Xenharmonic Wiki

Harmonotonic tuning: Difference between revisions

← Older editNewer edit →
VisualWikitext
Cmloegcmluin (talk | contribs)
autopatrolled, emailconfirmed
5,160 edits
→Example monotonic tuning charts and graphs for comparison: unreducing stuff at least in the frequency part of the table
Cmloegcmluin (talk | contribs)
autopatrolled, emailconfirmed
5,160 edits
Line 239: Line 239:


Typically, the undertone series is displayed moving from 1/1, descending in pitch. To better illustrate the analogies between these different types of monotonic tunings, however, all tunings are shown starting on 1/1 but ascending in pitch. This means that all utonal tunings here were necessary to truncate, so that the lowest pitch would be known, so that the pitches' order could be reversed and the lowest pitch repositioned to 1/1 and all other pitches relative to that instead.
Typically, the undertone series is displayed moving from 1/1, descending in pitch. To better illustrate the analogies between these different types of monotonic tunings, however, all tunings are shown starting on 1/1 but ascending in pitch. This means that all utonal tunings here were necessary to truncate, so that the lowest pitch would be known, so that the pitches' order could be reversed and the lowest pitch repositioned to 1/1 and all other pitches relative to that instead.
For a variety of perspectives, the frequency table values have been left unreduced, to illustrate the ideas behind the tuning better; the length values have been reduced.


{| class="wikitable"
{| class="wikitable"
|+comparison of example monotonic tunings
|+comparison of example monotonic tunings
! colspan="2" rowspan="2" |monotonic tuning
! colspan="2" rowspan="2" |monotonic tuning
! colspan="9" |frequency
! colspan="9" |frequency (f)
!
!
! colspan="9" |pitch
! colspan="9" |pitch (log₂f)
!
!
! colspan="9" |length
! colspan="9" |length (1/f)
|-
|-
! (0) !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8
! (0) !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8
! !! (0) !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8
! !! (0) !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8
! !! (0) !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8
! !! (0) !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8
|-
|-
! style="background-color: #00DB00;" |
! style="background-color: #00DB00;" |
! 4-ODO = 4th overtone mode = 4-ADO  
! 4-ODO = 4th overtone mode = 4-ADO
| (4/4) || 5/4 || 6/4 || 7/4 || 8/4 || ''5/2'' || ''3/1'' || ''7/2'' || ''4/1''
| (4/4) || 5/4 || 6/4 || 7/4 || 8/4 ||''5/2''||''3/1''||''7/2''||''4/1''
!  
!  
| 0.00 || 0.32 || 0.58 || 0.81 || 1.00 || ''1.32'' || ''1.58'' || ''1.81'' || ''2.00''
| 0.00 || 0.32 || 0.58 || 0.81 || 1.00 ||''1.32''||''1.58''||''1.81''||''2.00''
!  
!  
| 1/1 || 4/5 || 2/3 || 4/7 || 1/2 || ''2/5'' || ''1/3'' || ''2/7'' || ''1/4''
| 1/1 || 4/5 || 2/3 || 4/7 || 1/2 ||''2/5''||''1/3''||''2/7''||''1/4''
|-
|-
! style="background-color: #34A853;" |
! style="background-color: #34A853;" |
! 4-EDO = rank-1 temperament w/ generator 300¢ = APS⁴√2 ≈ APS1.189  
! 4-EDO = rank-1 temperament w/ generator 300¢ = APS⁴√2 ≈ APS1.189
| (2⁰⸍⁴) || 2¹⸍⁴ || 2²⸍⁴ || 2³⸍⁴ || 2⁴⸍⁴ || ''2.38'' || ''2.83'' || ''3.36'' || ''4''
| (2⁰⸍⁴) || 2¹⸍⁴ || 2²⸍⁴ || 2³⸍⁴ || 2⁴⸍⁴ ||''2.38''||''2.83''||''3.36''||''4''
!  
