16edo - Revision history

Vector at 09:36, 22 April 2026

2026-04-22T09:36:45Z

← Older revision		Revision as of 09:36, 22 April 2026
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	16edo does not approximate prime 3 very well, with its [[Perfect fifth\|fifth]] being over 25 cents flat. However, this is still within the range where it can function as a fifth in chords. This means that in a way, 16edo is sort of an opposite to [[12edo]] in the 7-limit, approximating 7 well, 5 decently, and 3 poorly, whereas 12edo approximates 3 well, 5 decently, and 7 poorly. 16edo's flat fifth means that the closest analog to a standard diatonic scale is [[antidiatonic]], with a pattern of 2-2-3-2-2-2-3. 16edo shares 12edo's minor third, which represents 6/5 in both systems (if 16edo's fifth is treated as 3/2).		16edo does not approximate prime 3 very well, with its [[Perfect fifth\|fifth]] being over 25 cents flat. However, this is still within the range where it can function as a fifth in chords. This means that in a way, 16edo is sort of an opposite to [[12edo]] in the 7-limit, approximating 7 well, 5 decently, and 3 poorly, whereas 12edo approximates 3 well, 5 decently, and 7 poorly. 16edo's flat fifth means that the closest analog to a standard diatonic scale is [[antidiatonic]], with a pattern of 2-2-3-2-2-2-3. 16edo shares 12edo's minor third, which represents 6/5 in both systems (if 16edo's fifth is treated as 3/2).

	As a multiple of 4, 16edo has an exceptionally accurate 19 at 300 cents,		As a multiple of 4, 16edo has an exceptionally accurate 19 at 300 cents, suggesting an asymmetric interpretation of the thirds as 5/4 and 19/16, stacking to a fifth of 95/64.

	16edo's fifth is almost exactly 1/3 of an eka off from just, making it suitable for [[erac]] theory (where two flat fifths and one sharp fifth stack to make an in-tune 27/16). The scale generated by this system does not resemble a diatonic scale. (Tripling 16edo to resolve this results in 48edo, which shares 12edo's fifth and 16edo's flat 5/4 and in-tune 7/4.)		16edo's fifth is almost exactly 1/3 of an eka off from just, making it suitable for [[erac]] theory (where two flat fifths and one sharp fifth stack to make an in-tune 27/16). The scale generated by this system does not resemble a diatonic scale. (Tripling 16edo to resolve this results in 48edo, which shares 12edo's fifth and 16edo's flat 5/4 and in-tune 7/4.)

Vector at 09:36, 22 April 2026

2026-04-22T09:36:00Z

← Older revision		Revision as of 09:36, 22 April 2026
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	=== JI approximation ===		=== JI approximation ===
	16edo does not approximate prime 3 very well, with its [[Perfect fifth\|fifth]] being over 25 cents flat. However, this is still within the range where it can function as a fifth in chords. This means that in a way, 16edo is sort of an opposite to [[12edo]] in the 7-limit, approximating 7 well, 5 decently, and 3 poorly, whereas 12edo approximates 3 well, 5 decently, and 7 poorly. 16edo's flat fifth means that the closest analog to a standard diatonic scale is [[antidiatonic]], with a pattern of 2-2-3-2-2-2-3. 16edo shares 12edo's minor third, which represents 6/5 in both systems (if 16edo's fifth is treated as 3/2).		16edo does not approximate prime 3 very well, with its [[Perfect fifth\|fifth]] being over 25 cents flat. However, this is still within the range where it can function as a fifth in chords. This means that in a way, 16edo is sort of an opposite to [[12edo]] in the 7-limit, approximating 7 well, 5 decently, and 3 poorly, whereas 12edo approximates 3 well, 5 decently, and 7 poorly. 16edo's flat fifth means that the closest analog to a standard diatonic scale is [[antidiatonic]], with a pattern of 2-2-3-2-2-2-3. 16edo shares 12edo's minor third, which represents 6/5 in both systems (if 16edo's fifth is treated as 3/2).

