196608edo - Revision history

Tristanbay: Added software tuning message

2025-09-08T18:02:04Z

Added software tuning message

← Older revision		Revision as of 18:02, 8 September 2025
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			{{Software tuning}}
	{{Infobox ET}}		{{Infobox ET}}
	{{ED intro}}		{{ED intro}}

ArrowHead294 at 22:37, 20 February 2025

2025-02-20T22:37:04Z

Show changes

Fredg999: /* External links */ Misc. edits

2024-07-07T22:46:45Z

External links: Misc. edits

← Older revision		Revision as of 22:46, 7 July 2024
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	== External links ==		== External links ==
	* [http://tonalsoft.com/enc/number/14mu.aspx ~~Tonalsoft Encyclopedia \| ''~~14mu / tetradekamu'']		* [http://tonalsoft.com/enc/number/14mu.aspx 14mu / tetradekamu] on [[Tonalsoft Encyclopedia]]

FloraC: Fix grammar

2023-11-07T08:19:49Z

Fix grammar

← Older revision		Revision as of 08:19, 7 November 2023
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	[[File:Tetradecamu Approximation Quality.png\|thumb\|Approximation quality of smaller edos in tetradecamu, maximum relative error against edo from 1 to 16808. ]]		[[File:Tetradecamu Approximation Quality.png\|thumb\|Approximation quality of smaller edos in tetradecamu, maximum relative error against edo from 1 to 16808. ]]

	A step of 196608edo is known as ~~the~~ '''MIDI Tuning Standard unit''' ('''MTSU''') or '''tetradecamu''' (fourteenth MIDI-resolution unit, 14mu, 2<sup>14</sup> = 16384 equal divisions of the [[12edo]] semitone). The 14mu is specified in the [[MIDI]] spec (1983) as the smallest increment available for the pitch-bend controller, and as the frequency data format for MTS (1999). The 14mu is the smallest unit of tuning resolution which has ever been put into common musical use, and provides extremely accurate tuning in microtonal electronic music.		A step of 196608edo is known as a '''MIDI Tuning Standard unit''' ('''MTSU''') or a '''tetradecamu''' (fourteenth MIDI-resolution unit, 14mu, 2<sup>14</sup> = 16384 equal divisions of the [[12edo]] semitone). The 14mu is specified in the [[MIDI]] spec (1983) as the smallest increment available for the pitch-bend controller, and as the frequency data format for MTS (1999). The 14mu is the smallest unit of tuning resolution which has ever been put into common musical use, and provides extremely accurate tuning in microtonal electronic music.

	The main application of 196608edo is thus not as a compositional device, but as a technical tuning standard. If we adopt direct approximation, some JI intervals are indeed improved, which makes sense since we are only quantizing JI to the grid of this edo.		The main application of 196608edo is thus not as a compositional device, but as a technical tuning standard. If we adopt direct approximation, some JI intervals are indeed improved, which makes sense since we are only quantizing JI to the grid of this edo.

FloraC: /* Selected intervals */ formatting

2023-11-07T08:18:36Z

Selected intervals: formatting

Show changes

FloraC: Move the 14mu info to the theory section; misc. cleanup

2023-11-07T08:09:42Z

Move the 14mu info to the theory section; misc. cleanup

← Older revision		Revision as of 08:09, 7 November 2023
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	{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro\|196608}} Its adjacent step is known as the '''MIDI Tuning Standard unit''' (MTSU) or '''Tetradecamu''' (fourteenth MIDI-resolution unit, 14mu, 2<sup>14</sup> = 16384 equal divisions of the [[12edo]] semitone). The 14mu is specified in the MIDI spec (1983) as the smallest increment available for the pitch-bend controller, and as the frequency data format for MTS (1999). The 14mu is the smallest unit of tuning resolution which has ever been put into common musical use, and provides extremely accurate tuning in microtonal electronic music.		{{EDO intro\|196608}}

