Father–3 equivalence continuum: Difference between revisions

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Remove the k-continuum since no one is actively arguing for it. Also remove the 3 & 33c temp, which is unenlighted result of looking at the continuum that way
 
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The '''chromatic-diatonic equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[25/24|chromatic semitones (25/24)]] with [[16/15|diatonic semitones (16/15)]].
The '''father–3 equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equate a number of [[16/15|classical diatonic semitones (16/15)]] with the [[32/27|Pythagorean minor third (32/27)]].


All temperaments in the continuum satisfy (25/24)<sup>''n''</sup> ~ 16/15. Varying ''n'' results in different temperaments listed in the table below. It converges to [[dicot]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[3edo]] (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 approximately 1.58097..., and temperaments having ''n'' near this value tend to be the most accurate ones.  
Note that because 3et is a record equal temperament in the [[2.5 subgroup]], the continuum can be conceptualized as the [[Father–3 equivalence continuum/Godtone's approach|''augmented–dicot equivalence continuum'']], which Godtone argues is easier to understand, with characteristic 2.5-subgroup [[comma]] [[128/125]] as the interval with a single factor of 3 is [[25/24]].


{| class="wikitable center-1 center-2"
All temperaments in the continuum satisfy {{nowrap|(16/15)<sup>''n''</sup> ~ 32/27}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[father]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[3edo]] 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 approximately 2.63252…, and temperaments having ''n'' near this value tend to be the most accurate ones.
|+ Temperaments in the continuum
 
32/27 is the characteristic 3-limit comma tempered out in 3edo. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1|3]] in the generator chain.
 
{| class="wikitable center-1"
|+ style="font-size: 105%;" | Temperaments with integer ''n''
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
Line 12: Line 16:
! Ratio
! Ratio
! Monzo
! Monzo
|-
| -1
| [[Yo]]
| [[10/9]]
| {{monzo|1 -2 1}}
|-
|-
| 0
| 0
| [[Father]]
| [[Very low accuracy temperaments #Alteraugment|Alteraugment]]
| [[16/15]]
| [[32/27]]
| {{monzo|4 -1 -1}}
| {{Monzo| 5 -3 }}
|-
|-
| 1
| 1
| [[Augmented]]
| [[Very low accuracy temperaments #Antonian|Antonian]]
| [[128/125]]
| [[10/9]]
| {{monzo|7 0 -3}}
| {{Monzo| 1 -2 1 }}
|-
|-
| 2
| 2
| [[Magic]]
| [[Dicot]]
| [[3125/3072]]
| [[25/24]]
| {{monzo|10 1 -5}}
| {{Monzo| -3 -1 2 }}
|-
|-
| 3
| 3
| [[Wesley family|Wesley]]
| [[Augmented (temperament)|Augmented]]
| 78125/73728
| [[128/125]]
| {{monzo| 13 2 -7}}
| {{Monzo| 7 0 -3 }}
|-
|-
| 4
| 4
| 3 & 33c
| [[Smate]]
| 1953125/1769472
| [[2048/1875]]
| {{monzo| 16 3 -9 }}
| {{Monzo| 11 -1 -4 }}
|-
|-
| …
| …
Line 49: Line 48:
|-
|-
| ∞
| ∞
| [[Father]]
| [[16/15]]
| {{Monzo| 4 -1 -1 }}
|}
We may invert the continuum by setting ''m'' such that {{nowrap| 1/''m'' + 1/''n'' {{=}} 1 }}. This may be called the ''antonian–3 equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.61255…
{| class="wikitable center-1"
|+ style="font-size: 105%;" | Temperaments with integer ''m''
|-
! rowspan="2" | ''m''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| 0
| [[Very low accuracy temperaments #Alteraugment|Alteraugment]]
| [[32/27]]
| {{Monzo| 5 -3 }}
|-
| 1
| [[Father]]
| [[16/15]]
| {{Monzo| 4 -1 -1 }}
|-
| 2
| [[Dicot]]
| [[Dicot]]
| [[25/24]]
| [[25/24]]
| {{monzo| -3 -1 2}}
| {{Monzo| -3 -1 2 }}
|-
| …
| …
| …
| …
|-
| ∞
| [[Very low accuracy temperaments #Antonian|Antonian]]
| [[10/9]]
| {{Monzo| 1 -2 1 }}
|}
 