!  
| 0.00 || 0.25 || 0.50 || 0.75 || 1.00 || ''1.25'' || ''1.50'' || ''1.75'' || ''2.00''
| 0.00 || 0.25 || 0.50 || 0.75 || 1.00 ||''1.25''||''1.50''||''1.75''||''2.00''
!  
!  
| 1.00 || 0.84 || 0.71 || 0.59 || 0.50 || ''0.42'' || ''0.35'' || ''0.30'' || ''0.25''
| 1.00 || 0.84 || 0.71 || 0.59 || 0.50 ||''0.42''||''0.35''||''0.30''||''0.25''
|-
|-
! style="background-color: #45818E;" |
! style="background-color: #45818E;" |
! 4-UDO = 4th undertone mode  
! 4-UDO = 4th undertone mode
| (8/8) || 8/7 || 8/6 || 8/5 || 8/4 || ''16/7'' || ''8/3'' || ''16/5'' || ''4/1''
| (8/8) || 8/7 || 8/6 || 8/5 || 8/4 ||''16/7''||''8/3''||''16/5''||''4/1''
!  
!  
| 0.00 || 0.19 || 0.42 || 0.68 || 1.00 || ''1.19'' || ''1.42'' || ''1.68'' || ''2.00''
| 0.00 || 0.19 || 0.42 || 0.68 || 1.00 ||''1.19''||''1.42''||''1.68''||''2.00''
!  
!  
| 1/1 || 7/8 || 3/4 || 5/8 || 1/2 || ''7/16'' || ''3/8'' || ''5/16'' || ''1/4''
| 1/1 || 7/8 || 3/4 || 5/8 || 1/2 ||''7/16''||''3/8''||''5/16''||''1/4''
|-
|-
! style="background-color: #B4A7D6;" |
! style="background-color: #B4A7D6;" |
! 4-EFDφ  
! 4-EFDφ
| (1) || 1.15 || 1.31 || 1.46 || φ || ''1.87'' || ''2.12'' || ''2.37'' || ''2.62''
| 1+(0/4)(φ-1) || 1+(1/4)(φ-1) || 1+(2/4)(φ-1) || 1+(3/4)(φ-1) || 1+(4/4)(φ-1) ||''1.87''||''2.12''||''2.37''||''2.62''
!  
!  
| 0.00 || 0.21 || 0.39 || 0.55 || 0.69 || ''0.90'' || ''1.08'' || ''1.24'' || ''1.39''
| 0.00 || 0.21 || 0.39 || 0.55 || 0.69 ||''0.90''||''1.08''||''1.24''||''1.39''
!  
!  
| 1.00 || 0.87 || 0.76 || 0.68 || 0.62 || ''0.54'' || ''0.47'' || ''0.42'' || ''0.38''
| 1.00 || 0.87 || 0.76 || 0.68 || 0.62 ||''0.54''||''0.47''||''0.42''||''0.38''
|-
|-
! style="background-color: #674EA7;" |
! style="background-color: #674EA7;" |
! 4-ELDφ  
! 4-ELDφ
| (1) || 1.11 || 1.24 || 1.40 || φ || ''1.79'' || ''2.00'' || ''2.27'' || ''2.62''
| (1) || 1.11 || 1.24 || 1.40 || φ ||''1.79''||''2.00''||''2.27''||''2.62''
!  
!  
| 0.00 || 0.14 || 0.31 || 0.49 || 0.69 || ''0.84'' || ''1.00'' || ''1.18'' || ''1.39''
| 0.00 || 0.14 || 0.31 || 0.49 || 0.69 ||''0.84''||''1.00''||''1.18''||''1.39''
!  
!  
| 1.00 || 0.90 || 0.81 || 0.71 || 0.62 || ''0.56'' || ''0.50'' || ''0.44'' || ''0.38''
| 1.00 || 0.90 || 0.81 || 0.71 || 0.62 ||''0.56''||''0.50''||''0.44''||''0.38''
|-
|-
! style="background-color: #D2EB00;" |
! style="background-color: #D2EB00;" |
! overtone series segment = 9-OS = 9-OD9  
! overtone series segment = 9-OS = 9-OD9
| (1/1) || 2/1 || 3/1 || 4/1 || 5/1 || 6/1 || 7/1 || 8/1 || 9/1  
| (1/1) || 2/1 || 3/1 || 4/1 || 5/1 || 6/1 || 7/1 || 8/1 || 9/1
!  
