Stretched and compressed tuning - Revision history

Fredg999: Text replacement - "[[The Riemann zeta function and tuning#" to "[[Riemann zeta function#"

2026-07-16T18:56:11Z

Text replacement - "[[The Riemann zeta function and tuning#" to "[[Riemann zeta function#"

← Older revision		Revision as of 18:56, 16 July 2026
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	* [[Slendro]]		* [[Slendro]]
	* [[Pelog]]		* [[Pelog]]
	* [[~~The~~ Riemann zeta function ~~and tuning~~#Optimal octave stretch\|Zeta optimal octave stretch]]		* [[Riemann zeta function#Optimal octave stretch\|Zeta optimal octave stretch]]
	* [[Stretched harmonic series]]		* [[Stretched harmonic series]]

	[[Category:Tuning]]		[[Category:Tuning]]

FloraC: I be braindead to be unaware of the correct inflection of shrink. Replace with squashed, attested in certain pages on this wiki. Misc. formatting improvements

2026-07-09T05:01:14Z

I be braindead to be unaware of the correct inflection of *shrink*. Replace with *squashed*, attested in certain pages on this wiki. Misc. formatting improvements

← Older revision		Revision as of 05:01, 9 July 2026
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	{{~~Wikipedia~~\|Stretched tuning}}		{{Interwiki
	~~{{interwiki~~		\| en = Stretched and compressed tuning
	\| de = Gestreckte/gestauchte_Stimmung		\| de = Gestreckte/gestauchte_Stimmung
	~~\| en = Stretched_and_compressed_tuning~~
	}}		}}
	[[Tuning]]s do not necessarily need [[equave]]s to be tuned to their exact [[ratio]]s, and in some cases, equaves (most often [[octave]]s) are best stretched or compressed. In '''stretched tuning''', two notes an [[equivalence]] apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly farther apart (a stretched equivalence). In '''compressed tuning''', also known as '''~~shrinked~~ tuning''' or '''shrunk tuning''', two notes an equivalence apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly closer together (a compressed ~~or shrinked~~ equivalence).		{{Wikipedia\|Stretched tuning}}
			[[Tuning]]s do not necessarily need [[equave]]s to be tuned to their exact [[ratio]]s, and in some cases, equaves (most often [[octave]]s) are best stretched or compressed. In '''stretched tuning''', two notes an [[equivalence]] apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly farther apart (a stretched equivalence). In '''compressed tuning''', also known as '''squashed tuning''' or '''shrunk tuning''', two notes an equivalence apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly closer together (a compressed equivalence).

	The most common goal of stretching or compressing the octave is to improve the intonation of some intervals, such as [[harmonic]]s, without sacrificing the melodic shape or harmonic structure of a [[temperament\|tempered]] tuning system. For example, [[19edo]] approximates ratios of 3, 5, 7, and 13 well, but tunes all of these harmonics flat, so it benefits from octave stretching. [[27edo]] approximates ratios of 3, 5, 7, and 13 well, but tunes these harmonics sharp, so it benefits from octave compression.		The most common goal of stretching or compressing the octave is to improve the intonation of some intervals, such as [[harmonic]]s, without sacrificing the melodic shape or harmonic structure of a [[temperament\|tempered]] tuning system. For example, [[19edo]] approximates ratios of 3, 5, 7, and 13 well, but tunes all of these harmonics flat, so it benefits from octave stretching. [[27edo]] approximates ratios of 3, 5, 7, and 13 well, but tunes these harmonics sharp, so it benefits from octave compression.

Hstraub: Link to German

2026-03-14T09:28:29Z

Link to German

← Older revision		Revision as of 09:28, 14 March 2026
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	{{Wikipedia\|Stretched tuning}}		{{Wikipedia\|Stretched tuning}}
			{{interwiki
			\| de = Gestreckte/gestauchte_Stimmung
			\| en = Stretched_and_compressed_tuning
			}}
	[[Tuning]]s do not necessarily need [[equave]]s to be tuned to their exact [[ratio]]s, and in some cases, equaves (most often [[octave]]s) are best stretched or compressed. In '''stretched tuning''', two notes an [[equivalence]] apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly farther apart (a stretched equivalence). In '''compressed tuning''', also known as '''shrinked tuning''' or '''shrunk tuning''', two notes an equivalence apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly closer together (a compressed or shrinked equivalence).		[[Tuning]]s do not necessarily need [[equave]]s to be tuned to their exact [[ratio]]s, and in some cases, equaves (most often [[octave]]s) are best stretched or compressed. In '''stretched tuning''', two notes an [[equivalence]] apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly farther apart (a stretched equivalence). In '''compressed tuning''', also known as '''shrinked tuning''' or '''shrunk tuning''', two notes an equivalence apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly closer together (a compressed or shrinked equivalence).

