Father–3 equivalence continuum/Godtone's approach - Revision history

FloraC: Fix linking & formatting

2025-09-11T06:08:40Z

Fix linking & formatting

Godtone: add some nowraps

2025-03-06T17:00:31Z

add some nowraps

← Older revision		Revision as of 17:00, 6 March 2025
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	* 128/125 is fundamental because it uniquely defines the relatively-very-accurate (strongly form-fitting) representation of the 2.5 subgroup in 3edo.		* 128/125 is fundamental because it uniquely defines the relatively-very-accurate (strongly form-fitting) representation of the 2.5 subgroup in 3edo.
	* 25/24 is fundamental because it gives the trivial way to relate ~5/4 = 1\3 to ~3/2 = 2\3 as 2 generators in 3edo. (By contrast, using 16/15 requires taking the octave-complement of one of the generators. There is also another stronger argument against using 16/15 detailed later in this list.)		* 25/24 is fundamental because it gives the trivial way to relate ~5/4 = 1\3 to ~3/2 = 2\3 as 2 generators in 3edo. (By contrast, using 16/15 requires taking the octave-complement of one of the generators. There is also another stronger argument against using 16/15 detailed later in this list.)
	* Using 25/24 is also useful because we then know how many intervals between ~6/5 and ~5/4 are guaranteed in a nontrivial tuning; because 25/24 is divided into ''n'' equal parts, the answer is ''n'' - 1. Meanwhile, if ''n'' = ''a''/''b'' is not an integer (meaning ''b'' > 1), then 25/24 is divided into ''a'' equal parts of ~(128/125)<sup>1/''b''</sup>, giving a clear meaning to the numerator and denominator (though more meanings are discussed later).		* Using 25/24 is also useful because we then know how many intervals between ~6/5 and ~5/4 are guaranteed in a nontrivial tuning; because 25/24 is divided into ''n'' equal parts, the answer is {{nowrap\|''n'' - 1}}. Meanwhile, if {{nowrap\|''n'' {{=}} ''a''/''b''}} is not an integer (meaning ''b'' > 1), then 25/24 is divided into ''a'' equal parts of {{nowrap\|~(128/125)<sup>1/''b''</sup>}}, giving a clear meaning to the numerator and denominator (though more meanings are discussed later).
	* Because {{nowrap\|25/24 {{=}} ([[25/16]])/([[3/2]])}}, this has the consequence of clearly relating the ''n'' in {{nowrap\|(128/125)<sup>n</sup> {{=}} 25/24}} with how many 5/4's are used to reach 3/2 (when octave-reduced):		* Because {{nowrap\|25/24 {{=}} ([[25/16]])/([[3/2]])}}, this has the consequence of clearly relating the ''n'' in {{nowrap\|(128/125)<sup>n</sup> {{=}} 25/24}} with how many 5/4's are used to reach 3/2 (when octave-reduced):
	: * If {{nowrap\|''n'' {{=}} 0}}, then it takes no 128/125's to reach 25/24, implying 25/24's size is 0 (so that it's tempered out), meaning that 3/2 is reached via (5/4)<sup>2</sup>.		: * If {{nowrap\|''n'' {{=}} 0}}, then it takes no 128/125's to reach 25/24, implying 25/24's size is 0 (so that it's tempered out), meaning that 3/2 is reached via (5/4)<sup>2</sup>.

Godtone: chromatic -> dicot

2025-03-04T21:36:51Z

chromatic -> dicot

← Older revision		Revision as of 21:36, 4 March 2025
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	{{Breadcrumb}}		{{Breadcrumb}}
	The '''~~augmented–chromatic~~ equivalence continuum''' is a [[equivalence continuum\|continuum]] of [[5-limit]] [[regular temperament\|temperaments]] which equates a number of [[128/125]]'s (augmented commas) with the chroma, [[25/24]]. As such, it represents the continuum of all 5-limit temperaments supported by [[3edo]].		The '''augmented–dicot equivalence continuum''' is a [[equivalence continuum\|continuum]] of [[5-limit]] [[regular temperament\|temperaments]] which equates a number of [[128/125]]'s (augmented commas) with the dicot comma (or chroma), [[25/24]]. As such, it represents the continuum of all 5-limit temperaments supported by [[3edo]].

