Xenharmonic Wiki

Orwell: Difference between revisions

← Older edit
VisualWikitext
mNo edit summary
m Undo revision 198110 by ArrowHead294 (talk)
Tag: Undo
 
(11 intermediate revisions by 4 users not shown)
Line 47: Line 47:
| 5
| 5
| 157.28
| 157.28
| 12/11, 11/10, 35/32
| 11/10, 12/11, 35/32
|-
|-
| 6
| 6
| 428.73
| 428.73
| 14/11, 9/7, 32/25
| 9/7, 14/11, 32/25
|-
|-
| 7
| 7
Line 63: Line 63:
| 9
| 9
| 43.10
| 43.10
| 49/48, 36/35, 33/32
| 33/32, 36/35, 49/48
|-
|-
| 10
| 10
Line 117: Line 117:
| 63/32
| 63/32
|}
|}
<nowiki/>* In 11-limit CWE tuning
<nowiki/>* In 11-limit CWE tuning, octave reduced


== Chords and harmony ==
== Chords and harmony ==
Line 123: Line 123:
{{See also| Functional harmony in rank-2 temperaments }}
{{See also| Functional harmony in rank-2 temperaments }}


The fundamental otonal consonance of orwell, voiced in a roughly {{w|tertian harmony|tertian}} manner, is 4:5:6:7:9:11. In terms of generator steps this is 0–(-3)–7–8–14–2, only available in a 22-tone mos. However, some subsets of this chord are way simpler, such as 8:11:12:14, which is 1–11/8–3/2–7/4 (0–2–7–8).  
The fundamental otonal consonance of orwell, voiced in a roughly {{w|tertian harmony|tertian}} manner, is 4:5:6:7:9:11. In terms of generator steps this is 0–(−3)–7–8–14–2, only available in a 22-tone mos. However, some subsets of this chord are way simpler, such as 8:11:12:14, which is 1–11/8–3/2–7/4 (0–2–7–8).  


The generator, ~7/6, is a septimal interval, so chords could instead be built around it as 1–7/6–3/2 (0–1–7), 1–7/4–3 (0–8–7), or tetrads such as 1–7/6–3/2–7/4 (0–1–7–8).  
The generator, ~7/6, is a septimal interval, so chords could instead be built around it as 1–7/6–3/2 (0–1–7), 1–7/4–3 (0–8–7), or tetrads such as 1–7/6–3/2–7/4 (0–1–7–8).  


To 1–7/6–3/2–7/4 we may add 11/8, or to 1–11/8–3/2–7/4 we may add 7/6, to form an essentially tempered pentad, 1–7/6–11/8–3/2–7/4 (0–1–2–7–8). Its inverse is 1–12/11–9/7–3/2–12/7 (0–5–6–7–(-1)), which can serve as a minor counterpart. This is similar, but also in clear contrast to the 1–5/4–3/2 (0–4–1) and 1–6/5–3/2 (0–(−3)–1) chords of [[meantone]]. Two approaches to functional harmony thus arise.  
To 1–7/6–3/2–7/4 we may add 11/8, or to 1–11/8–3/2–7/4 we may add 7/6, to form an essentially tempered pentad, 1–7/6–11/8–3/2–7/4 (0–1–2–7–8). Its inverse is 1–12/11–9/7–3/2–12/7 (0–5–6–7–(−1)), which can serve as a minor counterpart. This is similar, but also in clear contrast to the 1–5/4–3/2 (0–4–1) and 1–6/5–3/2 (0–(−3)–1) chords of [[meantone]]. Two approaches to functional harmony thus arise.  


First, we can treat the septimal chords above as the basis of harmony, but swapping the roles of 3 and 7 according to their temperamental complexities (number of generator steps). Thus a "dominant" chord is either 7/6 or 12/7 over tonic; a "subdominant" chord is either 7/6 or 12/7 under tonic. This leads to an approach closely adherent to mos scales. The 9-tone mos contains a tonic and a "dominant" triad. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions, triads to pentads alike.  
First, we can treat the septimal chords above as the basis of harmony, but swapping the roles of 3 and 7 according to their temperamental complexities (number of generator steps). Thus a "dominant" chord is either 7/6 or 12/7 over tonic; a "subdominant" chord is either 7/6 or 12/7 under tonic. This leads to an approach closely adherent to mos scales. The 9-tone mos contains a tonic and a "dominant" triad. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions, triads to pentads alike.  
Line 134: Line 134:


== Scales ==
== Scales ==
=== MOS scales ===
{{Main| Orwell scales }}
 
=== Mos scales ===
* [[Orwell5]]
* [[Orwell5]]