	== Theory ==		== Theory ==
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	[[File:Tetradecamu Approximation Quality.png\|thumb\|Approximation quality of smaller edos in tetradecamu, maximum relative error against edo from 1 to 16808. ]]		[[File:Tetradecamu Approximation Quality.png\|thumb\|Approximation quality of smaller edos in tetradecamu, maximum relative error against edo from 1 to 16808. ]]

	~~The main application~~ of 196608edo is ~~not~~ as ~~a compositional device~~, ~~but~~ as ~~a technical tuning standard~~. ~~If we adopt direct approximation, some JI intervals are indeed improved, which makes sense since we are only quantizing JI to~~ the ~~grid~~ of ~~this edo~~.		A step of 196608edo is known as the '''MIDI Tuning Standard unit''' ('''MTSU''') or '''tetradecamu''' (fourteenth MIDI-resolution unit, 14mu, 2<sup>14</sup> = 16384 equal divisions of the [[12edo]] semitone). The 14mu is specified in the [[MIDI]] spec (1983) as the smallest increment available for the pitch-bend controller, and as the frequency data format for MTS (1999). The 14mu is the smallest unit of tuning resolution which has ever been put into common musical use, and provides extremely accurate tuning in microtonal electronic music.

	Another usage that is not commonly seen in other edos is to approximate smaller edos. It cannot approximate any larger edos or any edos of the same order of magnitude. From the diagram we can observe the maximum relative errors of smallers edos are ~~most~~ linear with respect to the edo number. [[16808edo]], a notable zeta edo that is an order of magnitude below, is approximated with a ~4% maximum relative error.		The main application of 196608edo is thus not as a compositional device, but as a technical tuning standard. If we adopt direct approximation, some JI intervals are indeed improved, which makes sense since we are only quantizing JI to the grid of this edo.

			Another usage that is not commonly seen in other edos is to approximate smaller edos. It cannot approximate any larger edos or any edos of the same order of magnitude. From the diagram we can observe the maximum relative errors of smallers edos are mostly linear with respect to the edo number. [[16808edo]], a notable zeta edo that is an order of magnitude below, is approximated with a ~4% maximum relative error.

	=== Odd harmonics ===		=== Odd harmonics ===
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	\|}		\|}

	==See also==		== See also ==
	*[[Interval size measure]]		* [[Interval size measure]]
	*[[Equal-step tuning\|Equal multiplications]] of MIDI-resolution units		* [[Equal-step tuning\|Equal multiplications]] of MIDI-resolution units
	**[[24edo]] (1mu tuning)		** [[24edo]] (1mu tuning)
	**[[48edo]] (2mu tuning)		** [[48edo]] (2mu tuning)
	**[[96edo]] (3mu tuning)		** [[96edo]] (3mu tuning)
	**[[192edo]] (4mu tuning)		** [[192edo]] (4mu tuning)
	**[[384edo]] (5mu tuning)		** [[384edo]] (5mu tuning)
	**[[768edo]] (6mu tuning)		** [[768edo]] (6mu tuning)
	**[[1536edo]] (7mu tuning)		** [[1536edo]] (7mu tuning)
	**[[3072edo]] (8mu tuning)		** [[3072edo]] (8mu tuning)
	**[[6144edo]] (9mu tuning)		** [[6144edo]] (9mu tuning)
	**[[12288edo]] (10mu tuning)		** [[12288edo]] (10mu tuning)
	**[[24576edo]] (11mu tuning)		** [[24576edo]] (11mu tuning)
	**[[49152edo]] (12mu tuning)		** [[49152edo]] (12mu tuning)
	**[[98304edo]] (13mu tuning)		** [[98304edo]] (13mu tuning)

			== External links ==
			* [http://tonalsoft.com/enc/number/14mu.aspx Tonalsoft Encyclopedia \| ''14mu / tetradekamu'']