	This formulation has a number of specific reasons:		This formulation has a number of specific reasons:

Godtone: improve explanation of structure of 5/4 to 4/3 region in terms of n

2025-03-04T00:20:31Z

improve explanation of structure of 5/4 to 4/3 region in terms of n

← Older revision		Revision as of 00:20, 4 March 2025
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	* For ''n'' a nonnegative integer, half-integer or third-integer('''*'''), increasing ''n'' corresponds to increasingly sharp tunings of ~5/4. In the limit, as ''n'' goes to infinity, these all approach ~5/4 = 1\3, corresponding to [[augmented temperament]].		* For ''n'' a nonnegative integer, half-integer or third-integer('''*'''), increasing ''n'' corresponds to increasingly sharp tunings of ~5/4. In the limit, as ''n'' goes to infinity, these all approach ~5/4 = 1\3, corresponding to [[augmented temperament]].
	:: ('''*''' It is conjectured by [[User:Godtone]] that for a given choice of denominator ''b'' in ''n'' = ''a''/''b'', a larger value of ''a'' always corresponds to a sharper tuning of ~5/4, where the sharpness in a pure-octaves tuning is always strictly flat of 1\3, so that (more trivially) taking the limit as ''a'' goes to infinity, ~5/4 = 1\3. The intuition for why we might expect this to be true is that in a pure-2's pure-3's tuning, we are always constraining ~128/125's size to be equal to the appropriate relation to ~25/24, where as 2 and 3 are fixed, the ~5 is the only free variable and depending only on ''n'', with that ''n'' essentially indirectly specifying the degree of tempering.)		:: ('''*''' It is conjectured by [[User:Godtone]] that for a given choice of denominator ''b'' in ''n'' = ''a''/''b'', a larger value of ''a'' always corresponds to a sharper tuning of ~5/4, where the sharpness in a pure-octaves tuning is always strictly flat of 1\3, so that (more trivially) taking the limit as ''a'' goes to infinity, ~5/4 = 1\3. The intuition for why we might expect this to be true is that in a pure-2's pure-3's tuning, we are always constraining ~128/125's size to be equal to the appropriate relation to ~25/24, where as 2 and 3 are fixed, the ~5 is the only free variable and depending only on ''n'', with that ''n'' essentially indirectly specifying the degree of tempering.)
	* Also, if one is interested in what intervals are present between ~5/4 and ~4/3, it is simple to observe because (16/15)/(25/24) = 128/125, meaning for a nonnegative integer ''n'' there is exactly one more interval between ~5/4 and ~4/3 as between ~6/5 and ~5/4, and ~~more~~ generally, for rational ''n'' = ''a''/''b'', we have ''a'' - 1 intervals between ~6/5 and ~5/4 and because there is another ~128/125 between ~5/4 and ~4/3 we have ''a''/''b'' + 1 = ''a''/''b'' + ''b''/''b'' for the translated coordinates so that we have ''a'' + ''b'' - 1 intervals between ~5/4 and ~4/3, corresponding to splitting ~16/15 into ''a'' + ''b'' equal parts.		* Also, if one is interested in what intervals are present between ~5/4 and ~4/3, it is simple to observe because (16/15)/(25/24) = 128/125, meaning for a nonnegative integer ''n'' there is exactly one more interval between ~5/4 and ~4/3 as between ~6/5 and ~5/4. As there is ''n'' - 1 intervals between ~6/5 and ~5/4 (because of splitting ~25/24 into ''n'' parts), that means that (for nonnegative integer ''n'') there is exactly ''n'' intervals between ~5/4 and ~4/3. More generally, for rational ''n'' = ''a''/''b'', we have ''a'' - 1 intervals between ~6/5 and ~5/4 and because there is another ~128/125 between ~5/4 and ~4/3 we have ''a''/''b'' + 1 = ''a''/''b'' + ''b''/''b'' for the translated coordinates so that we have ''a'' + ''b'' - 1 intervals between ~5/4 and ~4/3, corresponding to splitting ~16/15 into ''a'' + ''b'' equal parts.

