Isoharmonic chord - Revision history

Cmloegcmluin: to help keep these closely intertwined concepts straight, consolidate information into one place

2024-02-21T03:46:14Z

to help keep these closely intertwined concepts straight, consolidate information into one place

Show changes

Inthar: /* See also */

2024-02-19T19:33:38Z

BudjarnLambeth: Added to category "xenharmonic series"

2023-04-26T05:17:10Z

Added to category "xenharmonic series"

← Older revision		Revision as of 05:17, 26 April 2023
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	[[Category:Harmonic]]		[[Category:Harmonic]]
	[[Category:Lists of scales]]		[[Category:Lists of scales]]
			[[Category:Xenharmonic series]]

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

2023-03-02T03:56:29Z

Removing from Category:Theory using Cat-a-lot

← Older revision		Revision as of 03:56, 2 March 2023
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	[[Category:Harmonic]]		[[Category:Harmonic]]
	[[Category:Lists of scales]]		[[Category:Lists of scales]]
	~~[[Category:Theory]]~~

Cmloegcmluin at 16:08, 13 April 2022

2022-04-13T16:08:32Z

← Older revision		Revision as of 16:08, 13 April 2022
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			== See also ==

			* [[Linear chord]]: a generalization of isoharmonic chord

	[[Category:Chords]]		[[Category:Chords]]

Fredg999: Reorganized lead section, mentioned isodifferential chords, added a Notation section, misc. edits

2021-10-07T02:19:09Z

Reorganized lead section, mentioned isodifferential chords, added a Notation section, misc. edits

← Older revision		Revision as of 02:19, 7 October 2021
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	=isoharmonic chords=		In [[just intonation]], an '''isoharmonic chord''' is a [[chord]] built by successive jumps up the [[harmonic series]] by some number of steps. Since the harmonic series is arranged such that each higher step is smaller than the one before it, all isoharmonic chords have this same shape—with diminishing step size as one ascends.

	~~In [[JustIntonation\|just intonation]],~~ '''~~Isoharmonic~~ chords''' are built by successive jumps up the [[OverToneSeries\|harmonic series]] by some number of steps. Since the harmonic series is arranged such that each higher step is smaller than the one before it, all isoharmonic chords have this same shape—with diminishing step size as one ascends. All isoharmonic chords are equal-hertz chords, meaning that the frequencies of the notes are in an arithmetic sequence with an equal difference in cycles per second between successive notes. However, not all ~~equal-hertz~~ chords are isoharmonic chords, since the ratios between the notes need not be ~~integers. An isoharmonic "chord" may function more like a "scale" than a chord (depending on the composition of course), but we will use the word "chord" on this page for consistency~~.		All isoharmonic chords are '''isodifferential chords''' (or '''equal-hertz chords'''), meaning that the frequencies of the notes are in an arithmetic sequence with an equal difference in cycles per second between successive notes. However, not all isodifferential chords are isoharmonic chords, since the ratios between the notes need not be rational numbers.

	==~~class~~ i==		An isoharmonic "chord" may function more like a "[[scale]]" than a chord (depending on the composition of course), but the word "chord" is used on this page for consistency.

			== Notation ==
			Some complex isoharmonic chords can be expressed with an offset from a simpler isoharmonic chord, so it is useful to notate them in a compact and readable way. For example, 41:51:61 is very similar to 4:5:6, so it can be notated as (4:5:6)[+0.1]. Similarly, (20:22:24:27:30:33:36)[+0.339] can be expanded to 20339:22339:24339:27339:30339:33339:36339.

			Irrational isodifferential chords can be expressed with the same notation by using irrational numbers within the square brackets. For example, the chord (1:2:3)[+φ] can be expanded to (1+φ):(2+φ):(3+φ), which is approximately equal to 1.618:2.618:3.618.

			== Classification ==
			=== Class i ===
	The simplest isoharmonic chords are built by stepping up the harmonic series by single steps (adjacent steps in the harmonic series). Take, for instance, 4:5:6:7, the harmonic seventh chord. We may call these class i isoharmonic chords. There is one class i series (the harmonic series), which looks like this:		The simplest isoharmonic chords are built by stepping up the harmonic series by single steps (adjacent steps in the harmonic series). Take, for instance, 4:5:6:7, the harmonic seventh chord. We may call these class i isoharmonic chords. There is one class i series (the harmonic series), which looks like this:

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	Some "scales" built this way: [[otones12-24]], [[otones20-40]]...		Some "scales" built this way: [[otones12-24]], [[otones20-40]]...

	==~~class~~ ii==		=== Class ii ===
	The next simplest isoharmonic chords are built by stepping up the harmonic series by two (skipping every other harmonic). This gives us chords such as 3:5:7:9 (the primary tetrad in the [[BP\|Bohlen-Pierce]] tuning system) and 9:11:13:15. Note that if you start on an even number, your chord is equivalent to a class i harmonic chord: 4:6:8:10 = 2:3:4:5. Thus, there is one class ii series (the series of all odd harmonics):		The next simplest isoharmonic chords are built by stepping up the harmonic series by two (skipping every other harmonic). This gives us chords such as 3:5:7:9 (the primary tetrad in the [[BP\|Bohlen-Pierce]] tuning system) and 9:11:13:15. Note that if you start on an even number, your chord is equivalent to a class i harmonic chord: 4:6:8:10 = 2:3:4:5. Thus, there is one class ii series (the series of all odd harmonics):