FloraC: Clarify the context

2023-02-20T09:19:52Z

Clarify the context

← Older revision		Revision as of 09:19, 20 February 2023
Line 8:		Line 8:
	[[File:Tetradecamu Approximation Quality.png\|thumb\|Approximation quality of smaller edos in tetradecamu, maximum relative error against edo from 1 to 16808. ]]		[[File:Tetradecamu Approximation Quality.png\|thumb\|Approximation quality of smaller edos in tetradecamu, maximum relative error against edo from 1 to 16808. ]]

	The main application of 196608edo is not as a compositional device, but as a technical tuning standard. If we adopt direct approximation, some JI intervals are indeed improved.		The main application of 196608edo is not as a compositional device, but as a technical tuning standard. If we adopt direct approximation, some JI intervals are indeed improved, which makes sense since we are only quantizing JI to the grid of this edo.

	Another usage that is not commonly seen in other edos is to approximate smaller edos~~, since they are quantized into the grid of this edo~~. It cannot approximate any larger edos or any edos of the same order of magnitude. From the diagram we can observe the maximum relative errors of smallers edos are most linear with respect to the edo number. [[16808edo]], a notable zeta edo that is an order of magnitude below, is approximated with a ~4% maximum relative error.		Another usage that is not commonly seen in other edos is to approximate smaller edos. It cannot approximate any larger edos or any edos of the same order of magnitude. From the diagram we can observe the maximum relative errors of smallers edos are most linear with respect to the edo number. [[16808edo]], a notable zeta edo that is an order of magnitude below, is approximated with a ~4% maximum relative error.

	=== Odd harmonics ===		=== Odd harmonics ===
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	**[[49152edo]] (12mu tuning)		**[[49152edo]] (12mu tuning)
	**[[98304edo]] (13mu tuning)		**[[98304edo]] (13mu tuning)

	~~[[Category:Equal divisions of the octave\|######]] <!-- 6-digit number -->~~

FloraC at 08:05, 20 February 2023

2023-02-20T08:05:12Z

← Older revision		Revision as of 08:05, 20 February 2023
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	=== As a tuning standard ===		=== As a tuning standard ===
	The main application of 196608edo is not as a compositional device, but as a technical tuning standard. If we adopt direct approximation, some JI intervals are indeed improved. Another usage that is not commonly seen in other edos is to approximate smaller edos, since they are quantized into the grid of this edo. It cannot approximate any larger edos or any edos of the same order of magnitude.		[[File:Tetradecamu Approximation Quality.png\|thumb\|Approximation quality of smaller edos in tetradecamu, maximum relative error against edo from 1 to 16808. ]]

			The main application of 196608edo is not as a compositional device, but as a technical tuning standard. If we adopt direct approximation, some JI intervals are indeed improved.

			Another usage that is not commonly seen in other edos is to approximate smaller edos, since they are quantized into the grid of this edo. It cannot approximate any larger edos or any edos of the same order of magnitude. From the diagram we can observe the maximum relative errors of smallers edos are most linear with respect to the edo number. [[16808edo]], a notable zeta edo that is an order of magnitude below, is approximated with a ~4% maximum relative error.

	=== Odd harmonics ===		=== Odd harmonics ===

FloraC: Cleanup and expansion

2023-02-20T07:51:18Z

Cleanup and expansion

← Older revision		Revision as of 07:51, 20 February 2023
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	~~'''196608EDO''' is the [[~~EDO\|~~equal division of the octave]] into~~ 196608 ~~parts of exact 0.006103515625 cents each, which is to say 2<sup>(1/196608)</sup> as a frequency ratio.~~ Its adjacent step is known as the '''MIDI Tuning Standard unit''' (MTSU) or '''Tetradecamu''' (fourteenth MIDI-resolution unit, 14mu, 2<sup>14</sup> = 16384 equal divisions of the [[12edo]] semitone). The 14mu is specified in the MIDI spec (1983) as the smallest increment available for the pitch-bend controller, and as the frequency data format for MTS (1999). The 14mu is the smallest unit of tuning resolution which has ever been put into common musical use, and provides extremely accurate tuning in microtonal electronic music.		{{EDO intro\|196608}} Its adjacent step is known as the '''MIDI Tuning Standard unit''' (MTSU) or '''Tetradecamu''' (fourteenth MIDI-resolution unit, 14mu, 2<sup>14</sup> = 16384 equal divisions of the [[12edo]] semitone). The 14mu is specified in the MIDI spec (1983) as the smallest increment available for the pitch-bend controller, and as the frequency data format for MTS (1999). The 14mu is the smallest unit of tuning resolution which has ever been put into common musical use, and provides extremely accurate tuning in microtonal electronic music.
	~~{{harmonics in equal\|196608\|start=2\|intervals=prime}}~~