	Therefore, if ''n'' = ''a''/''b'' is a rational with ''b'' > 1 and ''b'' not a multiple of 3 (so that 3''a''/''b'' + 2 doesn't simplify), we reach prime 3 in a fractional number of generators of ~5/4, which means that the generator is not ~5/4 but rather ''b'' equal divisions of some octave-equivalent of ~5/4 or ~8/5, which as a result means that we reach prime 3 in ''b''(3''a''/''b'' + 2) = 3''a'' + 2''b'' generators, and also means that ~128/125 is split into ''b'' equal parts.		Therefore, if ''n'' = ''a''/''b'' is a rational with ''b'' > 1 and ''b'' not a multiple of 3 (so that 3''a''/''b'' + 2 doesn't simplify), we reach prime 3 in a fractional number of generators of ~5/4, which means that the generator is not ~5/4 but rather ''b'' equal divisions of some octave-equivalent of ~5/4 or ~8/5, which as a result means that we reach prime 3 in ''b''(3''a''/''b'' + 2) = 3''a'' + 2''b'' generators, and also means that ~128/125 is split into ''b'' equal parts.

Godtone: correct syntax of join

2025-03-03T00:47:15Z

correct syntax of join

← Older revision		Revision as of 00:47, 3 March 2025
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	\|-		\|-
	\| 10/3		\| 10/3
	\| {{nowrap\|147c & 150c = 291cc & 297cc }} {{nowrap\|(generator {{=}} negative ~[[magus comma]])}}		\| {{nowrap\|147c & 150c {{=}} 291cc & 297cc }} {{nowrap\|(generator {{=}} negative ~[[magus comma]])}}
	\| [[16320498564797493858533376/14551915228366851806640625\|(52 digits)]]		\| [[16320498564797493858533376/14551915228366851806640625\|(52 digits)]]
	\| {{ monzo\| 79 3 -36 }}		\| {{ monzo\| 79 3 -36 }}

Godtone: finish elaborating the precise meaning of n = 12/7 for the 3 & 612 microtemp

2025-03-03T00:43:20Z

finish elaborating the precise meaning of n = 12/7 for the 3 & 612 microtemp

← Older revision		Revision as of 00:43, 3 March 2025
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	The 3 & 118 microtemperament [[squarschmidt]] is at ''n'' = 7/4. Its generator is approximately 397{{cent}} so that four generators reaches 5/2, corresponding to the denominator of 4. The number of generators of ~(5/2)<sup>1/4</sup> needed to find prime 3 is thus four times the result of plugging ''n'' = 7/4 into 3''n'' + 2 , which is 3(7/4) + 2 = 21/4 + 8/4 = 29/4, that is, 29 generators.		The 3 & 118 microtemperament [[squarschmidt]] is at ''n'' = 7/4. Its generator is approximately 397{{cent}} so that four generators reaches 5/2, corresponding to the denominator of 4. The number of generators of ~(5/2)<sup>1/4</sup> needed to find prime 3 is thus four times the result of plugging ''n'' = 7/4 into 3''n'' + 2 , which is 3(7/4) + 2 = 21/4 + 8/4 = 29/4, that is, 29 generators.