!  
| 0.00 || 1.00 || 1.58 || 2.00 || 2.32 || 2.58 || 2.81 || 3.00 || 3.17  
| 0.00 || 1.00 || 1.58 || 2.00 || 2.32 || 2.58 || 2.81 || 3.00 || 3.17
!  
!  
| 1/1 || 1/2 || 1/3 || 1/4 || 1/5 || 1/6 || 1/7 || 1/8 || 1/9
| 1/1 || 1/2 || 1/3 || 1/4 || 1/5 || 1/6 || 1/7 || 1/8 || 1/9
|-
|-
! style="background-color: #A1A200;" |
! style="background-color: #A1A200;" |
! undertone series segment = 9-US = 9-UD9 = 9-EDL  
! undertone series segment = 9-US = 9-UD9 = 9-EDL
| (9/9) || 9/8 || 9/7 || 9/6 || 9/5 || 9/4 || 9/3 || 9/2 || 9/1  
| (9/9) || 9/8 || 9/7 || 9/6 || 9/5 || 9/4 || 9/3 || 9/2 || 9/1
!  
!  
| 0.00 || 0.17 || 0.36 || 0.58 || 0.85 || 1.17 || 1.58 || 2.17 || 3.17  
| 0.00 || 0.17 || 0.36 || 0.58 || 0.85 || 1.17 || 1.58 || 2.17 || 3.17
!  
!  
| 1/1 || 8/9 || 7/9 || 2/3 || 5/9 || 4/9 || 1/3 || 2/9 || 1/9
| 1/1 || 8/9 || 7/9 || 2/3 || 5/9 || 4/9 || 1/3 || 2/9 || 1/9
|-
|-
! style="background-color: #FF7700;" |
! style="background-color: #FF7700;" |
! 9-OS(3/4)  
! 9-OS(3/4)
| (4/4) || 7/4 || 10/4 || 13/4 || 16/4 || 19/4 || 22/4 || 25/4 || 28/4
| (4/4) || 7/4 || 10/4 || 13/4 || 16/4 || 19/4 || 22/4 || 25/4 || 28/4
!  
!  
| 0.00 || 0.81 || 1.32 || 1.70 || 2.00 || 2.25 || 2.46 || 2.64 || 2.81  
| 0.00 || 0.81 || 1.32 || 1.70 || 2.00 || 2.25 || 2.46 || 2.64 || 2.81
!  
!  
| 1/1 || 4/7 || 2/5 || 4/13 || 1/4 || 4/19 || 2/11 || 4/25 || 1/7
| 1/1 || 4/7 || 2/5 || 4/13 || 1/4 || 4/19 || 2/11 || 4/25 || 1/7
|-
|-
! style="background-color: #C95E00;" |
! style="background-color: #C95E00;" |
! 9-US(3/4)  
! 9-US(3/4)
| (28/28) || 28/25 || 28/22 || 28/19 || 28/16 || 28/13 || 28/10 || 28/7 || 28/4
| (28/28) || 28/25 || 28/22 || 28/19 || 28/16 || 28/13 || 28/10 || 28/7 || 28/4
!  
!  
| 0.00 || 0.16 || 0.35 || 0.56 || 0.81 || 1.11 || 1.49 || 2.00 || 2.81  
| 0.00 || 0.16 || 0.35 || 0.56 || 0.81 || 1.11 || 1.49 || 2.00 || 2.81
!  