	==~~Intervals~~==		== Theory ==
			196608edo is [[enfactoring\|enfactored]] in the 17-limit, having the same tuning as [[98304edo]], which is quite an efficient system in itself. In that regard, 196608edo provides barely anything new apart from most characteristics of what it doubles.

			=== As a tuning standard ===
			The main application of 196608edo is not as a compositional device, but as a technical tuning standard. If we adopt direct approximation, some JI intervals are indeed improved. Another usage that is not commonly seen in other edos is to approximate smaller edos, since they are quantized into the grid of this edo. It cannot approximate any larger edos or any edos of the same order of magnitude.

			=== Odd harmonics ===
			{{Harmonics in equal\|196608}}

			== Selected intervals ==
	Below is a list for just intervals.		Below is a list for just intervals.
	{\| class="wikitable"		{\| class="wikitable"
	\|-		\|-
	! rowspan="2"\| ~~196608EDO <br>steps~~		! rowspan="2"\| #
	! rowspan="2"\| ~~cents <br>value~~		! rowspan="2"\| Cents
	! colspan="2"\| JI ~~interval~~		! colspan="2"\| JI Interval
	! rowspan="2"\| ~~error~~ <br>(~~steps~~)		! rowspan="2"\| Error <br>(Steps)
	! rowspan="2"\| ~~error~~ <br>(~~cents~~)		! rowspan="2"\| Error <br>(Cents)
	\|-		\|-
	! ~~\| ratio~~		! Ratio
	! ~~\| cents~~		! Cents
	\|-		\|-
	\| \| 0		\| \| 0

Royalmilktea: harmonics table

2022-10-07T07:14:27Z

harmonics table

← Older revision		Revision as of 07:14, 7 October 2022
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	'''196608EDO''' is the [[EDO\|equal division of the octave]] into 196608 parts of exact 0.006103515625 cents each, which is to say 2<sup>(1/196608)</sup> as a frequency ratio. Its adjacent step is known as the '''MIDI Tuning Standard unit''' (MTSU) or '''Tetradecamu''' (fourteenth MIDI-resolution unit, 14mu, 2<sup>14</sup> = 16384 equal divisions of the [[12edo]] semitone). The 14mu is specified in the MIDI spec (1983) as the smallest increment available for the pitch-bend controller, and as the frequency data format for MTS (1999). The 14mu is the smallest unit of tuning resolution which has ever been put into common musical use, and provides extremely accurate tuning in microtonal electronic music.		'''196608EDO''' is the [[EDO\|equal division of the octave]] into 196608 parts of exact 0.006103515625 cents each, which is to say 2<sup>(1/196608)</sup> as a frequency ratio. Its adjacent step is known as the '''MIDI Tuning Standard unit''' (MTSU) or '''Tetradecamu''' (fourteenth MIDI-resolution unit, 14mu, 2<sup>14</sup> = 16384 equal divisions of the [[12edo]] semitone). The 14mu is specified in the MIDI spec (1983) as the smallest increment available for the pitch-bend controller, and as the frequency data format for MTS (1999). The 14mu is the smallest unit of tuning resolution which has ever been put into common musical use, and provides extremely accurate tuning in microtonal electronic music.
			{{harmonics in equal\|196608\|start=2\|intervals=prime}}

	==Intervals==		==Intervals==