			As a multiple of 4, 16edo has an exceptionally accurate 19 at 300 cents,

	16edo's fifth is almost exactly 1/3 of an eka off from just, making it suitable for [[erac]] theory (where two flat fifths and one sharp fifth stack to make an in-tune 27/16). The scale generated by this system does not resemble a diatonic scale. (Tripling 16edo to resolve this results in 48edo, which shares 12edo's fifth and 16edo's flat 5/4 and in-tune 7/4.)		16edo's fifth is almost exactly 1/3 of an eka off from just, making it suitable for [[erac]] theory (where two flat fifths and one sharp fifth stack to make an in-tune 27/16). The scale generated by this system does not resemble a diatonic scale. (Tripling 16edo to resolve this results in 48edo, which shares 12edo's fifth and 16edo's flat 5/4 and in-tune 7/4.)
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	=== Scales and chords ===		=== Scales and chords ===
	The [[antidiatonic]] scale can be understood as somewhat of an "inverted" counterpart of the standard major scale. In antidiatonic, minor thirds are evenly divisible in two while major thirds are not, which is backwards from 12edo. Additionally, there are perfect and augmented fifths rather than perfect and diminished.		The [[antidiatonic]] scale is 16edo's primary functional feature, and 16edo is one of only a small number of edos in which the generator of antidiatonic can be understood to be a perfect fifth (these are also the edos that, alongside [[7edo]] and [[2edo]], support mavila temperament). The antidiatonic scale can be understood as somewhat of an "inverted" counterpart of the standard major scale. In antidiatonic, minor thirds are evenly divisible in two while major thirds are not, which is backwards from 12edo. Additionally, there are perfect and augmented fifths rather than perfect and diminished. If the chain of fifths is used to render standard diatonic songs in antidiatonic, notes that are flat in diatonic become sharp, and interval qualities switch places.

	Due to 16edo dividing the fifth into three parts, 16edo supports slendric [[Collection of chords#Slendric tetrad\|arto and tendo]] triads alongside its major and minor triads. These may be treated as forms of suspended triads or as augmented and diminished thirds, depending on the framework being used.		Due to 16edo dividing the fifth into three parts, 16edo supports slendric [[Collection of chords#Slendric tetrad\|arto and tendo]] triads alongside its major and minor triads. These may be treated as forms of suspended triads or as augmented and diminished thirds, depending on the framework being used.

	16edo shares its mapping of 5/4, 8/7, and 3/2 in terms of edosteps with [[15edo]] ~~and with~~ the [[Carlos Alpha]] scale.		16edo shares its mapping of 5/4, 8/7, and 3/2 in terms of edosteps with [[15edo]], as both function as the extreme tunings of the [[Carlos Alpha]] scale.

	=== Practicality ===		=== Practicality ===
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	=== Regular temperaments ===		=== Regular temperaments ===
	16edo shares [[Mabilic]] with 25edo and Jubilic with 26edo.		16edo shares [[Mabilic]] with 25edo and Jubilic with 26edo. In particular, it is the intersection of [[Mavila]] and [[Trismegistus]], two extensions of Mabilic that map 3/2 to 1 generator and -15 generators respectively (hence 16edo).

	== Notation ==		== Notation ==
	The most common notation system for 16edo is [[Diatonic notation#Melodic antidiatonic notation\|melodic antidiatonic notation]], where the seven nominals are the antidiatonic scale, and # and b raise and lower by one eka respectively. Armodue notation can also be used, where the numbers 1-9 are used for notes of the 9-note [[armotonic]] scale, and each staff corresponds to a single octave (indicated by a numeric clef).		The most common notation system for 16edo is [[Diatonic notation#Melodic antidiatonic notation\|melodic antidiatonic notation]], where the seven nominals are the antidiatonic scale, and # and b raise and lower by one eka respectively. Armodue notation can also be used, where the numbers 1-9 are used for notes of the 9-note [[armotonic]] scale, and each staff corresponds to a single octave (indicated by a numeric clef). Additionally, especially when retuning songs from diatonic systems to 16edo, it is useful to allow # to stand for a lowering in pitch and b to stand for a raising in pitch; this is however quite unintuitive and is not widely used.

	== Armodue theory ==		== Armodue theory ==
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	\|Tenth		\|Tenth
	\|}		\|}
			A notable characteristic of 16edo's selection of intervals is that despite the apparent interordinals in the 450- and 225-cent intervals, 16edo in fact has no true interordinals, because the fourth is not evenly divided. This is a consequence of the "switching places" behavior of major and minor intervals in antidiatonic tunings. An interordinal third-fourth in 16edo would be 412.5 cents, an interval which only exists in 32edo (which has a more accurate fifth regardless).