	Finally, the 3 & 612 microtemperament at ''n'' = 12/7 is extremely complex, because to find prime 5, you need 7 times 3(12/7) + 2 = 36/7 + 14/7 = 50/7, that is, 50 generators, and is noted only because of being extremely close to the JIP and being supported by the 5-limit microtemperament [[612edo]]. The denominator of 7 indicates that 128/125 is split into 7 equal parts.		Finally, the 3 & 612 microtemperament at ''n'' = 12/7 is extremely complex, because to find prime 5, you need 7 times 3(12/7) + 2 = 36/7 + 14/7 = 50/7, that is, 50 generators, and is noted only because of being extremely close to the JIP and being supported by the 5-limit microtemperament [[612edo]]. The denominator (7) indicates that 128/125 is split into 7 equal parts, while the numerator indicates that each (128/125)<sup>1/7</sup> part represents (25/24)<sup>1/12</sup>, that is, a twelfth of 25/24.

	{\| class="wikitable center-1"		{\| class="wikitable center-1"

Godtone: make capitalization of temperaments consistent

2025-03-03T00:19:04Z

make capitalization of temperaments consistent

← Older revision		Revision as of 00:19, 3 March 2025
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	\|-		\|-
	\| 7/4		\| 7/4
	\| [[~~squarschmidt~~]] ({{nowrap\|3 & 118}})		\| [[Squarschmidt]] ({{nowrap\|3 & 118}})
	\| [[186773283746309210112/186264514923095703125\|(42 digits)]]		\| [[186773283746309210112/186264514923095703125\|(42 digits)]]
	\| {{ monzo\| 61 4 -29 }}		\| {{ monzo\| 61 4 -29 }}

Godtone: fix accidental breaking of table formatting

2025-03-03T00:12:58Z

fix accidental breaking of table formatting

← Older revision		Revision as of 00:12, 3 March 2025
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	Therefore, if ''n'' = ''a''/''b'' is a rational with ''b'' > 1 and ''b'' not a multiple of 3 (so that 3''a''/''b'' + 2 doesn't simplify), we reach prime 3 in a fractional number of generators of ~5/4, which means that the generator is not ~5/4 but rather ''b'' equal divisions of some octave-equivalent of ~5/4 or ~8/5, which as a result means that we reach prime 3 in ''b''(3''a''/''b'' + 2) = 3''a'' + 2''b'' generators, and also means that ~128/125 is split into ''b'' equal parts.		Therefore, if ''n'' = ''a''/''b'' is a rational with ''b'' > 1 and ''b'' not a multiple of 3 (so that 3''a''/''b'' + 2 doesn't simplify), we reach prime 3 in a fractional number of generators of ~5/4, which means that the generator is not ~5/4 but rather ''b'' equal divisions of some octave-equivalent of ~5/4 or ~8/5, which as a result means that we reach prime 3 in ''b''(3''a''/''b'' + 2) = 3''a'' + 2''b'' generators, and also means that ~128/125 is split into ''b'' equal parts.

	The just value of ''n'' is {{nowrap\|log(25/24) / log(128/125) {{=}} 1.72125…}} where {{nowrap\|''n'' {{=}} 2}} corresponds to the [[Würschmidt comma]].{\| class="wikitable center-1"		The just value of ''n'' is {{nowrap\|log(25/24) / log(128/125) {{=}} 1.72125…}} where {{nowrap\|''n'' {{=}} 2}} corresponds to the [[Würschmidt comma]].

			{\| class="wikitable center-1"
	\|+ style="font-size: 105%;" \| Temperaments with integer ''n''		\|+ style="font-size: 105%;" \| Temperaments with integer ''n''
	\|-		\|-

Godtone: clarify it's a repetition of information to remind the reader

2025-03-03T00:11:17Z

clarify it's a repetition of information to remind the reader

← Older revision		Revision as of 00:11, 3 March 2025
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	\| {{ monzo\| -7 0 3 }}		\| {{ monzo\| -7 0 3 }}
	\|}		\|}
	Notice that as ''n'' increases, we temper ~5/4 sharper and ~128/125 flatter (closer to unison), so that as ''n'' goes to infinity, ~5/4 goes to 1\3.		Notice that (as mentioned), as ''n'' increases, we temper ~5/4 sharper and ~128/125 flatter (closer to unison), so that as ''n'' goes to infinity, ~5/4 goes to 1\3.

