Father–3 equivalence continuum/Godtone's approach: Difference between revisions
the father-3 continuum is so bad that i didnt realise it was supposed to be equivalent to this continuum until now. i propose replacing it with this continuum and reworking the temperaments from there into here, but if that isnt acceptable i still believe this continuum should be documented and named as such |
m add two missing exotemperaments and add a high-accuracy temperaments section (neither mutt nor the 3&118 temperament is covered in the original page) |
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| Line 18: | Line 18: | ||
! Ratio | ! Ratio | ||
! Monzo | ! Monzo | ||
|- | |||
| -2 | |||
| [[Smate]] (14 & 17c) | |||
| [[2048/1875]] | |||
| {{monzo| 11 -1 -4 }} | |||
|- | |||
| -1 | |||
| [[Father]] (5 & 8) | |||
| [[16/15]] | |||
| {{monzo| 4 -1 -1 }} | |||
|- | |- | ||
| 0 | | 0 | ||
| Line 70: | Line 80: | ||
! Monzo | ! Monzo | ||
|- | |- | ||
| -1/2 | |||
| [[Very low accuracy temperaments#Yo (2c&3)|Yo]] | |||
| [[10/9]] | |||
| {{monzo| 1 -2 1 }} | |||
|- | |- | ||
| 1/2 | | 1/2 | ||
| Line 91: | Line 105: | ||
| {{ monzo| 55 2 -25 }} | | {{ monzo| 55 2 -25 }} | ||
|} | |} | ||
If we approximate the [[JIP]] with increasing accuracy, (that is, using ''n'' a rational that is an increasingly good approximation of 1.72125...) we find these high-accuracy temperaments: | |||
{| class="wikitable center-1" | |||
|+ style="font-size: 105%;" | Microtemperaments with fractional ''n'' | |||
|- | |||
! rowspan="2" | ''n'' | |||
! rowspan="2" | Temperament | |||
! colspan="2" | Comma | |||
|- | |||
! Ratio | |||
! Monzo | |||
|- | |||
| 5/3 | |||
| [[Mutt]] (84 & 87) | |||
| [[mutt comma]] | |||
| {{ monzo| -44 -3 21 }} | |||
|- | |||
| 7/4 | |||
| 3 & 118 | |||
| (42 digits) | |||
| {{ monzo| 61 4 -29 }} | |||
|} | |||
The simplest of these is [[mutt]] and has interesting properties discussed there. | |||
Revision as of 00:08, 27 February 2025
The augmented-chromatic equivalence continuum is a continuum of 5-limit 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.
This formulation has a specific reason: 128/125 is significantly smaller than 25/24, so that it makes sense to equate some number of 128/125's with 25/24, but because 25/24 = (25/16)/(3/2), this has the consequence of clearly relating the n in (128/125)n = 25/24 with how many 5/4's are used to reach 3/2 (when octave-reduced):
If 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)2.
For integer n > 0, we always reach 25/24 via (25/16)/(128/125)n because of (128/125)n ~ 25/24 by definition, meaning that we reach 3/2 at 3n + 2 generators of ~5/4, octave-reduced.
The just value of n is log2(25/24) / log2(128/125) = 1.72125... where n = 2 corresponds to Würschmidt's comma.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -2 | Smate (14 & 17c) | 2048/1875 | [11 -1 -4⟩ |
| -1 | Father (5 & 8) | 16/15 | [4 -1 -1⟩ |
| 0 | Dicot (7 & 10) | 25/24 | [-3 -1 2⟩ |
| 1 | Magic (19 & 22) | 3125/3072 | [-10 -1 5⟩ |
| 2 | Würschmidt (31 & 34) | 393216/390625 | [17 1 -8⟩ |
| 3 | Magus (43 & 46) | 50331648/48828125 | [24 1 -11⟩ |
| 4 | Supermagus (55 & 58) | 6442450944/6103515625 | [31 1 -14⟩ |
| 5 | Ultramagus (67 & 70) | 824633720832/762939453125 | [38 1 -17⟩ |
| … | … | … | … |
| ∞ | Augmented (12 & 15) | 128/125 | [-7 0 3⟩ |
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -1/2 | Yo | 10/9 | [1 -2 1⟩ |
| 1/2 | Wesley (26 & 29) | 78125/73728 | [13 2 -7⟩ |
| 3/2 | Ditonic (50 & 53) | 1220703125/1207959552 | [-27 -2 13⟩ |
| 5/2 | Novamajor (77 & 80) | 19791209299968/19073486328125 | [41 2 -19⟩ |
| 7/2 | 3 & 101 | (36 digits) | [55 2 -25⟩ |
If we approximate the JIP with increasing accuracy, (that is, using n a rational that is an increasingly good approximation of 1.72125...) we find these high-accuracy temperaments:
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 5/3 | Mutt (84 & 87) | mutt comma | [-44 -3 21⟩ |
| 7/4 | 3 & 118 | (42 digits) | [61 4 -29⟩ |
The simplest of these is mutt and has interesting properties discussed there.