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{{About|the regular temperament|the scale structure sometimes associated with it|5L 3s}}
{{About|the regular temperament|the scale structure sometimes associated with it|5L 3s}}
{{Infobox regtemp
| Title = Father
| Subgroups = 2.3.5, 2.3.5.7
| Comma basis = [[16/15]] (5-limit) <br>16/15, [[28/27]] (7-limit)
| Edo join 1 = 2(d) | Edo join 2 = 3(d)
| Mapping = 1; 1 -1 3
| Generators = 3/2 | Generators tuning = 738.443 | Optimization method = CWE
| MOS scales = [[1L 1s]], [[2L 1s]], [[3L 2s]]
| Pergen = (P8, P5)
| Odd limit 1 = 5 | Mistuning 1 = 55.9 | Complexity 1 = 3
| Odd limit 2 = 7 | Mistuning 2 = 68.1 | Complexity 2 = 5
}}
'''Father''' is a very coarse, simplistic, and inaccurate [[exotemperament]]. It [[tempering out|tempers out]] [[16/15]], the classical diatonic semitone. This means the [[5/4|classical major third (5/4)]] is conflated with the [[4/3|perfect fourth (4/3)]], making it one that challenges the very notion of JI approximation, and playing harmony in it, it sounds only remotely reminiscent of the [[5-limit]] no matter how it is tuned. If one could get their head around this way of hearing intervals, they may as well take a look at the 7-limit interpretation, where it tempers out [[28/27]] and [[36/35]].


'''Father''' is a very coarse, simplistic, and inaccurate [[exotemperament]]. It [[Tempering out|tempers out]] [[16/15]], the classical diatonic semitone. This means the [[5/4|classical major third (5/4)]] is conflated with the [[4/3|perfect fourth (4/3)]], making it one that challenges the very notion of JI approximation, and playing harmony in it, it sounds only remotely reminiscent of the [[5-limit]] no matter how it is tuned. If one could get their head around this way of hearing intervals, they may as well take a look at the 7-limit interpretation, where it tempers out [[28/27]] and [[36/35]].  
The main interest in this temperament is its [[mos scale]]s, as [[3L 2s|antipentic (3L 2s)]] and [[5L 3s|oneirotonic (5L 3s)]] are often chosen first, and only later is each step associated with a ratio consistent with this temperament. Another potential reason to choose this temperament is to equate suspended chords and more conventional tertian chords (though options like [[trienstonian]] (4/3~9/7), [[blackwood]] (4/3~81/64), and [[fendo]] (4/3~13/10) are more accurate).  


As an exotemperament, there are a variety of acceptable tunings for father, ranging from roughly 720¢ to 800¢. However, since the main interest in an exotemperament is usually its [[mos scale]]s, [[3L 2s|antipentic (3L 2s)]] and [[5L 3s|oneirotonic (5L 3s)]] are often chosen first, and only later is each step associated with a ratio consistent with this temperament; this means that the most common tunings of father are between 720 and 750 cents. A potential reason to choose father as a temperament is to equate suspended chords and more conventional tertian chords (though options like [[Trienstonic clan|trienstonic]] (4/3~9/7), [[blackwood]] (4/3~81/64), and [[fendo]] (4/3~13/10) are more accurate).    
As an exotemperament, it has a large range of acceptable tunings, from roughly [[5edo|3\5]] (720{{c}}) to [[3edo|2\3]] (800{{c}}). However, only tunings between 3\5 and [[8edo|5\8]] (750{{c}}) generate oneirotonic scales, which are the primary structure it represents as an exotemperament.  


See [[Father family #Father]] and [[Trienstonic clan #Father]] for technical details.  
See [[Father family #Father]] for technical details.  


== Interval chain ==
== Interval chain ==
Line 12: Line 25:
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
! &#35; !! Cents* !! Approximate Ratios
! # !! Cents* !! Approximate ratios
|-
|-
| 0 || 0.0 || '''1/1'''
| 0 || 0.0 || '''1/1'''
|-
|-
| 1 || 727.9 || '''3/2''', '''8/5''', 14/9
| 1 || 738.4 || '''3/2''', '''8/5''', 14/9
|-
|-
| 2 || 255.7 || 6/5, 7/6, '''9/8'''
| 2 || 276.9 || 6/5, 7/6, '''9/8'''
|-
|-
| 3 || 983.6 || '''7/4''', 9/5
| 3 || 1015.3 || '''7/4''', 9/5
|-
|-
| 4 || 511.4 || 7/5
| 4 || 553.8 || 7/5
|-
| 5 || 92.2 || 21/20
|}
|}
<nowiki />* In 7-limit CTE tuning
<nowiki />* In 7-limit CWE tuning