Line 192: Line 194:
; 13-tone scales (LsLLsLLLsLLsL, improper)  
; 13-tone scales (LsLLsLLLsLLsL, improper)  
* [[Orwell13]] – 84edo tuning
* [[Orwell13]] – 84edo tuning
* [[Orwellwoo13]] – [6 5/2] eigenmonzo (unchanged-interval) tuning
* [[Orwellwoo13]] – [6 5/2] unchanged-interval (eigenmonzo) tuning


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 255: Line 257:
; 22-tone scales
; 22-tone scales
* [[Orwell22]]
* [[Orwell22]]
* [[Orwellwoo22]] – [6 5/2] eigenmonzo (unchanged-interval) tuning
* [[Orwellwoo22]] – [6 5/2] unchanged-interval (eigenmonzo) tuning


=== Transversal scales ===
=== Transversal scales ===
Line 268: Line 270:
* [[Orwell-graham]] – 13-tone modmos in 53edo tuning
* [[Orwell-graham]] – 13-tone modmos in 53edo tuning
* [[Orwell13-modmos-containing-minerva12]] – 13-tone modmos in POTE tuning
* [[Orwell13-modmos-containing-minerva12]] – 13-tone modmos in POTE tuning
* [[Minerva12-orwell-tempered]] – minerva[12] tempered to orwell
* [[Minerva12-orwell-tempered]] – Minerva[12] tempered to orwell


== Tunings ==
== Tunings ==
Line 277: Line 279:
! colspan="2" | Euclidean
! colspan="2" | Euclidean
|-
|-
! Unskewed
! Constrained
! Skewed
! Constrained & skewed
|-
|-
! Equilateral
! Equilateral
| CEE: ~7/6 = 271.3553¢
| CEE: ~7/6 = 271.3553{{c}}
| CSEE: ~7/6 = 271.3339¢
| CSEE: ~7/6 = 271.3339{{c}}
|-
|-
! Tenney
! Tenney
| CTE: ~7/6 = 271.5130¢
| CTE: ~7/6 = 271.5130{{c}}
| CWE: ~7/6 = 271.5097¢
| CWE: ~7/6 = 271.5097{{c}}
|-
|-
! Benedetti, <br>Wilson
! Benedetti, <br>Wilson
| CBE: ~7/6 = 271.5725¢
| CBE: ~7/6 = 271.5725{{c}}
| CSBE: ~7/6 = 271.5741¢
| CSBE: ~7/6 = 271.5741{{c}}
|}
|}


Line 299: Line 301:
! colspan="2" | Euclidean
! colspan="2" | Euclidean
|-
|-
! Unskewed
! Constrained
! Skewed
! Constrained & skewed
|-
|-
! Equilateral
! Equilateral
| CEE: ~7/6 = 271.4920¢
| CEE: ~7/6 = 271.4920{{c}}
| CSEE: ~7/6 = 271.3038¢
| CSEE: ~7/6 = 271.3038{{c}}
|-
|-
! Tenney
! Tenney
| CTE: ~7/6 = 271.5597¢
| CTE: ~7/6 = 271.5597{{c}}
| CWE: ~7/6 = 271.4552¢
| CWE: ~7/6 = 271.4552{{c}}
|-
|-
! Benedetti, <br>Wilson
! Benedetti, <br>Wilson
| CBE: ~7/6 = 271.5915¢
| CBE: ~7/6 = 271.5915{{c}}
| CSBE: ~7/6 = 271.5302¢
| CSBE: ~7/6 = 271.5302{{c}}
|}
|}


Line 320: Line 322:
! Optimized chord !! Generator value !! Polynomial !! Further notes
! Optimized chord !! Generator value !! Polynomial !! Further notes
|-
|-
| 3:4:5 (+1 +1) || ~7/6 = 272.890 || ''f''<sup>10</sup> &minus; 8''f''<sup>3</sup> + 8 = 0 || 1–3–5 equal-beating tuning
| 3:4:5 (+1 +1) || ~7/6 = 272.890{{c}} || ''f''<sup>10</sup> &minus; 8''f''<sup>3</sup> + 8 = 0 || 1–3–5 equal-beating tuning
|-
|-
| 4:5:6 (+1 +1) || ~7/6 = 271.508 || ''f''<sup>10</sup> + 2''f''<sup>3</sup> - 8 = 0 || 1–3–5 equal-beating tuning
| 4:5:6 (+1 +1) || ~7/6 = 271.508{{c}} || ''f''<sup>10</sup> + 2''f''<sup>3</sup> - 8 = 0 || 1–3–5 equal-beating tuning
|}
|}


Line 329: Line 331:
|-
|-
! Edo<br>generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 652: Line 654:
[[File:orwell13_axis49.png|alt=orwell13_axis49.png|orwell13_axis49.png]]
[[File:orwell13_axis49.png|alt=orwell13_axis49.png|orwell13_axis49.png]]


[[Category:Temperaments]]
[[Category:Orwell| ]] <!-- main article -->
[[Category:Orwell| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Orwell"