ArrowHead294 at 14:30, 19 September 2025

2025-09-19T14:30:37Z

← Older revision		Revision as of 14:30, 19 September 2025
Line 1:		Line 1:
	{{Wikipedia\|Stretched tuning}}		{{Wikipedia\|Stretched tuning}}
	[[Tuning]]s do not necessarily need [[equave]]s to be tuned to their exact [[ratio]]s, and in some cases, equaves (most often [[octave]]s) are best stretched or compressed. In '''stretched tuning''', two notes an [[equivalence]] apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly farther apart (a stretched equivalence). In '''compressed tuning''', also known as '''shrinked tuning''', two notes an equivalence apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly closer together (a compressed or shrinked equivalence).		[[Tuning]]s do not necessarily need [[equave]]s to be tuned to their exact [[ratio]]s, and in some cases, equaves (most often [[octave]]s) are best stretched or compressed. In '''stretched tuning''', two notes an [[equivalence]] apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly farther apart (a stretched equivalence). In '''compressed tuning''', also known as '''shrinked tuning''' or '''shrunk tuning''', two notes an equivalence apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly closer together (a compressed or shrinked equivalence).

	The most common goal of stretching or compressing the octave is to improve the intonation of some intervals, such as [[harmonic]]s, without sacrificing the melodic shape or harmonic structure of a [[temperament\|tempered]] tuning system. For example, [[19edo]] approximates ratios of 3, 5, 7, and 13 well, but tunes all of these harmonics flat, so it benefits from octave stretching. [[27edo]] approximates ratios of 3, 5, 7, and 13 well, but tunes these harmonics sharp, so it benefits from octave compression.		The most common goal of stretching or compressing the octave is to improve the intonation of some intervals, such as [[harmonic]]s, without sacrificing the melodic shape or harmonic structure of a [[temperament\|tempered]] tuning system. For example, [[19edo]] approximates ratios of 3, 5, 7, and 13 well, but tunes all of these harmonics flat, so it benefits from octave stretching. [[27edo]] approximates ratios of 3, 5, 7, and 13 well, but tunes these harmonics sharp, so it benefits from octave compression.

FloraC: Make the implicit assumption of tempered systems explicit

2025-09-19T11:16:19Z

Make the implicit assumption of tempered systems explicit

← Older revision		Revision as of 11:16, 19 September 2025
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	[[Tuning]]s do not necessarily need [[equave]]s to be tuned to their exact [[ratio]]s, and in some cases, equaves (most often [[octave]]s) are best stretched or compressed. In '''stretched tuning''', two notes an [[equivalence]] apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly farther apart (a stretched equivalence). In '''compressed tuning''', also known as '''shrinked tuning''', two notes an equivalence apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly closer together (a compressed or shrinked equivalence).		[[Tuning]]s do not necessarily need [[equave]]s to be tuned to their exact [[ratio]]s, and in some cases, equaves (most often [[octave]]s) are best stretched or compressed. In '''stretched tuning''', two notes an [[equivalence]] apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly farther apart (a stretched equivalence). In '''compressed tuning''', also known as '''shrinked tuning''', two notes an equivalence apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly closer together (a compressed or shrinked equivalence).