== Tunings ==
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 737.469{{c}}
| CWE: ~3/2 = 742.290{{c}}
| POTE: ~3/2 = 743.986{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 727.855{{c}}
| CWE: ~3/2 = 738.443{{c}}
| POTE: ~3/2 = 742.002{{c}}
|}
=== Tuning spectrum ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br />Generator !! Eigenmonzo<br />(Unchanged-interval)* !! Generator (¢) !! Comments
! Edo<br>generator !! Eigenmonzo<br>(Unchanged-interval)* !! Generator (¢) !! Comments
|-
|-
| 1\2 ||  || 600.0 || Lower bound of 5-odd-limit diamond monotone
| 1\2 ||  || 600.0 || Lower bound of 5-odd-limit diamond monotone
Line 50: Line 97:
| 5\8 ||  || 750.0 || Upper bound of 7-odd-limit diamond monotone
| 5\8 ||  || 750.0 || Upper bound of 7-odd-limit diamond monotone
|-
|-
| || 5/3 || 757.8 || 1/2-comma, 5-odd-limit minimax
| || 5/3 || 757.8 || 1/2-comma, 5-odd-limit minimax, 5-limit CEE & CSEE
|-
|-
| || 9/7 || 764.9 || 9-odd-limit minimax
| || 9/7 || 764.9 || 9-odd-limit minimax
Line 63: Line 110:
* ''[[Noodles adorno foucault]]''
* ''[[Noodles adorno foucault]]''


[[Category:Temperaments]]
[[Category:Father| ]] <!-- Main article -->
[[Category:Father| ]] <!-- Main article -->
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Exotemperaments]]
[[Category:Father family]]
[[Category:Father family]]
[[Category:Trienstonic clan]]
[[Category:Trienstonic clan]]
[[Category:Mint temperaments]]
[[Category:Mint temperaments]]
[[Category:Exotemperaments]]

Latest revision as of 05:14, 14 February 2026

This page is about the regular temperament. For the scale structure sometimes associated with it, see 5L 3s.
Father
Subgroups 2.3.5, 2.3.5.7
Comma basis 16/15 (5-limit)
16/15, 28/27 (7-limit)
Reduced mapping ⟨1; 1 -1 3]
ET join 2(d) & 3(d)
Generators (CWE) ~3/2 = 738.443 ¢
MOS scales 1L 1s, 2L 1s, 3L 2s
Ploidacot monocot
Pergen (P8, P5)
Minimax error 5-odd-limit: 55.9 ¢;
7-odd-limit: 68.1 ¢
Target scale size 5-odd-limit: 3 notes;
7-odd-limit: 5 notes

Father is a very coarse, simplistic, and inaccurate exotemperament. It tempers out 16/15, the classical diatonic semitone. This means the classical major third (5/4) is conflated with the perfect fourth (4/3), making it one that challenges the very notion of JI approximation, and playing harmony in it, it sounds only remotely reminiscent of the 5-limit no matter how it is tuned. If one could get their head around this way of hearing intervals, they may as well take a look at the 7-limit interpretation, where it tempers out 28/27 and 36/35.

The main interest in this temperament is its mos scales, as antipentic (3L 2s) and oneirotonic (5L 3s) are often chosen first, and only later is each step associated with a ratio consistent with this temperament. Another potential reason to choose this temperament is to equate suspended chords and more conventional tertian chords (though options like trienstonian (4/3~9/7), blackwood (4/3~81/64), and fendo (4/3~13/10) are more accurate).

As an exotemperament, it has a large range of acceptable tunings, from roughly 3\5 (720 ¢) to 2\3 (800 ¢). However, only tunings between 3\5 and 5\8 (750 ¢) generate oneirotonic scales, which are the primary structure it represents as an exotemperament.

See Father family #Father for technical details.

Interval chain

In the following table, odd harmonics 1–9 are labeled in bold.

# Cents* Approximate ratios
0 0.0 1/1
1 738.4 3/2, 8/5, 14/9
2 276.9 6/5, 7/6, 9/8
3 1015.3 7/4, 9/5
4 553.8 7/5
5 92.2 21/20

* In 7-limit CWE tuning

Tunings

5-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 737.469 ¢ CWE: ~3/2 = 742.290 ¢ POTE: ~3/2 = 743.986 ¢
7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 727.855 ¢ CWE: ~3/2 = 738.443 ¢ POTE: ~3/2 = 742.002 ¢

Tuning spectrum

Edo
generator		 Eigenmonzo
(Unchanged-interval)*		 Generator (¢) Comments
1\2 600.0 Lower bound of 5-odd-limit diamond monotone
3/2 702.0 Pythagorean tuning
3\5 720.0 Lower bound of 7-odd-limit diamond monotone
9-odd-limit diamond monotone (singleton)
7/4 722.9
7/6 733.4
8\13 738.5
9/5 739.2 1/3-comma
7/5 745.6 7-odd-limit minimax
5\8 750.0 Upper bound of 7-odd-limit diamond monotone
5/3 757.8 1/2-comma, 5-odd-limit minimax, 5-limit CEE & CSEE
9/7 764.9 9-odd-limit minimax
2\3 800.0 Upper bound of 5-odd-limit diamond monotone
5/4 813.7 Full-comma

* Besides the octave

Music

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