	{{Cat\|Edos}}{{Navbox EDO}}		{{Cat\|Edos}}{{Navbox EDO}}

Vector at 09:25, 22 April 2026

2026-04-22T09:25:40Z

← Older revision		Revision as of 09:25, 22 April 2026
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	16edo does not approximate prime 3 very well, with its [[Perfect fifth\|fifth]] being over 25 cents flat. However, this is still within the range where it can function as a fifth in chords. This means that in a way, 16edo is sort of an opposite to [[12edo]] in the 7-limit, approximating 7 well, 5 decently, and 3 poorly, whereas 12edo approximates 3 well, 5 decently, and 7 poorly. 16edo's flat fifth means that the closest analog to a standard diatonic scale is [[antidiatonic]], with a pattern of 2-2-3-2-2-2-3. 16edo shares 12edo's minor third, which represents 6/5 in both systems (if 16edo's fifth is treated as 3/2).		16edo does not approximate prime 3 very well, with its [[Perfect fifth\|fifth]] being over 25 cents flat. However, this is still within the range where it can function as a fifth in chords. This means that in a way, 16edo is sort of an opposite to [[12edo]] in the 7-limit, approximating 7 well, 5 decently, and 3 poorly, whereas 12edo approximates 3 well, 5 decently, and 7 poorly. 16edo's flat fifth means that the closest analog to a standard diatonic scale is [[antidiatonic]], with a pattern of 2-2-3-2-2-2-3. 16edo shares 12edo's minor third, which represents 6/5 in both systems (if 16edo's fifth is treated as 3/2).

	16edo's fifth is almost exactly 1/3 of an eka off from just, making it suitable for [[erac]] theory (where two flat fifths and one sharp fifth stack to make an in-tune 27/16). The scale generated by this system does not resemble a diatonic scale. (Tripling 16edo to resolve this results in 48edo, which shares 12edo's fifth and 16edo's flat 5/4 and 7/4.)		16edo's fifth is almost exactly 1/3 of an eka off from just, making it suitable for [[erac]] theory (where two flat fifths and one sharp fifth stack to make an in-tune 27/16). The scale generated by this system does not resemble a diatonic scale. (Tripling 16edo to resolve this results in 48edo, which shares 12edo's fifth and 16edo's flat 5/4 and in-tune 7/4.)
	{{Harmonics in ED\|16\|31}}		{{Harmonics in ED\|16\|31}}

Vector at 09:25, 22 April 2026

2026-04-22T09:25:29Z

← Older revision		Revision as of 09:25, 22 April 2026
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	16edo does not approximate prime 3 very well, with its [[Perfect fifth\|fifth]] being over 25 cents flat. However, this is still within the range where it can function as a fifth in chords. This means that in a way, 16edo is sort of an opposite to [[12edo]] in the 7-limit, approximating 7 well, 5 decently, and 3 poorly, whereas 12edo approximates 3 well, 5 decently, and 7 poorly. 16edo's flat fifth means that the closest analog to a standard diatonic scale is [[antidiatonic]], with a pattern of 2-2-3-2-2-2-3. 16edo shares 12edo's minor third, which represents 6/5 in both systems (if 16edo's fifth is treated as 3/2).		16edo does not approximate prime 3 very well, with its [[Perfect fifth\|fifth]] being over 25 cents flat. However, this is still within the range where it can function as a fifth in chords. This means that in a way, 16edo is sort of an opposite to [[12edo]] in the 7-limit, approximating 7 well, 5 decently, and 3 poorly, whereas 12edo approximates 3 well, 5 decently, and 7 poorly. 16edo's flat fifth means that the closest analog to a standard diatonic scale is [[antidiatonic]], with a pattern of 2-2-3-2-2-2-3. 16edo shares 12edo's minor third, which represents 6/5 in both systems (if 16edo's fifth is treated as 3/2).

	16edo's fifth is almost exactly 1/3 of an eka off from just, making it suitable for [[erac]] theory (where two flat fifths and one sharp fifth stack to make an in-tune 27/16). The scale generated by this system does not resemble a diatonic scale.		16edo's fifth is almost exactly 1/3 of an eka off from just, making it suitable for [[erac]] theory (where two flat fifths and one sharp fifth stack to make an in-tune 27/16). The scale generated by this system does not resemble a diatonic scale. (Tripling 16edo to resolve this results in 48edo, which shares 12edo's fifth and 16edo's flat 5/4 and 7/4.)
	{{Harmonics in ED\|16\|31}}		{{Harmonics in ED\|16\|31}}

