Septimal-diatonic equivalence continuum - Revision history

ArrowHead294: Changed redirect target from Archytas-diatonic equivalence continuum to Archytas–diatonic equivalence continuum

2024-12-18T14:45:22Z

Changed redirect target from Archytas-diatonic equivalence continuum to Archytas–diatonic equivalence continuum

← Older revision		Revision as of 14:45, 18 December 2024
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	#redirect [[~~Archytas-diatonic~~ equivalence continuum]]		#redirect [[Archytas–diatonic equivalence continuum]]

FloraC: Redirected page to Archytas-diatonic equivalence continuum

2022-04-19T13:01:50Z

Redirected page to Archytas-diatonic equivalence continuum

← Older revision		Revision as of 13:01, 19 April 2022
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	~~The '''septimal-diatonic equivalence continuum''' is a continuum of temperaments which equate a number of~~ [[~~Septimal comma\|septimal commas (64/63)]] with the [[256/243\|limma (256/243)]].~~		#redirect [[Archytas-diatonic equivalence continuum]]

	All temperaments in the continuum satisfy (64/63)<sup>''n''</sup> ~ 256/243. Varying ''n'' results in different temperaments listed in the table below. It converges to [[Archy]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 2.3.7 subgroup temperaments supported by [[5edo]] due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of ''n'' is 3.3093…, and temperaments near this tend to be the most accurate ones.

	256/243 is the characteristic [[3-limit]] comma tempered out in [[5edo]]. In each case, we notice that ''n'' equals the order of harmonic 7 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the MOS scale. However, if we let ''k'' = ''n'' - 1 (meaning ''n'' = ''k'' + 1) so that ''k'' = 0 means ''n'' = 1, ''k'' = -1 means ''n'' = 0, etc. then the continuum corresponds to (64/63)<sup>''k''</sup> = 28/27, which might be a preferred way of conceptualising it because:
	*28/27 is the classic 2.3.7 diatonic semitone, notable in 2.3.7 as the difference between 9/8 and 7/6, so this shifted continuum could also logically be termed the "septimal-diatonic equivalence continuum". This means that at ''k'' = 0, 9/8, and 7/6 are mapped to the same interval while 64/63 becomes independent of 28/27 (meaning 64/63 may or may not be tempered) because the relation becomes (64/63)<sup>0</sup> ~ 1/1 ~ 28/27.
	*''k'' = 1 and upwards (up to a point) represent temperaments with (the potential for) reasonably good accuracy as equating at least one 64/63 with 28/27 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be ''k'' = 2, with the only exception being Archy (''n'' = ''k'' = (unsigned) infinity). (Temperaments corresponding to ''k'' = 0, -1, -2 are comparatively low-accuracy to the point of developing various intriguing structures and consequences.)
	*28/27 is the simplest ratio to be tempered in the continuum, the second simplest being 49/48 at k = -2, which together are the two smallest 2.3.7 [[List of superparticular intervals\|superparticular intervals]] and, along with 64/63, the only superparticular intervals in the continuum.
	~~{\| class="wikitable center-1 center-2"~~
	~~\|+Temperaments in the continuum~~
	\|-
	~~! rowspan="2" \|''k'' = ''n'' − 1~~
	~~! rowspan="2" \|''n'' = ''k'' + 2~~
	~~! rowspan="2" \|Temperament~~
	~~! colspan="2" \|Comma~~
	\|-
	~~!Ratio~~
	~~!Monzo~~
	\|-
	~~\| -2~~
	\|0
	\|
	~~\|[[256/243~~]]
	~~\|[8, -5⟩~~
	\|-
	~~\| -1~~
	\|1
	~~\|[[Trienstonic clan#No-fives trienstonic\|No-fives trienstonic]]~~
	~~\|[[28/27]]~~
	~~\|[2, -3, 1⟩~~
	\|-
	\|0
	\|2
	~~\|[[Slendro clan#Semaphore\|Semaphore]]~~
	~~\|[[49/48]]~~
	~~\|[-4, -1, 2⟩~~
	\|-
	\|1
	\|3
	~~\|[[Slendric]]~~
	~~\|[[1029/1024]]~~
	~~\|[-10, 1, 3⟩~~
	\|-
	\|2
	\|4
	~~\|[http://x31eq.com/cgi-bin/rt.cgi?ets=5_58&limit=2_3_7 2.3.7 Vulture]~~
	~~\|65536/64827~~
	~~\|[16, -3, -4⟩~~
	\|-
	\|3
	\|5
	~~\|[http://x31eq.com/cgi-bin/rt.cgi?ets=5_70&limit=2_3_7 2.3.7 Saquinru]~~
	~~\|4194304/4084101~~
	~~\|[22, -5, -5⟩~~
	\|-
	\|…
	\|…
	\|…
	\|…
	\|
	\|-
	\|∞
	\|∞
	~~\|[[Archytas clan#Archy\|Archy]]~~
	~~\|[[64/63]]~~
	~~\|[6, -2, -1⟩~~
	\|}