	The most common goal of stretching or compressing the octave is to improve the intonation of some intervals, such as ~~harmonics~~, without sacrificing the melodic shape or harmonic structure of ~~the~~ tuning system. For example, [[19edo]] approximates ratios of 3, 5, 7, and 13 well, but tunes all of these harmonics flat, so it benefits from octave stretching. 27edo approximates ratios of 3, 5, 7, and 13 well, but tunes these harmonics sharp, so it benefits from octave compression.		The most common goal of stretching or compressing the octave is to improve the intonation of some intervals, such as [[harmonic]]s, without sacrificing the melodic shape or harmonic structure of a [[temperament\|tempered]] tuning system. For example, [[19edo]] approximates ratios of 3, 5, 7, and 13 well, but tunes all of these harmonics flat, so it benefits from octave stretching. [[27edo]] approximates ratios of 3, 5, 7, and 13 well, but tunes these harmonics sharp, so it benefits from octave compression.

	Optimal octave stretches measured by [[path-based goodness]] can be found on that page.		Optimal octave stretches measured by [[path-based goodness]] can be found on that page.

FloraC: Narrowed -> shrinked. I've never seen anyone use the term "narrowed tuning"

2025-09-19T11:13:57Z

Narrowed -> shrinked. I've never seen anyone use the term "narrowed tuning"

← Older revision		Revision as of 11:13, 19 September 2025
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	{{Wikipedia\|Stretched tuning}}		{{Wikipedia\|Stretched tuning}}
	[[Tuning]]s do not necessarily need [[equave]]s to be tuned to their exact [[ratio]]s, and in some cases, equaves (most often [[octave]]s) are best stretched or compressed. In '''stretched tuning''', two notes an [[equivalence]] apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly farther apart (a stretched equivalence). In '''compressed tuning''', also known as ~~narrowed~~ tuning, two notes an equivalence apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly closer together (a compressed or ~~narrowed~~ equivalence).		[[Tuning]]s do not necessarily need [[equave]]s to be tuned to their exact [[ratio]]s, and in some cases, equaves (most often [[octave]]s) are best stretched or compressed. In '''stretched tuning''', two notes an [[equivalence]] apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly farther apart (a stretched equivalence). In '''compressed tuning''', also known as '''shrinked tuning''', two notes an equivalence apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly closer together (a compressed or shrinked equivalence).

	The most common goal of stretching or compressing the octave is to improve the intonation of some intervals, such as harmonics, without sacrificing the melodic shape or harmonic structure of the tuning system. For example, [[19edo]] approximates ratios of 3, 5, 7, and 13 well, but tunes all of these harmonics flat, so it benefits from octave stretching. 27edo approximates ratios of 3, 5, 7, and 13 well, but tunes these harmonics sharp, so it benefits from octave compression.		The most common goal of stretching or compressing the octave is to improve the intonation of some intervals, such as harmonics, without sacrificing the melodic shape or harmonic structure of the tuning system. For example, [[19edo]] approximates ratios of 3, 5, 7, and 13 well, but tunes all of these harmonics flat, so it benefits from octave stretching. 27edo approximates ratios of 3, 5, 7, and 13 well, but tunes these harmonics sharp, so it benefits from octave compression.

Hkm at 00:17, 7 August 2025

2025-08-07T00:17:04Z

← Older revision		Revision as of 00:17, 7 August 2025
Line 4:		Line 4:
	The most common goal of stretching or compressing the octave is to improve the intonation of some intervals, such as harmonics, without sacrificing the melodic shape or harmonic structure of the tuning system. For example, [[19edo]] approximates ratios of 3, 5, 7, and 13 well, but tunes all of these harmonics flat, so it benefits from octave stretching. 27edo approximates ratios of 3, 5, 7, and 13 well, but tunes these harmonics sharp, so it benefits from octave compression.		The most common goal of stretching or compressing the octave is to improve the intonation of some intervals, such as harmonics, without sacrificing the melodic shape or harmonic structure of the tuning system. For example, [[19edo]] approximates ratios of 3, 5, 7, and 13 well, but tunes all of these harmonics flat, so it benefits from octave stretching. 27edo approximates ratios of 3, 5, 7, and 13 well, but tunes these harmonics sharp, so it benefits from octave compression.

	Optimal octave stretches measured by [[~~Path~~-based goodness]] can be found on that page.		Optimal octave stretches measured by [[path-based goodness]] can be found on that page.

