Xenharmonic Wiki

Father–3 equivalence continuum/Godtone's approach: Difference between revisions

← Older editNewer edit →
VisualWikitext
Godtone (talk | contribs)
autopatrolled, emailconfirmed
2,289 edits
mNo edit summary
Godtone (talk | contribs)
autopatrolled, emailconfirmed
2,289 edits
add third-integer continuum and more explanations
Line 70: Line 70:
| {{ 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.


{| class="wikitable center-1"
{| class="wikitable center-1"
Line 131: Line 132:
| {{ monzo| 61 4 -29 }}
| {{ monzo| 61 4 -29 }}
|}
|}
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.
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 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.


Also note that at {{nowrap|''n'' {{=}} −{{frac|2|3}}}}, we find the exotemperament tempering out [[32/27]].
{| class="wikitable center-1"
|+ style="font-size: 105%;" | Temperaments with third-integer ''n''<br />aka temperaments with 1\3 period
|-
! rowspan="2" | ''n''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| -2/3
| [[Very low accuracy temperaments#Alteraugment|32/27]] ({{nowrap|3 & 3c}}) {{nowrap|(generator {{=}} [[16/15|father comma]])}}
| 32/27
| {{ monzo| 5 0 -3 }}
|-
| -1/3
| {{nowrap|9c &amp; 12c}} {{nowrap|(generator {{=}} negative [[25/24|dicot comma]])}}
| 125/108
| {{ monzo| -2 -3 3 }}
|-
| 1/3
| {{nowrap|33c &amp; 36c}} {{nowrap|(generator {{=}} ~[[25/24|dicot comma]])}}
| 1953125/1769472
| {{ monzo| -16 -3 9 }}
|-
| 2/3
| {{nowrap|48 &amp; 51}} {{nowrap|(generator {{=}} negative ~[[magic comma]])}}
| 244140625/226492416
| {{ monzo| -23 -3 12 }}
|-
| 4/3
| {{nowrap|72 &amp; 75}} {{nowrap|(generator {{=}} ~[[magic comma]])}}
| 3814697265625/3710851743744
| {{ monzo| 41 2 -19 }}
|-
| 5/3
| [[Mutt]] ({{nowrap|84 &amp; 87}}) {{nowrap|(generator {{=}} ~[[Würschmidt's comma]])}}
| [[476837158203125/474989023199232|mutt comma]]
| {{ monzo| -44 -3 21 }}
|-
| 7/3
| {{nowrap|108 &amp; 111}} {{nowrap|(generator {{=}} negative ~[[Würschmidt's comma]])}}
| [[7782220156096217088/7450580596923828125|(38 digits)]]
| {{ monzo| 58 3 -27 }}
|-
| 8/3
| {{nowrap|120 &amp; 123}} {{nowrap|(generator {{=}} ~[[magus comma]])}}
| [[996124179980315787264/931322574615478515625|(42 digits)]]
| {{ monzo| 65 3 -30 }}
|-
| 10/3
| {{nowrap|120 &amp; 123}} {{nowrap|(generator {{=}} negative ~[[magus comma]])}}
| [[16320498564797493858533376/14551915228366851806640625|(52 digits)]]
| {{ monzo| 79 3 -36 }}
|-
| 11/3
| {{nowrap|156c &amp; 159c}} {{nowrap|(generator {{=}} ~[[6442450944/6103515625|supermagus comma]])}}
| [[2089023816294079213892272128/1818989403545856475830078125|(56 digits)]]
| {{ monzo| 86 3 -39 }}
|}
Notice the alternating pattern of comma offsets from 1\3, where those commas are themselves in the pattern present in the continuum of integer ''n''.
 
Also notice that we always find ~5/4 in terms of 1\3 minus the generator, which is a tempered version of the aforementioned comma offset, which is either positive or negative, and that as ''n'' grows, the generator becomes smaller so that ~5/4 becomes sharper.
 


[[Category:3edo]]
[[Category:3edo]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]
Retrieved from "https://en.xen.wiki/w/Father–3_equivalence_continuum/Godtone%27s_approach"