!  
| 1/1 || 25/28 || 11/14 || 19/28 || 4/7 || 13/28 || 5/14 || 1/4 || 1/7
| 1/1 || 25/28 || 11/14 || 19/28 || 4/7 || 13/28 || 5/14 || 1/4 || 1/7
|-
|-
! style="background-color: #FFC000;" |
! style="background-color: #FFC000;" |
! (1/⁴√2)-shifted overtone series segment = 9-AFS(1/⁴√2)  
! (1/⁴√2)-shifted overtone series segment = 9-AFS(1/⁴√2)
| (1) || 1.84 || 2.68 || 3.52 || 4.36 || 5.20 || 6.05 || 6.89 || 7.73  
| (1) || 1.84 || 2.68 || 3.52 || 4.36 || 5.20 || 6.05 || 6.89 || 7.73
!  
!  
| 0.00 || 0.88 || 1.42 || 1.82 || 2.13 || 2.38 || 2.60 || 2.78 || 2.95  
| 0.00 || 0.88 || 1.42 || 1.82 || 2.13 || 2.38 || 2.60 || 2.78 || 2.95
!  
!  
| 1.00 || 0.54 || 0.37 || 0.28 || 0.23 || 0.19 || 0.17 || 0.15 || 0.13
| 1.00 || 0.54 || 0.37 || 0.28 || 0.23 || 0.19 || 0.17 || 0.15 || 0.13
|-
|-
! style="background-color: #C59500;" |
! style="background-color: #C59500;" |
! (1/⁴√2)-shifted undertone series segment = 9-ALS(1/⁴√2)  
! (1/⁴√2)-shifted undertone series segment = 9-ALS(1/⁴√2)
| (1) || 1.12 || 1.28 || 1.48 || 1.77 || 2.19 || 2.88 || 4.20 || 7.73  
| (1) || 1.12 || 1.28 || 1.48 || 1.77 || 2.19 || 2.88 || 4.20 || 7.73
!  
!  
| 0.00 || 0.17 || 0.35 || 0.57 || 0.82 || 1.13 || 1.53 || 2.07 || 2.95  
| 0.00 || 0.17 || 0.35 || 0.57 || 0.82 || 1.13 || 1.53 || 2.07 || 2.95
!  
!  
| 1.00 || 0.89 || 0.78 || 0.67 || 0.56 || 0.46 || 0.35 || 0.24 || 0.13
| 1.00 || 0.89 || 0.78 || 0.67 || 0.56 || 0.46 || 0.35 || 0.24 || 0.13
|-
|-
! style="background-color: #00CFA9;" |
! style="background-color: #00CFA9;" |
! AS5/4 = 1D JI lattice of 5/4 = 5/4 chain  
! AS5/4 = 1D JI lattice of 5/4 = 5/4 chain
| (5⁰/4⁰) || 5¹/4¹ || 5²/4² || 5³/4³ || 5⁴/4⁴ || 5⁵/4⁵ || 5⁶/4⁶ || 5⁷/4⁷ || 5⁸/4⁸
| (5⁰/4⁰) || 5¹/4¹ || 5²/4² || 5³/4³ || 5⁴/4⁴ || 5⁵/4⁵ || 5⁶/4⁶ || 5⁷/4⁷ || 5⁸/4⁸
!  
!  
| 0.00 || 0.32 || 0.64 || 0.97 || 1.29 || 1.61 || 1.93 || 2.25 || 2.58  
| 0.00 || 0.32 || 0.64 || 0.97 || 1.29 || 1.61 || 1.93 || 2.25 || 2.58
!  
!  
| 1/1 || 4/5 || 16/25 || 64/125 || 256/625 || 1024/3125 || 4096/15625 || 16384/78125 || 65536/390625
| 1/1 || 4/5 || 16/25 || 64/125 || 256/625 || 1024/3125 || 4096/15625 || 16384/78125 || 65536/390625
|-
|-
! style="background-color: #F4CCCC;" |
! style="background-color: #F4CCCC;" |
! 9 pitches of 1/2-powharmonic series = harmonic series compressed by 1/2  
! 9 pitches of 1/2-powharmonic series = harmonic series compressed by 1/2
| √1 || √2 || √3 || √4 || √5 || √6 || √7 || √8 || √9
| √1 || √2 || √3 || √4 || √5 || √6 || √7 || √8 || √9
!  