	{\| class="wikitable center-1"		{\| class="wikitable center-1"

Godtone: 3 & 118 is named squarschmidt

2025-03-02T23:49:43Z

3 & 118 is named squarschmidt

← Older revision		Revision as of 23:49, 2 March 2025
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	\|-		\|-
	\| 7/4		\| 7/4
	\| {{nowrap\|3 & 118}}		\| [[squarschmidt]] ({{nowrap\|3 & 118}})
	\| [[186773283746309210112/186264514923095703125\|(42 digits)]]		\| [[186773283746309210112/186264514923095703125\|(42 digits)]]
	\| {{ monzo\| 61 4 -29 }}		\| {{ monzo\| 61 4 -29 }}
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	The simplest of these is [[mutt]] which has interesting properties discussed there. In regards to mutt, the fact that the denominator of ''n'' is a multiple of 3 tells us that it has a 1\3 period because it's contained in 3edo. The fact that the numerator is 5 tells us that 25/24 is split into 5 parts. From {{nowrap\|(128/125)<sup>n</sup> {{=}} 25/24}} we can thus deduce that each part is thus equal to ~cbrt(128/125) = (128/125)<sup>1/3</sup>, so that ~5/4 is found at 1\3 minus a third of a diesis, so that ~125/64 is found at thrice that. This observation is more general, leading to consideration of temperaments of third-integer ''n''.		The simplest of these is [[mutt]] which has interesting properties discussed there. In regards to mutt, the fact that the denominator of ''n'' is a multiple of 3 tells us that it has a 1\3 period because it's contained in 3edo. The fact that the numerator is 5 tells us that 25/24 is split into 5 parts. From {{nowrap\|(128/125)<sup>n</sup> {{=}} 25/24}} we can thus deduce that each part is thus equal to ~cbrt(128/125) = (128/125)<sup>1/3</sup>, so that ~5/4 is found at 1\3 minus a third of a diesis, so that ~125/64 is found at thrice that. This observation is more general, leading to consideration of temperaments of third-integer ''n''.

	The 3 & 118 microtemperament is at ''n'' = 7/4. Its generator is approximately 397{{cent}} so that four generators reaches 5/2, corresponding to the denominator of 4. The number of generators of ~(5/2)<sup>1/4</sup> needed to find prime 3 is thus four times the result of plugging ''n'' = 7/4 into 3''n'' + 2 , which is 3(7/4) + 2 = 21/4 + 8/4 = 29/4, that is, 29 generators.		The 3 & 118 microtemperament [[squarschmidt]] is at ''n'' = 7/4. Its generator is approximately 397{{cent}} so that four generators reaches 5/2, corresponding to the denominator of 4. The number of generators of ~(5/2)<sup>1/4</sup> needed to find prime 3 is thus four times the result of plugging ''n'' = 7/4 into 3''n'' + 2 , which is 3(7/4) + 2 = 21/4 + 8/4 = 29/4, that is, 29 generators.

	Finally, the 3 & 612 microtemperament at ''n'' = 12/7 is extremely complex, because to find prime 5, you need 7 times 3(12/7) + 2 = 36/7 + 14/7 = 50/7, that is, 50 generators, and is noted only because of being extremely close to the JIP and being supported by the 5-limit microtemperament [[612edo]]. The denominator of 7 indicates that 128/125 is split into 7 equal parts.		Finally, the 3 & 612 microtemperament at ''n'' = 12/7 is extremely complex, because to find prime 5, you need 7 times 3(12/7) + 2 = 36/7 + 14/7 = 50/7, that is, 50 generators, and is noted only because of being extremely close to the JIP and being supported by the 5-limit microtemperament [[612edo]]. The denominator of 7 indicates that 128/125 is split into 7 equal parts.