Vector at 01:17, 14 February 2026

2026-02-14T01:17:28Z

← Older revision		Revision as of 01:17, 14 February 2026
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	\|2		\|2
	\|150		\|150
	\|~~Pentaminor~~ second		\|Nearminor second
	\|Major second		\|Major second
	\|-		\|-
	\|3		\|3
	\|225		\|225
	\|~~Pentamajor~~ second		\|Nearmajor second
	\|Minor third		\|Minor third
	\|-		\|-
	\|4		\|4
	\|300		\|300
	\|~~Pentaminor~~ third		\|Nearminor third
	\|Major third		\|Major third
	\|-		\|-
	\|5		\|5
	\|375		\|375
	\|~~Pentamajor~~ third		\|Nearmajor third
	\|Minor fourth		\|Minor fourth
	\|-		\|-
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	\|11		\|11
	\|825		\|825
	\|~~Pentaminor~~ sixth		\|Nearminor sixth
	\|Major seventh		\|Major seventh
	\|-		\|-
	\|12		\|12
	\|900		\|900
	\|~~Pentamajor~~ sixth		\|Nearmajor sixth
	\|Minor eighth		\|Minor eighth
	\|-		\|-
	\|13		\|13
	\|975		\|975
	\|~~Pentaminor~~ seventh		\|Nearminor seventh
	\|Major eighth		\|Major eighth
	\|-		\|-
	\|14		\|14
	\|1050		\|1050
	\|~~Pentamajor~~ seventh		\|Nearmajor seventh
	\|Minor ninth		\|Minor ninth
	\|-		\|-
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	\|Tenth		\|Tenth
	\|}		\|}
	{{Cat\|Edos}}		{{Cat\|Edos}}{{Navbox EDO}}

Vector at 01:22, 8 February 2026

2026-02-08T01:22:43Z

← Older revision		Revision as of 01:22, 8 February 2026
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			[[File:Valentine.png\|thumb\|A comparison of 15edo and 16edo, showing their similarities (see [[Carlos Alpha]]) and 16edo's antidiatonic scale.]]
	'''16edo''' is the tuning which divides the octave into 16 equal parts of 75 cents each. It is notable for its antidiatonic scale.		'''16edo''' is the tuning which divides the octave into 16 equal parts of 75 cents each. It is notable for its antidiatonic scale.

Inthar: /* Table of intervals */

2026-01-25T17:09:05Z

Table of intervals

← Older revision		Revision as of 17:09, 25 January 2026
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	\|Tenth		\|Tenth
	\|}		\|}
			{{Cat\|Edos}}

Inthar: /* Regular temperaments */

2026-01-11T13:10:44Z

Regular temperaments

← Older revision		Revision as of 13:10, 11 January 2026
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	=== Regular temperaments ===		=== Regular temperaments ===
	16edo shares [[~~mabilic~~]] with 25edo and ~~jubilic~~ with 26edo.		16edo shares [[Mabilic]] with 25edo and Jubilic with 26edo.

	== Notation ==		== Notation ==

Lériendil: /* Armodue theory */

2026-01-11T06:55:38Z

Armodue theory

← Older revision		Revision as of 06:55, 11 January 2026
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	Functional consonance is assigned linearly based on interval size, from the closest to the unison/octave being the most dissonant to the closest to the tritone being the most consonant, with the exception of the tritone and octave themselves, which serve as the most dissonant and consonant intervals respectively.		Functional consonance is assigned linearly based on interval size, from the closest to the unison/octave being the most dissonant to the closest to the tritone being the most consonant, with the exception of the tritone and octave themselves, which serve as the most dissonant and consonant intervals respectively.
	{\| class="wikitable"		{\| class="wikitable center-1 center-2"
	\|+		\|+
	! colspan="2" \|Interval		! colspan="2" \|Interval
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	\|		\|
	\|-		\|-
	\| colspan="2" \|6		\| colspan="2" \|8
	\|Unstable dissonance		\|Unstable dissonance
	\|Any chord containing this interval is considered unstable, meaning that the standard 4:5:6:7 harmonic tetrad is an unstable chord, and omitting the 5 or 7 can lead to a stabilized sound.		\|Any chord containing this interval is considered unstable, meaning that the standard 4:5:6:7 harmonic tetrad is an unstable chord, and omitting the 5 or 7 can lead to a stabilized sound.

Vector at 23:04, 10 January 2026

2026-01-10T23:04:26Z

← Older revision		Revision as of 23:04, 10 January 2026
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	=== Regular temperaments ===		=== Regular temperaments ===
	16edo shares mabilic with 25edo,		16edo shares [[mabilic]] with 25edo and jubilic with 26edo.

	== Notation ==		== Notation ==