	== Inharmonicity compensation for string instruments ==		== Inharmonicity compensation for string instruments ==

Hkm at 23:11, 6 August 2025

2025-08-06T23:11:06Z

← Older revision		Revision as of 23:11, 6 August 2025
Line 3:		Line 3:

	The most common goal of stretching or compressing the octave is to improve the intonation of some intervals, such as harmonics, without sacrificing the melodic shape or harmonic structure of the tuning system. For example, [[19edo]] approximates ratios of 3, 5, 7, and 13 well, but tunes all of these harmonics flat, so it benefits from octave stretching. 27edo approximates ratios of 3, 5, 7, and 13 well, but tunes these harmonics sharp, so it benefits from octave compression.		The most common goal of stretching or compressing the octave is to improve the intonation of some intervals, such as harmonics, without sacrificing the melodic shape or harmonic structure of the tuning system. For example, [[19edo]] approximates ratios of 3, 5, 7, and 13 well, but tunes all of these harmonics flat, so it benefits from octave stretching. 27edo approximates ratios of 3, 5, 7, and 13 well, but tunes these harmonics sharp, so it benefits from octave compression.

			Optimal octave stretches measured by [[Path-based goodness]] can be found on that page.

	== Inharmonicity compensation for string instruments ==		== Inharmonicity compensation for string instruments ==
	In general, stretched tunings detune intervals in the same way as compressed tunings do, so for instance a 5{{c}} compressed octave should sound as out-of-tune as a 5{{c}} stretched octave. On mechanoacoustic instruments, in particular string instruments, that is not true, as the overtones of the strings tend ~~slighly~~ sharp from their ideal natural harmonics and do not exactly line up with the [[harmonic series]] (especially on spinet pianos with their rather short strings). Because the octave is the most significant partial on the strings, by stretching the octaves we can better match our tuning systems with the timbre.		In general, stretched tunings detune intervals in the same way as compressed tunings do, so for instance a 5{{c}} compressed octave should sound as out-of-tune as a 5{{c}} stretched octave. On mechanoacoustic instruments, in particular string instruments, that is not true, as the overtones of the strings tend slightly sharp from their ideal natural harmonics and do not exactly line up with the [[harmonic series]] (especially on spinet pianos with their rather short strings). Because the octave is the most significant partial on the strings, by stretching the octaves we can better match our tuning systems with the timbre.

	The significance of the match is a point of debate. For some, the match but shifts the problem from the timbre to the tuning systems, so no stretch or less stretch than the timbre will be alright. However, with compressed-octave tunings, the discrepancy between our compressed octave and the timbre octave will be larger. For that reason, compressed-octave tunings tend to sound more out of tune, with some going so far to say that octave compression should be avoided no matter what other reasons there may be going for them.		The significance of the match is a point of debate. For some, the match but shifts the problem from the timbre to the tuning systems, so no stretch or less stretch than the timbre will be alright. However, with compressed-octave tunings, the discrepancy between our compressed octave and the timbre octave will be larger. For that reason, compressed-octave tunings tend to sound more out of tune, with some going so far to say that octave compression should be avoided no matter what other reasons there may be going for them.

Sintel: /* See also */ cleanup

2025-04-30T17:48:25Z

ArrowHead294 at 15:05, 25 February 2025

2025-02-25T15:05:10Z

← Older revision		Revision as of 15:05, 25 February 2025
Line 2:		Line 2:
	[[Tuning]]s do not necessarily need [[equave]]s to be tuned to their exact [[ratio]]s, and in some cases, equaves (most often [[octave]]s) are best stretched or compressed. In '''stretched tuning''', two notes an [[equivalence]] apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly farther apart (a stretched equivalence). In '''compressed tuning''', also known as narrowed tuning, two notes an equivalence apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly closer together (a compressed or narrowed equivalence).		[[Tuning]]s do not necessarily need [[equave]]s to be tuned to their exact [[ratio]]s, and in some cases, equaves (most often [[octave]]s) are best stretched or compressed. In '''stretched tuning''', two notes an [[equivalence]] apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly farther apart (a stretched equivalence). In '''compressed tuning''', also known as narrowed tuning, two notes an equivalence apart, whose fundamental frequencies theoretically have an exact ratio, are tuned slightly closer together (a compressed or narrowed equivalence).