{| class="wikitable"
|+ style="font-size: 105%;" | Temperaments with fractional ''n'' and ''m''
|-
! ''n'' !! ''m''!! Temperament || Comma
|-
| 7/3 = 2.{{overline|3}} || 7/4 = 1.75 || [[Wesley]] || {{monzo| -13 -2 7 }}
|-
| 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Magic]] || {{monzo| -10 -1 5 }}
|-
| 21/8 = 2.625 || 21/13 = 1.{{overline|615384}} || [[Mutt]] || {{monzo| -44 -3 21 }}
|-
| 29/11 = 2.{{overline|63}} || 29/18 = 1.6{{overline|1}} || [[Squarschmidt]] || {{monzo| 61 4 -29 }}
|-
| 8/3 = 2.{{overline|6}} || 8/5 = 1.6 || [[Würschmidt]] || {{monzo| 17 1 -8 }}
|-
| 19/7 = 2.{{overline|714285}} || 19/12 = 1.58{{overline|3}} || [[#Isnes|Isnes]] || {{monzo| 41 2 -19 }}
|-
| 11/4 = 2.75 || 11/7 = 1.{{overline|571428}} || [[Magus]] || {{monzo| 24 1 -11 }}
|}
|}


Examples of temperaments with fractional values of ''n'':
== Mutt (5-limit) ==
* [[Very low accuracy temperaments#Alteraugment|Alteraugment]] (''n'' = -0.5)
{{Main| Mutt }}
* [[Symbolic]] (''n'' = 0.5)
: ''For extensions, see [[Horwell temperaments #Mutt]].''  
* [[Würschmidt]] (''n'' = 1.5)
 
* Isnes (''n'' = 1.6)
[[Subgroup]]: 2.3.5
* [[Sensamagic clan#Magus|Magus]] (''n'' = 5/3 = 1.{{Overline|6}})
 
[[Comma list]]: {{monzo| -44 -3 21 }}


== 3 & 33c ==
{{Mapping|legend=1| 3 -2 6 | 0 7 1 }}
: mapping generators: ~98304/78125, ~5/4


Comma list: {{monzo| 16 3 -9 }}
[[Optimal tuning]]s:
* [[WE]]: ~98304/78125 = 400.0227{{c}}, ~5/4 = 386.0017{{c}} (~393216/390625 = 14.0210{{c}})
: [[error map]]: {{val| +0.068 +0.012 -0.176 }}
* [[CWE]]: ~98304/78125 = 400.0000{{c}}, ~5/4 = 385.9858{{c}} (~393216/390625 = 14.0142{{c}})
: error map: {{val| 0.000 -0.055 -0.328 }}


POTE generator: 34.0971 cents
{{Optimal ET sequence|legend=1| 84, 87, 171, 771, 942, 1113, 1284, 1455, 4194cc, 5649cc }}


Mapping: [{{val| 3 5 7 }}, {{val| 0 -3 -1 }}]
[[Badness]] (Sintel): 3.81


{{Val list|legend=1| 3, 6, 9b, 33c }}
== Isnes ==
Isnes is so called because the generator is half of a [[5/2]] major tenth, in a similar way that [[sensi]] has a generator of half a [[5/3]] major sixth. This corresponds to {{nowrap|''n'' {{=}} 19/7 }} and {{nowrap| ''m'' {{=}} 19/12 }}.


