Harmonotonic tuning: Difference between revisions

{| class="wikitable"

|+comparison of example monotonic tunings

! colspan="9" |frequency

! colspan="9" |pitch

! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9

|-

! 4-ODO = 4th overtone mode = 4-ADO  

| 1/1 || 5/4 || 3/2 || 7/4 || 2/1 || 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 || 1/1 || 4/5 || 2/3 || 4/7 || 1/2 || 2/5 || 1/3 || 2/7 || 1/4

|-

! 4-EDO = rank-1 temperament w/ generator 300¢ = APS⁴√2 ≈ APS1.189  

| 1.00 || 1.19 || 1.41 || 1.68 || 2.00 || 2.38 || 2.83 || 3.36 || 4.00 || 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

|-

! 4-UDO = 4th undertone mode  

| 1/1 || 8/7 || 4/3 || 8/5 || 2/1 || 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 || 1/1 || 7/8 || 3/4 || 5/8 || 1/2 || 7/16 || 3/8 || 5/16 || 1/4

|-

! 4-EFDφ  

| 1.00 || 1.15 || 1.31 || 1.46 || 1.62 || 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 || 1.00 || 0.87 || 0.76 || 0.68 || 0.62 || 0.54 || 0.47 || 0.42 || 0.38

|-

! 4-ELDφ  

| 1.00 || 1.11 || 1.24 || 1.40 || 1.62 || 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 || 1.00 || 0.90 || 0.81 || 0.71 || 0.62 || 0.56 || 0.50 || 0.44 || 0.38

|-

! 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 || 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

|-

! undertone series segment = 9-US = 9-UD9 = 9-EDL  

| 1/1 || 9/8 || 9/7 || 3/2 || 9/5 || 9/4 || 3/1 || 9/2 || 9/1 || 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

|-

! 9-OS(3/4)  

| 1/1 || 7/4 || 5/2 || 13/4 || 4/1 || 19/4 || 11/2 || 25/4 || 7/1 || 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

|-

! 9-US(3/4)  

| 1/1 || 28/25 || 14/11 || 28/19 || 7/4 || 28/13 || 14/5 || 4/1 || 7/1 || 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

|-

! (1/⁴√2)-shifted overtone series segment = 9-AFS(1/⁴√2)  

| 1.00 || 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 || 1.00 || 0.54 || 0.37 || 0.28 || 0.23 || 0.19 || 0.17 || 0.15 || 0.13

|-

! (1/⁴√2)-shifted undertone series segment = 9-ALS(1/⁴√2)  

| 1.00 || 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 || 1.00 || 0.89 || 0.78 || 0.67 || 0.56 || 0.46 || 0.35 || 0.24 || 0.13

|-

! AS5/4 = 1D JI lattice of 5/4 = 5/4 chain  

| 1/1 || 5/4 || 25/16 || 125/64 || 625/256 || 3125/1024 || 15625/4096 || 78125/16384 || 390625/65536 || 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

|-

! 9 pitches of 1/2-powharmonic series = harmonic series compressed by 1/2  

| 1.00 || 1.41 || 1.73 || 2.00 || 2.24 || 2.45 || 2.65 || 2.83 || 3.00 || 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

|-

! 9 pitches of 1/2-subpowharmonic = subharmonic series compressed by 1/2  

| 1.00 || 1.06 || 1.13 || 1.22 || 1.34 || 1.50 || 1.73 || 2.12 || 3.00 || 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

|-

! 9 pitches of 2-logharmonic series  

| 1.00 || 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 || 1.00 || 0.63 || 0.50 || 0.43 || 0.39 || 0.36 || 0.33 || 0.32 || 0.30

|-

! 9 pitches of 2-sublogharmonic series  

| 1.00 || 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 || 1.00 || 0.95 || 0.90 || 0.85 || 0.78 || 0.70 || 0.60 || 0.48 || 0.30