Xenharmonic Wiki

Decimal: Difference between revisions

← Older edit
VisualWikitext
Wikispaces>keenanpepper
**Imported revision 309959420 - Original comment: **
Cleanup on infobox
 
(33 intermediate revisions by 8 users not shown)
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox regtemp
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| Title = Decimal
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2012-03-12 01:36:45 UTC</tt>.<br>
| Subgroups = 2.3.5.7
: The original revision id was <tt>309959420</tt>.<br>
| Comma basis = [[25/24]], [[49/48]]
: The revision comment was: <tt></tt><br>
| Edo join 1 = 4 | Edo join 2 = 6
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| Mapping = 2; 2 1 1
<h4>Original Wikitext content:</h4>
| Generators = 7/4 | Generators tuning = 951.0 | Optimization method = CWE
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">See [[Dicot family#Decimal]].</pre></div>
| MOS scales = [[4L 2s]], [[4L 6s]], [[10L 4s]]
<h4>Original HTML content:</h4>
| Pergen = (P8/2, P4/2)
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Decimal&lt;/title&gt;&lt;/head&gt;&lt;body&gt;See &lt;a class="wiki_link" href="/Dicot%20family#Decimal"&gt;Dicot family&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
| Odd limit 1 = 7 | Mistuning 1 = 35.3 | Complexity 1 = 6
| Odd limit 2 = 7-limit 21 | Mistuning 2 = 35.3 | Complexity 2 = 10
}}
'''Decimal''' is an [[exotemperament]] in the [[dicot family]], [[semaphoresmic clan]], and [[jubilismic clan]] of [[regular temperament|temperaments]]. It is a [[weak extension|weak]] [[extension]] of [[dicot]], the [[5-limit]] temperament tempering out [[25/24]], splitting the octave in two parts, each representing [[7/5]][[~]][[10/7]]. It is also the prototypical fully [[hemipyth]] temperament, with [[sqrt(2)]] representing 7/5~10/7, [[sqrt(3)]] representing [[7/4]]~[[12/7]], [[sqrt(3/2)]] representng [[5/4]]~[[6/5]], and [[sqrt(4/3)]] representing [[7/6]]~[[8/7]], with a [[pergen]] of (P8/2, P4/2), splitting all Pythagorean intervals in two.
 
More precisely, it is the [[7-limit]] temperament that [[tempering out|tempers out]] both [[25/24]], the classic chromatic semitone, and [[49/48]], the septimal diesis. These two intervals have a rather similar function separating close intervals and creating "major" and "minor" triads (either pental ones splitting the perfect fifth or septimal ones splitting the perfect fourth), and tempering them out allows 5/4~6/5 to be sqrt(3/2) a neutral third and 7/6~8/7 to be a sqrt(4/3) neutral semifourth. This equates the [[4:5:6]] and [[6:7:8]] triads with their inverses, therefore also equating the [[4:5:6:7]] major tetrad with the [[70:84:105:120|1/(12:10:8:7)]] minor tetrad. The neutral third and semifourth can be equated (far more accurately) to [[11/9]] and [[15/13]] respectively, tempering out [[243/242]] and [[676/675]] and extending this temperament to the [[13-limit]]. Since {{nowrap|(25/24)/(49/48) {{=}} [[50/49]] }}, it also tempers that out, splitting the octave in two equal parts. As both the generator and period are half that of the diatonic scale, this means it forms mos scales of 4, 6, 10, 14, 24, … tones.
 
Decimal can serve as a structural archetype for a [[decatonic]] system that views the [[4:5:6]] and [[10:12:15|1/(4:5:6)]] chords as a major–minor pair (which is equated in decimal temperament as 25/24 is tempered out), and the [[6:7:8]] and [[21:24:28|1/(6:7:8)]] chords as another major–minor pair, neutralized in decimal via vanishing of 49/48.
 
A more accurate system based on 10 interval classes that distinguishes major and minor chords is [[pajara]], where 50/49 remains tempered and 49/48 is equated to 25/24. An even more accurate one is [[miracle]], which equates 50/49 with 49/48 to half of 25/24 by tempering out [[2401/2400]], though its structure is more complex than that of pajara. Both of these temperaments also temper out the marvel comma, [[225/224]].
 
{{Tdlink|Dicot family #Decimal}}
 
== Interval chain ==
In the following table, odd harmonics 1–9 and their inverses are in '''bold'''.
 
{| class="wikitable center-1 right-2 right-4"
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 1
|-
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
|-
| 0
| 0.0
| '''1/1'''
| 600.0
| 7/5, 10/7
|-
| 1
| 351.0
| '''5/4''', 6/5
| 951.0
| '''7/4''', 12/7
|-
| 2
| 701.9
| '''3/2'''
| 101.9
| 15/14, 21/20
|-
| 3
| 1052.9
| 9/5, 15/8
| 452.9
| 9/7, 21/16
|-
| 4
| 203.8
| '''9/8'''
| 803.8
| 45/28, 54/35
|-
| 5
| 554.8
| 27/20, 45/32
| 1154.8
| 27/14, 63/32
|}
<nowiki/>* In 7-limit CWE tuning, octave reduced
 
One can see that the 10-note mos of the decimal temperament contains the 7-odd-limit [[tonality diamond]].
 
== Tunings ==
{| 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: ~7/4 = 955.608{{c}}
| CWE: ~7/4 = 950.957{{c}}
| POTE: ~7/4 = 948.443{{c}}
|}
 
[[Category:Decimal| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Exotemperaments]]
[[Category:Jubilismic clan]]
[[Category:Dicot family]]
[[Category:Semaphoresmic clan]]
Retrieved from "https://en.xen.wiki/w/Decimal"