	The most common goal of stretching or compressing the octave is to improve the intonation of some intervals, such as harmonics, without sacrificing the melodic shape or harmonic structure of the tuning system. For example, [[19edo]] ~~benefits from a stretch as it tunes harmonics~~ 3, 5, 7, and 13 all ~~flat. The stretch makes~~ these harmonics ~~less~~ flat. 27edo ~~benefits from a compression as it tunes harmonics~~ 3, 5, 7, and 13 ~~all sharp. The compression makes~~ these harmonics ~~less~~ sharp.		The most common goal of stretching or compressing the octave is to improve the intonation of some intervals, such as harmonics, without sacrificing the melodic shape or harmonic structure of the tuning system. For example, [[19edo]] approximates ratios of 3, 5, 7, and 13 well, but tunes all of these harmonics flat, so it benefits from octave stretching. 27edo approximates ratios of 3, 5, 7, and 13 well, but tunes these harmonics sharp, so it benefits from octave compression.

	== Inharmonicity compensation for string instruments ==		== Inharmonicity compensation for string instruments ==
	In general, stretched tunings detune intervals in the same way as compressed tunings do, so for instance a 5~~-cent~~ compressed octave should sound as out-of-tune as a 5~~-cent~~ stretched octave. On mechanoacoustic instruments, in particular string instruments, that is not true, as the overtones of the strings tend slighly sharp from their ideal natural harmonics and do not exactly line up with the [[harmonic series]] (especially on spinet pianos with their rather short strings). Because the octave is the most significant partial on the strings, by stretching the octaves we can better match our tuning systems with the timbre.		In general, stretched tunings detune intervals in the same way as compressed tunings do, so for instance a 5{{c}} compressed octave should sound as out-of-tune as a 5{{c}} stretched octave. On mechanoacoustic instruments, in particular string instruments, that is not true, as the overtones of the strings tend slighly sharp from their ideal natural harmonics and do not exactly line up with the [[harmonic series]] (especially on spinet pianos with their rather short strings). Because the octave is the most significant partial on the strings, by stretching the octaves we can better match our tuning systems with the timbre.

	The significance of the match is a point of debate. For some, the match but shifts the problem from the timbre to the tuning systems, so no stretch or less stretch than the timbre will be alright. However, with compressed-octave tunings, the discrepancy between our compressed octave and the timbre octave will be larger. For that reason, compressed-octave tunings tend to sound more out of tune~~. Some would go~~ so far to say that ~~they~~ should be avoided no matter what other reasons there may be going for them.		The significance of the match is a point of debate. For some, the match but shifts the problem from the timbre to the tuning systems, so no stretch or less stretch than the timbre will be alright. However, with compressed-octave tunings, the discrepancy between our compressed octave and the timbre octave will be larger. For that reason, compressed-octave tunings tend to sound more out of tune, with some going so far to say that octave compression should be avoided no matter what other reasons there may be going for them.

	== See also ==		== See also ==
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	* [[Pelog]]		* [[Pelog]]
	* [[Mavila]]		* [[Mavila]]
	** [[2L 5s]]		** [[2L 5s]]
	** [[7L 2s]]		** [[7L 2s]]
	* [[Phoenix]] tuning		* [[Phoenix]] tuning
	* [[Zeta peak index]] tunings		* [[Zeta peak index]] tunings

← Older revision		Revision as of 17:48, 30 April 2025
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	== See also ==		== See also ==
	* [[23edo and octave stretching]]
	* [[Musical cells]]
	* [[Slendro]]		* [[Slendro]]
	* [[Pelog]]		* [[Pelog]]
	* [[Mavila]]		* [[The Riemann zeta function and tuning#Optimal octave stretch\|Zeta optimal octave stretch]]
	** [[2L 5s]]
	** [[7L 2s]]
	* [[Phoenix]] tuning
	* [[Zeta peak index]] tunings
	** [[The Riemann zeta function and tuning#Optimal octave stretch\|~~Optimal~~ octave stretch]] ~~(zeta stretch)~~
	** [[5- to 10-tone scales in 47zpi]]
	* [[Ed257/128]]s
	* [[Ed255/128]]s
	* [[Stretched harmonic series]]		* [[Stretched harmonic series]]

	[[Category:Tuning]]		[[Category:Tuning]]