!  
| 0.00 || 0.50 || 0.79 || 1.00 || 1.16 || 1.29 || 1.40 || 1.50 || 1.58  
| 0.00 || 0.50 || 0.79 || 1.00 || 1.16 || 1.29 || 1.40 || 1.50 || 1.58
!  
!  
| 1.00 || 0.71 || 0.58 || 0.50 || 0.45 || 0.41 || 0.38 || 0.35 || 0.33
| 1.00 || 0.71 || 0.58 || 0.50 || 0.45 || 0.41 || 0.38 || 0.35 || 0.33
|-
|-
! style="background-color: #E06666;" |
! style="background-color: #E06666;" |
! 9 pitches of 1/2-subpowharmonic = subharmonic series compressed by 1/2  
! 9 pitches of 1/2-subpowharmonic = subharmonic series compressed by 1/2
| (1) || 1.06 || 1.13 || 1.22 || 1.34 || 1.50 || 1.73 || 2.12 || 3
| (1) || 1.06 || 1.13 || 1.22 || 1.34 || 1.50 || 1.73 || 2.12 || 3
!  
!  
| 0.00 || 0.08 || 0.18 || 0.29 || 0.42 || 0.58 || 0.79 || 1.08 || 1.58  
| 0.00 || 0.08 || 0.18 || 0.29 || 0.42 || 0.58 || 0.79 || 1.08 || 1.58
!  
!  
| 1.00 || 0.94 || 0.88 || 0.82 || 0.75 || 0.67 || 0.58 || 0.47 || 0.33
| 1.00 || 0.94 || 0.88 || 0.82 || 0.75 || 0.67 || 0.58 || 0.47 || 0.33
|-
|-
! style="background-color: #00B6FF;" |
! style="background-color: #00B6FF;" |
! 9 pitches of 2-logharmonic series  
! 9 pitches of 2-logharmonic series
| (1) || 1.58 || 2.00 || 2.32 || 2.58 || 2.81 || 3.00 || 3.17 || 3.32  
| (1) || 1.58 || 2.00 || 2.32 || 2.58 || 2.81 || 3.00 || 3.17 || 3.32
!  
!  
| 0.00 || 0.66 || 1.00 || 1.22 || 1.37 || 1.49 || 1.58 || 1.66 || 1.73  
| 0.00 || 0.66 || 1.00 || 1.22 || 1.37 || 1.49 || 1.58 || 1.66 || 1.73
!  
!  
| 1.00 || 0.63 || 0.50 || 0.43 || 0.39 || 0.36 || 0.33 || 0.32 || 0.30
| 1.00 || 0.63 || 0.50 || 0.43 || 0.39 || 0.36 || 0.33 || 0.32 || 0.30
|-
|-
! style="background-color: #006D99;" |
! style="background-color: #006D99;" |
! 9 pitches of 2-sublogharmonic series  
! 9 pitches of 2-sublogharmonic series
| (1) || 1.05 || 1.11 || 1.18 || 1.29 || 1.43 || 1.66 || 2.10 || 3.32  
| (1) || 1.05 || 1.11 || 1.18 || 1.29 || 1.43 || 1.66 || 2.10 || 3.32
!  
!  
| 0.00 || 0.07 || 0.15 || 0.24 || 0.36 || 0.52 || 0.73 || 1.07 || 1.73  
| 0.00 || 0.07 || 0.15 || 0.24 || 0.36 || 0.52 || 0.73 || 1.07 || 1.73
!  
!  
| 1.00 || 0.95 || 0.90 || 0.85 || 0.78 || 0.70 || 0.60 || 0.48 || 0.30
| 1.00 || 0.95 || 0.90 || 0.85 || 0.78 || 0.70 || 0.60 || 0.48 || 0.30
Retrieved from "https://en.xen.wiki/w/Harmonotonic_tuning"