Lhearne: created page - copyed from diatonic-chromatic equivalence continuum page, and edited

2022-04-18T17:17:45Z

created page - copyed from diatonic-chromatic equivalence continuum page, and edited

New page

The '''septimal-diatonic equivalence continuum''' is a continuum of temperaments which equate a number of [[Septimal comma|septimal commas (64/63)]] with the [[256/243|limma (256/243)]].

All temperaments in the continuum satisfy (64/63)''n'' ~ 256/243. Varying ''n'' results in different temperaments listed in the table below. It converges to [[Archy]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 2.3.7 subgroup temperaments supported by [[5edo]] due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of ''n'' is 3.3093…, and temperaments near this tend to be the most accurate ones.

256/243 is the characteristic [[3-limit]] comma tempered out in [[5edo]]. In each case, we notice that ''n'' equals the order of harmonic 7 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the MOS scale. However, if we let ''k'' = ''n'' - 1 (meaning ''n'' = ''k'' + 1) so that ''k'' = 0 means ''n'' = 1, ''k'' = -1 means ''n'' = 0, etc. then the continuum corresponds to (64/63)''k'' = 28/27, which might be a preferred way of conceptualising it because:
*28/27 is the classic 2.3.7 diatonic semitone, notable in 2.3.7 as the difference between 9/8 and 7/6, so this shifted continuum could also logically be termed the "septimal-diatonic equivalence continuum". This means that at ''k'' = 0, 9/8, and 7/6 are mapped to the same interval while 64/63 becomes independent of 28/27 (meaning 64/63 may or may not be tempered) because the relation becomes (64/63)0 ~ 1/1 ~ 28/27.
*''k'' = 1 and upwards (up to a point) represent temperaments with (the potential for) reasonably good accuracy as equating at least one 64/63 with 28/27 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be ''k'' = 2, with the only exception being Archy (''n'' = ''k'' = (unsigned) infinity). (Temperaments corresponding to ''k'' = 0, -1, -2 are comparatively low-accuracy to the point of developing various intriguing structures and consequences.)
*28/27 is the simplest ratio to be tempered in the continuum, the second simplest being 49/48 at k = -2, which together are the two smallest 2.3.7 [[List of superparticular intervals|superparticular intervals]] and, along with 64/63, the only superparticular intervals in the continuum.
{| class="wikitable center-1 center-2"
|+Temperaments in the continuum
|-
! rowspan="2" |''k'' = ''n'' − 1
! rowspan="2" |''n'' = ''k'' + 2
! rowspan="2" |Temperament
! colspan="2" |Comma
|-
!Ratio
!Monzo
|-
| -2
|0
|
|[[256/243]]
|[8, -5⟩
|-
| -1
|1
|[[Trienstonic clan#No-fives trienstonic|No-fives trienstonic]]
|[[28/27]]
|[2, -3, 1⟩
|-
|0
|2
|[[Slendro clan#Semaphore|Semaphore]]
|[[49/48]]
|[-4, -1, 2⟩
|-
|1
|3
|[[Slendric]]
|[[1029/1024]]
|[-10, 1, 3⟩
|-
|2
|4
|[http://x31eq.com/cgi-bin/rt.cgi?ets=5_58&limit=2_3_7 2.3.7 Vulture]
|65536/64827
|[16, -3, -4⟩
|-
|3
|5
|[http://x31eq.com/cgi-bin/rt.cgi?ets=5_70&limit=2_3_7 2.3.7 Saquinru]
|4194304/4084101
|[22, -5, -5⟩
|-
|…
|…
|…
|…
|
|-
|∞
|∞
|[[Archytas clan#Archy|Archy]]
|[[64/63]]
|[6, -2, -1⟩
|}