', the result will always be a chord made up of just intervals that share a common denominator. Therefore, any such chord is a local subset of the harmonic series at its proper position.

Holger Stoltenberg: Link formatted

2025-02-12T09:58:33Z

Link formatted

← Older revision		Revision as of 09:58, 12 February 2025
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	== Find out more about tonal space… ==		== Find out more about tonal space… ==
	====[[Expanding tonal space/planar extensions\|Part II: Planar extensions]]====		====[[Expanding tonal space/planar extensions\|Part II: <span style="font-weight:normal">Planar extensions</span>]]====

	==See also…==		==See also…==

Holger Stoltenberg: /* General applicability */

2025-02-09T13:42:42Z

General applicability

← Older revision		Revision as of 13:42, 9 February 2025
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	The model of tonal space is well suited for the design and comparison of chords. No matter what intervals we '''mark on any horizontal line''', the result will always be a chord made up of just intervals that share a common denominator. Therefore, any such chord is a local subset of the harmonic series at its proper position.		The model of tonal space is well suited for the design and comparison of chords. No matter what intervals we '''mark on any horizontal line''', the result will always be a chord made up of just intervals that share a common denominator. Therefore, any such chord is a local subset of the harmonic series at its proper position.

	A final example: If we want to create a major ''b''7 chord, we will find four suitable pitches in the horizontal Mode 4-line (Fig.3) from m=0 to m=3. ~~If we want to~~ replace the upper 7/4 interval with, say, a 9/5 interval, ~~find~~ the ''Least Common Denominator'' (''LCM'', which is 4*5=20 in this case)~~, and we~~ get a 20:25:30:36 chord, which lives in Mode 20 (not shown) ~~and~~ sounds noticeably more dissonant.		A final example: If we want to create a major ''b''7 chord, we will find four suitable pitches in the horizontal Mode 4-line (Fig.3) from m=0 to m=3. To replace the upper 7/4 interval with, say, a 9/5 interval, we determine the ''Least Common Denominator'' (''LCM'', which is 4*5=20 in this case) to get a 20:25:30:36 chord, which lives in Mode 20 (not shown) and sounds noticeably more dissonant.

	== Find out more about tonal space… ==		== Find out more about tonal space… ==

Holger Stoltenberg: /* Polar projection of tonal space */

2025-02-09T13:33:11Z

Polar projection of tonal space

← Older revision		Revision as of 13:33, 9 February 2025
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	==Polar projection of tonal space==		==Polar projection of tonal space==
	In polar projection, the horizontal ''mode'' lines form circles:		In polar projection the horizontal ''mode'' lines form circles:
	[[File:Fig-4 tonal space POLAR 149.png\|496px\|center]]		[[File:Fig-4 tonal space POLAR 149.png\|496px\|center]]
	<center><small><u>Fig.4</u>: A plane of tonal space in polar projection (up to Mode 16)</small></center>		<center><small><u>Fig.4</u>: A plane of tonal space in polar projection (up to Mode 16)</small></center>
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	The labeled spokes reveal a hidden symmetry of tonal space that is not apparent in Cartesian projection.		The labeled spokes reveal a hidden symmetry of tonal space that is not apparent in Cartesian projection.

	Note that a line drawn from the center to the south (at an angle of π) represents an irrational interval with a frequency ratio of √2 ≈ 1.414..., a tritone of 600 ¢. Unsurprisingly, no rational interval fits exactly on this line.		Note that a line drawn from the center to the south (at an angle of π) represents an irrational interval with a frequency ratio of √2 ≈ 1.414..., a tritone of 600 ¢. Unsurprisingly, no rational interval fits exactly on this line.

	==General applicability==		==General applicability==
	In the model discussed so far a ''chord'' is composed of at least two stacked intervals with frequency ratios taken from the harmonic series in ascending order. The chord should be footed on the tonic of the particular mode. Skipped harmonics within a chord may remain mute. Fig.5 shows a comparison of four augmented chords that sound quite different:		In the model discussed so far a ''chord'' is composed of at least two stacked intervals with frequency ratios taken from the harmonic series in ascending order. The chord should be footed on the tonic of the particular mode. Skipped harmonics within a chord may remain mute. Fig.5 shows a comparison of four augmented chords that sound quite different: