17/16 - Revision history

Overthink: /* Temperaments */ + august

2025-11-29T21:14:03Z

Temperaments: + august

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	== Temperaments ==		== Temperaments ==
	The 17/16 interval generates [[srutal archagall]] with a half-octave period, where it also represents [[16/15]] and [[18/17]]. With a full-octave period it generates [[septendesemi]], the 80 & 103 temperament. It is also a good generator for [[lithium]] with a 1/3-octave period.		The 17/16 interval generates [[srutal archagall]] with a half-octave period, where it also represents [[16/15]] and [[18/17]]. With a full-octave period it generates [[septendesemi]], the 80 & 103 temperament. It is also a good generator for [[lithium]] with a 1/3-octave period, as well as the less accurate [[august]].

	It is very close to 7 steps of [[80edo]], and is tempered as such in many [[80th-octave temperaments]].		It is very close to 7 steps of [[80edo]], and is tempered as such in many [[80th-octave temperaments]].

Overthink: + Temperaments section

2025-11-29T21:13:16Z

+ Temperaments section

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	== Approximation ==		== Approximation ==
	{{Interval edo approximation\|17/16}}		{{Interval edo approximation\|17/16}}

			== Temperaments ==
			The 17/16 interval generates [[srutal archagall]] with a half-octave period, where it also represents [[16/15]] and [[18/17]]. With a full-octave period it generates [[septendesemi]], the 80 & 103 temperament. It is also a good generator for [[lithium]] with a 1/3-octave period.

			It is very close to 7 steps of [[80edo]], and is tempered as such in many [[80th-octave temperaments]].

	== See also ==		== See also ==

FloraC: Text replacement - " {{Interval_Edo_Approximation | " to "{{Interval edo approximation|"

2025-11-03T13:08:09Z

Text replacement - " {{Interval_Edo_Approximation | " to "{{Interval edo approximation|"

← Older revision		Revision as of 13:08, 3 November 2025
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	The term ''large septendecimal semitone'' omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal semitone of 18/17.		The term ''large septendecimal semitone'' omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal semitone of 18/17.
	== Approximation ==		== Approximation ==
	{{~~Interval_Edo_Approximation~~ \| 17/16}}		{{Interval edo approximation\|17/16}}

	== See also ==		== See also ==

Pailiaq at 09:01, 3 November 2025

2025-11-03T09:01:19Z

← Older revision		Revision as of 09:01, 3 November 2025
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	The term ''large septendecimal semitone'' omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal semitone of 18/17.		The term ''large septendecimal semitone'' omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal semitone of 18/17.
			== Approximation ==
			{{Interval_Edo_Approximation \| 17/16}}

	== See also ==		== See also ==

VectorGraphics at 09:05, 17 March 2025

2025-03-17T09:05:56Z

← Older revision		Revision as of 09:05, 17 March 2025
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	* In [[Functional Just System]], it is a diatonic semitone, separated by [[4131/4096]] from the [[256/243\|Pythagorean minor second (256/243)]]. It is also called the '''minor diatonic semitone''', which contrasts the [[5-limit]] major diatonic semitone of [[16/15]] by [[256/255]], about 6.8{{c}}.		* In [[Functional Just System]], it is a diatonic semitone, separated by [[4131/4096]] from the [[256/243\|Pythagorean minor second (256/243)]]. It is also called the '''minor diatonic semitone''', which contrasts the [[5-limit]] major diatonic semitone of [[16/15]] by [[256/255]], about 6.8{{c}}.
	* In [[Helmholtz–Ellis notation]], it is a chromatic semitone, separated by [[2187/2176]] from the [[2187/2048\|Pythagorean augmented unison (2187/2048)]].		* In [[Helmholtz–Ellis notation]], it is a chromatic semitone, separated by [[2187/2176]] from the [[2187/2048\|Pythagorean augmented unison (2187/2048)]].
	It could also be reasonable to treat 17/16 as the ~~accidental~~ for prime 17 in its own right, as it is roughly the same size as the 3-limit accidental 2187/2048.		It could also be reasonable to treat 17/16 as the formal comma for prime 17 in its own right, as it is roughly the same size as the 3-limit accidental 2187/2048.

	The term ''large septendecimal semitone'' omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal semitone of 18/17.		The term ''large septendecimal semitone'' omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal semitone of 18/17.

VectorGraphics at 09:05, 17 March 2025

2025-03-17T09:05:44Z

← Older revision		Revision as of 09:05, 17 March 2025
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	* In [[Functional Just System]], it is a diatonic semitone, separated by [[4131/4096]] from the [[256/243\|Pythagorean minor second (256/243)]]. It is also called the '''minor diatonic semitone''', which contrasts the [[5-limit]] major diatonic semitone of [[16/15]] by [[256/255]], about 6.8{{c}}.		* In [[Functional Just System]], it is a diatonic semitone, separated by [[4131/4096]] from the [[256/243\|Pythagorean minor second (256/243)]]. It is also called the '''minor diatonic semitone''', which contrasts the [[5-limit]] major diatonic semitone of [[16/15]] by [[256/255]], about 6.8{{c}}.
	* In [[Helmholtz–Ellis notation]], it is a chromatic semitone, separated by [[2187/2176]] from the [[2187/2048\|Pythagorean augmented unison (2187/2048)]].		* In [[Helmholtz–Ellis notation]], it is a chromatic semitone, separated by [[2187/2176]] from the [[2187/2048\|Pythagorean augmented unison (2187/2048)]].
			It could also be reasonable to treat 17/16 as the accidental for prime 17 in its own right, as it is roughly the same size as the 3-limit accidental 2187/2048.

	The term ''large septendecimal semitone'' omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal semitone of 18/17.		The term ''large septendecimal semitone'' omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal semitone of 18/17.

ArrowHead294 at 01:57, 25 February 2025

2025-02-25T01:57:37Z

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	}}		}}
	{{Wikipedia\|Minor diatonic semitone}}		{{Wikipedia\|Minor diatonic semitone}}
	In [[17-limit]] [[just intonation]], '''17/16''' is the 17th [[harmonic]], [[octave reduced]], and may be called the '''large septendecimal semitone'''. Measuring about ~~105¢~~, it is close to the [[12edo]] semitone of ~~100¢~~, and thus 12edo can be said to approximate it closely, although an even better approximation is available in [[23edo]]. In a chord, it can function similarly to a jazz "minor ninth" ~~– for~~ instance, 8:10:12:14:17 (although here the interval is [[17/8]], which is a little less harsh sounding than 17/16). In 17-limit JI, [[17/1]] is treated as the next basic consonance after [[13/1\|13]] and [[15/1\|15]].		In [[17-limit]] [[just intonation]], '''17/16''' is the 17th [[harmonic]], [[octave reduced]], and may be called the '''large septendecimal semitone'''. Measuring about 105{{cent}}, it is close to the [[12edo]] semitone of 100{{cent}}, and thus 12edo can be said to approximate it closely, although an even better approximation is available in [[23edo]]. In a chord, it can function similarly to a jazz "minor ninth"—for instance, 8:10:12:14:17 (although here the interval is [[17/8]], which is a little less harsh sounding than 17/16). In 17-limit JI, [[17/1]] is treated as the next basic consonance after [[13/1\|13]] and [[15/1\|15]].

	17/16 is one of two [[superparticular]] semitones in the 17-limit; the other is [[18/17]], the small septendecimal semitone, which measures about ~~99¢~~. The difference between them is [[289/288]], about 6¢. If 12edo is treated as a harmonic system approximating 9 and 17, then 289/288 is tempered out.		17/16 is one of two [[superparticular]] semitones in the 17-limit; the other is [[18/17]], the small septendecimal semitone, which measures about 99{{c}}. The difference between them is [[289/288]], about 6{{c}}. If 12edo is treated as a harmonic system approximating 9 and 17, then 289/288 is tempered out.

	17/16 is almost exactly 1/3 of the [[6/5]] minor third. The difference between 6/5 and three 17/16 semitones is [[24576/24565]], an interval of approximately 0.8¢. 17/16 is also almost exactly 1/8 of [[13/8]], with the difference between 13/8 and (17/16)<sup>8</sup> being approximately 0.9¢. The difference between ten 17/16's and [[11/6]] is approximately 0.2¢, while the difference between thirteen 17/16's and [[11/5]] is approximately 0.6¢.		17/16 is almost exactly 1/3 of the [[6/5]] minor third. The difference between 6/5 and three 17/16 semitones is [[24576/24565]], an interval of approximately 0.8{{c}}. 17/16 is also almost exactly 1/8 of [[13/8]], with the difference between 13/8 and (17/16)<sup>8</sup> being approximately 0.9{{c}}. The difference between ten 17/16's and [[11/6]] is approximately 0.2{{c}}, while the difference between thirteen 17/16's and [[11/5]] is approximately 0.6{{c}}.

	== Terminology and notation ==		== Terminology and notation ==
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	For 17/16 specifically:		For 17/16 specifically:
	* In [[Functional Just System]], it is a diatonic semitone, separated by [[4131/4096]] from the [[256/243\|Pythagorean minor second (256/243)]]. It is also called the '''minor diatonic semitone''', which contrasts the [[5-limit]] major diatonic semitone of [[16/15]] by [[256/255]], about 6.8¢.		* In [[Functional Just System]], it is a diatonic semitone, separated by [[4131/4096]] from the [[256/243\|Pythagorean minor second (256/243)]]. It is also called the '''minor diatonic semitone''', which contrasts the [[5-limit]] major diatonic semitone of [[16/15]] by [[256/255]], about 6.8{{c}}.
	* In [[~~Helmholtz-Ellis~~ notation]], it is a chromatic semitone, separated by [[2187/2176]] from the [[2187/2048\|Pythagorean augmented unison (2187/2048)]].		* In [[Helmholtz–Ellis notation]], it is a chromatic semitone, separated by [[2187/2176]] from the [[2187/2048\|Pythagorean augmented unison (2187/2048)]].

	The term ''large septendecimal semitone'' omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal semitone of 18/17.		The term ''large septendecimal semitone'' omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal semitone of 18/17.

CellularAutomaton: minor formatting fixes

2025-02-24T23:30:07Z

minor formatting fixes

← Older revision		Revision as of 23:30, 24 February 2025
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	* In [[Functional Just System]], it is a diatonic semitone, separated by [[4131/4096]] from the [[256/243\|Pythagorean minor second (256/243)]]. It is also called the '''minor diatonic semitone''', which contrasts the [[5-limit]] major diatonic semitone of [[16/15]] by [[256/255]], about 6.8¢.		* In [[Functional Just System]], it is a diatonic semitone, separated by [[4131/4096]] from the [[256/243\|Pythagorean minor second (256/243)]]. It is also called the '''minor diatonic semitone''', which contrasts the [[5-limit]] major diatonic semitone of [[16/15]] by [[256/255]], about 6.8¢.
	* In [[Helmholtz-Ellis notation]], it is a chromatic semitone, separated by [[2187/2176]] from the [[2187/2048\|Pythagorean augmented unison (2187/2048)]].		* In [[Helmholtz-Ellis notation]], it is a chromatic semitone, separated by [[2187/2176]] from the [[2187/2048\|Pythagorean augmented unison (2187/2048)]].
	*

	The term ''large septendecimal semitone'' omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal semitone of 18/17.		The term ''large septendecimal semitone'' omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal semitone of 18/17.

	== See also ==		== See also ==

CellularAutomaton: removed editorializing

2025-02-24T23:29:21Z

removed editorializing

← Older revision		Revision as of 23:29, 24 February 2025
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	*		*

	The term ''large septendecimal semitone'' omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal semitone of 18/17. It is objectively correct, as in a scale-agnostic context such as pure JI, and especially for an octave-reduced prime that isn't defined by separating any other intervals, the diatonic/chromatic distinction is meaningless.		The term ''large septendecimal semitone'' omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal semitone of 18/17.

	== See also ==		== See also ==

VectorGraphics at 20:19, 24 February 2025

2025-02-24T20:19:31Z

← Older revision		Revision as of 20:19, 24 February 2025
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	* In [[Functional Just System]], it is a diatonic semitone, separated by [[4131/4096]] from the [[256/243\|Pythagorean minor second (256/243)]]. It is also called the '''minor diatonic semitone''', which contrasts the [[5-limit]] major diatonic semitone of [[16/15]] by [[256/255]], about 6.8¢.		* In [[Functional Just System]], it is a diatonic semitone, separated by [[4131/4096]] from the [[256/243\|Pythagorean minor second (256/243)]]. It is also called the '''minor diatonic semitone''', which contrasts the [[5-limit]] major diatonic semitone of [[16/15]] by [[256/255]], about 6.8¢.
	* In [[Helmholtz-Ellis notation]], it is a chromatic semitone, separated by [[2187/2176]] from the [[2187/2048\|Pythagorean augmented unison (2187/2048)]].		* In [[Helmholtz-Ellis notation]], it is a chromatic semitone, separated by [[2187/2176]] from the [[2187/2048\|Pythagorean augmented unison (2187/2048)]].
			*

	The term ''large septendecimal semitone'' omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal semitone of 18/17.		The term ''large septendecimal semitone'' omits the diatonic/chromatic part and only describes its melodic property i.e. the size. It is said in contrast to the small septendecimal semitone of 18/17. It is objectively correct, as in a scale-agnostic context such as pure JI, and especially for an octave-reduced prime that isn't defined by separating any other intervals, the diatonic/chromatic distinction is meaningless.

	== See also ==		== See also ==