[http://x31eq.com/cgi-bin/rt.cgi?ets=3_33c&limit=5 The temperament finder - 5-limit 3 & 33c]
[[Subgroup]]: 2.3.5


== Symbolic ==
[[Comma list]]: {{monzo| 41 2 -19 }}


Comma list: [[2048/1875]]
{{Mapping|legend=1| 1 -11 1 | 0 19 2 }}
: mapping generators: ~2, ~3145728/1953125


POTE generator: ~5/4 = 420.8548 cents
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2782{{c}}, ~3145728/1953125 = 794.4174{{c}}
: [[error map]]: {{val| -0.722 -0.090 +1.799 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3145728/1953125 = 794.8728{{c}}
: error map: {{val| 0.000 +0.628 +3.432 }}


Mapping: [{{val| 1 3 2 }}, {{val| 0 -4 1 }}]
{{Optimal ET sequence|legend=1| 3, 71b, 74, 77, 157, 548ccc }}


{{Val list|legend=1| 3, 6, 11, 14, 17c }}
[[Badness]] (Sintel): 30.4


[http://x31eq.com/cgi-bin/rt.cgi?ets=3_14p&limit=5 The temperament finder - 5-limit symbolic]
== Squarschmidt (5-limit) ==
: ''For extensions, see [[Hemimage temperaments #Squarschmidt]].''


== Isnes ==
A generator for the squarschmidt temperament is the fourth root of [[5/2]], (5/2)<sup>1/4</sup>, tuned around 396.6 cents.


So called because the generator is half of a [[8/5]] minor sixth, in a similar way that [[sensi]] has a generator of half a [[5/3]].
[[Subgroup]]: 2.3.5


Comma list: {{Monzo|41 2 -19}}
[[Comma list]]: {{monzo| 61 4 -29 }}


POTE generator: 405.1047 cents
{{Mapping|legend=1| 1 -8 1 | 0 29 4 }}
: mapping generators: ~2, ~98304/78125


Mapping: [{{val| 1 8 3 }}, {{val| 0 -19 -2 }}]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9653{{c}}, ~98304/78125 = 396.6094{{c}}
: [[error map]]: {{val| -0.099 +0.543 +0.029 -0.719 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~98304/78125 = 396.6201{{c}}
: error map: {{val| 0.000 +0.653 +0.253 -0.552 }}


{{Val list|legend=1| 3, 74, 77, 80, 83, 154, 157, 160 }}
{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947, 9588cc, 10535cc, 11482ccc }}


[http://x31eq.com/cgi-bin/rt.cgi?ets=3_77&limit=5 The temperament finder - 5-limit 3 & 77]
[[Badness]] (Sintel): 5.12


[[Category:3edo]]
[[Category:3edo]]
[[Category:Theory]]
[[Category:Temperament]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]

Latest revision as of 14:43, 22 June 2026

The father–3 equivalence continuum is a continuum of 5-limit temperaments which equate a number of classical diatonic semitones (16/15) with the Pythagorean minor third (32/27).

Note that because 3et is a record equal temperament in the 2.5 subgroup, the continuum can be conceptualized as the augmented–dicot equivalence continuum, which Godtone argues is easier to understand, with characteristic 2.5-subgroup comma 128/125 as the interval with a single factor of 3 is 25/24.

All temperaments in the continuum satisfy (16/15)n ~ 32/27. Varying n results in different temperaments listed in the table below. It converges to father as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 3edo 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 approximately 2.63252…, and temperaments having n near this value tend to be the most accurate ones.

32/27 is the characteristic 3-limit comma tempered out in 3edo. In each case, n equals the order of harmonic 5 in the corresponding comma, and equals the number of steps to obtain the interval class of 3 in the generator chain.