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	==~~class~~ iii==		=== Class iii ===
	Class iii isoharmonic chords are less common and more complex sounding. They include chords such as 7:10:13:16 and 14:17:20:23. Note that if you start on a number divisible by three, you'll again get a chord reducible to class i (e.g. 9:12:15 = 3:4:5). There are two series for class iii:		Class iii isoharmonic chords are less common and more complex sounding. They include chords such as 7:10:13:16 and 14:17:20:23. Note that if you start on a number divisible by three, you'll again get a chord reducible to class i (e.g. 9:12:15 = 3:4:5). There are two series for class iii:

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	==~~class~~ iv==		=== Class iv ===

	{\| class="wikitable"		{\| class="wikitable"
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	==~~class~~ v==		=== Class v ===

	{\| class="wikitable"		{\| class="wikitable"
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	[[Category:Chords]]		[[Category:Chords]]
	[[Category:~~Isoharmonic~~]]		[[Category:Harmonic]]
	[[Category:~~List]]~~		[[Category:Lists of scales]]
	~~[[Category:Overtone~~]]
	[[Category:Theory]]		[[Category:Theory]]

Fredg999: Fredg999 moved page Isoharmonic chords to Isoharmonic chord: Singular title by convention

2021-09-05T17:10:20Z

Fredg999 moved page Isoharmonic chords to Isoharmonic chord: Singular title by convention

← Older revision	Revision as of 17:10, 5 September 2021
(No difference)

ResonantFrequencies: re-categorisation (AWB), typos fixed: eg. → e.g.

2019-06-30T00:38:18Z

re-categorisation (AWB), typos fixed: eg. → e.g.

← Older revision		Revision as of 00:38, 30 June 2019
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	=isoharmonic chords=		=isoharmonic chords=

	In [[JustIntonation\|just intonation]], Isoharmonic chords are built by successive jumps up the [[OverToneSeries\|harmonic series]] by some number of steps. Since the harmonic series is arranged such that each higher step is smaller than the one before it, all isoharmonic chords have this same ~~shape -- with~~ diminishing step size as one ascends. All isoharmonic chords are equal-hertz chords, meaning that the frequencies of the notes are in an arithmetic sequence with an equal difference in cycles per second between successive notes. However, not all equal-hertz chords are isoharmonic chords, since the ratios between the notes need not be integers. An isoharmonic "chord" may function more like a "scale" than a chord (depending on the composition of course), but we will use the word "chord" on this page for consistency.		In [[JustIntonation\|just intonation]], '''Isoharmonic chords''' are built by successive jumps up the [[OverToneSeries\|harmonic series]] by some number of steps. Since the harmonic series is arranged such that each higher step is smaller than the one before it, all isoharmonic chords have this same shape—with diminishing step size as one ascends. All isoharmonic chords are equal-hertz chords, meaning that the frequencies of the notes are in an arithmetic sequence with an equal difference in cycles per second between successive notes. However, not all equal-hertz chords are isoharmonic chords, since the ratios between the notes need not be integers. An isoharmonic "chord" may function more like a "scale" than a chord (depending on the composition of course), but we will use the word "chord" on this page for consistency.

	===class i===		==class i==
	The simplest isoharmonic chords are built by stepping up the harmonic series by single steps (adjacent steps in the harmonic series). Take, for instance, 4:5:6:7, the harmonic seventh chord. We may call these class i isoharmonic chords. There is one class i series (the harmonic series), which looks like this:		The simplest isoharmonic chords are built by stepping up the harmonic series by single steps (adjacent steps in the harmonic series). Take, for instance, 4:5:6:7, the harmonic seventh chord. We may call these class i isoharmonic chords. There is one class i series (the harmonic series), which looks like this:

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	Some "scales" built this way: [[~~otones12-24\|~~otones12-24]], [[~~otones20-40\|~~otones20-40]]...		Some "scales" built this way: [[otones12-24]], [[otones20-40]]...

	===class ii===		==class ii==
	The next simplest isoharmonic chords are built by stepping up the harmonic series by two (skipping every other harmonic). This gives us chords such as 3:5:7:9 (the primary tetrad in the [[BP\|Bohlen-Pierce]] tuning system) and 9:11:13:15. Note that if you start on an even number, your chord is equivalent to a class i harmonic chord: 4:6:8:10 = 2:3:4:5. Thus, there is one class ii series (the series of all odd harmonics):		The next simplest isoharmonic chords are built by stepping up the harmonic series by two (skipping every other harmonic). This gives us chords such as 3:5:7:9 (the primary tetrad in the [[BP\|Bohlen-Pierce]] tuning system) and 9:11:13:15. Note that if you start on an even number, your chord is equivalent to a class i harmonic chord: 4:6:8:10 = 2:3:4:5. Thus, there is one class ii series (the series of all odd harmonics):

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	\|}		\|}

	===class iii===		==class iii==
	Class iii isoharmonic chords are less common and more complex sounding. They include chords such as 7:10:13:16 and 14:17:20:23. Note that if you start on a number divisible by three, you'll again get a chord reducible to class i (eg. 9:12:15 = 3:4:5). There are two series for class iii:		Class iii isoharmonic chords are less common and more complex sounding. They include chords such as 7:10:13:16 and 14:17:20:23. Note that if you start on a number divisible by three, you'll again get a chord reducible to class i (e.g. 9:12:15 = 3:4:5). There are two series for class iii:

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

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

	{\| class="wikitable"		{\| class="wikitable"
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	\|}		\|}
	[[Category:~~chord~~]]		[[Category:Chords]]
	[[Category:~~isoharmonic~~]]		[[Category:Isoharmonic]]
	[[Category:~~list~~]]		[[Category:List]]
	[[Category:~~overtone~~]]		[[Category:Overtone]]
	[[Category:~~theory~~]]		[[Category:Theory]]

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z