Temperaments with integer n
n Temperament Comma
Ratio Monzo
0 Alteraugment 32/27 [5 -3⟩
1 Antonian 10/9 [1 -2 1⟩
2 Dicot 25/24 [-3 -1 2⟩
3 Augmented 128/125 [7 0 -3⟩
4 Smate 2048/1875 [11 -1 -4⟩
… … … …
∞ Father 16/15 [4 -1 -1⟩

We may invert the continuum by setting m such that 1/m + 1/n = 1. This may be called the antonian–3 equivalence continuum, which is essentially the same thing. The just value of m is 1.61255…

Temperaments with integer m
m Temperament Comma
Ratio Monzo
0 Alteraugment 32/27 [5 -3⟩
1 Father 16/15 [4 -1 -1⟩
2 Dicot 25/24 [-3 -1 2⟩
… … … …
∞ Antonian 10/9 [1 -2 1⟩
Temperaments with fractional n and m
n m Temperament Comma
7/3 = 2.3 7/4 = 1.75 Wesley [-13 -2 7⟩
5/2 = 2.5 5/3 = 1.6 Magic [-10 -1 5⟩
21/8 = 2.625 21/13 = 1.615384 Mutt [-44 -3 21⟩
29/11 = 2.63 29/18 = 1.61 Squarschmidt [61 4 -29⟩
8/3 = 2.6 8/5 = 1.6 Würschmidt [17 1 -8⟩
19/7 = 2.714285 19/12 = 1.583 Isnes [41 2 -19⟩
11/4 = 2.75 11/7 = 1.571428 Magus [24 1 -11⟩

Mutt (5-limit)

Main article: Mutt
For extensions, see Horwell temperaments #Mutt.

Subgroup: 2.3.5

Comma list: [-44 -3 21⟩

Mapping: [⟨3 -2 6], ⟨0 7 1]]

mapping generators: ~98304/78125, ~5/4

Optimal tunings:

  • WE: ~98304/78125 = 400.0227 ¢, ~5/4 = 386.0017 ¢ (~393216/390625 = 14.0210 ¢)
error map: ⟨+0.068 +0.012 -0.176]
  • CWE: ~98304/78125 = 400.0000 ¢, ~5/4 = 385.9858 ¢ (~393216/390625 = 14.0142 ¢)
error map: ⟨0.000 -0.055 -0.328]

Optimal ET sequence: 84, 87, 171, 771, 942, 1113, 1284, 1455, 4194cc, 5649cc

Badness (Sintel): 3.81

Isnes

Isnes is so called because the generator is half of a 5/2 major tenth, in a similar way that sensi has a generator of half a 5/3 major sixth. This corresponds to n = 19/7 and m = 19/12.

Subgroup: 2.3.5

Comma list: [41 2 -19⟩

Mapping: [⟨1 -11 1], ⟨0 19 2]]

mapping generators: ~2, ~3145728/1953125

Optimal tunings:

  • WE: ~2 = 1199.2782 ¢, ~3145728/1953125 = 794.4174 ¢
error map: ⟨-0.722 -0.090 +1.799]
  • CWE: ~2 = 1200.0000 ¢, ~3145728/1953125 = 794.8728 ¢
error map: ⟨0.000 +0.628 +3.432]

Optimal ET sequence: 3, 71b, 74, 77, 157, 548ccc

Badness (Sintel): 30.4

Squarschmidt (5-limit)

For extensions, see Hemimage temperaments #Squarschmidt.

A generator for the squarschmidt temperament is the fourth root of 5/2, (5/2)1/4, tuned around 396.6 cents.

Subgroup: 2.3.5

Comma list: [61 4 -29⟩

Mapping: [⟨1 -8 1], ⟨0 29 4]]

mapping generators: ~2, ~98304/78125

Optimal tunings:

  • WE: ~2 = 1199.9653 ¢, ~98304/78125 = 396.6094 ¢
error map: ⟨-0.099 +0.543 +0.029 -0.719]
  • CWE: ~2 = 1200.0000 ¢, ~98304/78125 = 396.6201 ¢
error map: ⟨0.000 +0.653 +0.253 -0.552]

Optimal ET sequence: 118, 593, 711, 829, 947, 9588cc, 10535cc, 11482ccc

Badness (Sintel): 5.12

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