The Template - A Unified Mathematical, Geometrical and Musical Instrument
Unified Fractal Calculator v1.08 beta
Views / Perspectives (Select a Unison / Fundamental Frequency first on the Grid to activate the disabled tabs)
Data Tables
Pythagorean Lambda Tuning Grid (Template)
Select a Unison on the grid to highlight it's Pythagorean Chromatic Octave:
Reference Unison/Octave: Awaiting selection
Unison Coordinate (y, x): Awaiting selection
Transformation Chain: No transforms in force
The template lattice is generated from unity, so (0, 0) = 1 Hz.
Legend
Legend: Pythagorean Intervals & Ratios
Reference Unison & Octave Anchors
Unison
Ratio to Unison: 1:1
Frequency: Awaiting selection
Octave
Ratio to Unison: 2:1
Frequency: Awaiting selection
Pythagorean Comma Offsets
Diminished 2nd / Unison - Comma
Ratio to Unison: 524288:531441
Frequency: Awaiting selection
Unison + Comma
Ratio to Unison: 531441:524288
Frequency: Awaiting selection
Augmented 7th / Octave + Comma
Ratio to Unison: 531441:262144
Frequency: Awaiting selection
Diminished 9th / Octave - Comma
Ratio to Unison: 1048576:531441
Frequency: Awaiting selection
Positive Intervals - Ascending from Unison (to Octave)
Negative Intervals - Descending from Octave (to Unison)
Perfect 5th
Ratio to Unison: 3:2
Frequency: Awaiting selection
Perfect 4th
Ratio to Octave: 3:4
Frequency: Awaiting selection
Major 2nd
Ratio to Unison: 9:8
Frequency: Awaiting selection
Minor 7th
Ratio to Octave: 9:16
Frequency: Awaiting selection
Major 6th
Ratio to Unison: 27:16
Frequency: Awaiting selection
Minor 3rd
Ratio to Octave: 27:32
Frequency: Awaiting selection
Major 3rd
Ratio to Unison: 81:64
Frequency: Awaiting selection
Minor 6th
Ratio to Octave: 81:128
Frequency: Awaiting selection
Major 7th
Ratio to Unison: 243:128
Frequency: Awaiting selection
Minor 2nd
Ratio to Octave: 243:256
Frequency: Awaiting selection
Augmented 4th
Ratio to Unison: 729:512
Frequency: Awaiting selection
Diminished 5th
Ratio to Octave: 729:1024
Frequency: Awaiting selection
Augmented 5th
Ratio to Unison: 6561:4096
Frequency: Awaiting selection
Diminished 4th
Ratio to Octave: 6561:8192
Frequency: Awaiting selection
Augmented 2nd
Ratio to Unison: 19683:16384
Frequency: Awaiting selection
Diminished 7th
Ratio to Octave: 19683:32768
Frequency: Awaiting selection
Augmented 6th
Ratio to Unison: 59049:32768
Frequency: Awaiting selection
Diminished 3rd
Ratio to Octave: 59049:65536
Frequency: Awaiting selection
Augmented 3rd
Ratio to Unison: 177147:131072
Frequency: Awaiting selection
Diminished 6th
Ratio to Octave: 177147:262144
Frequency: Awaiting selection
Augmented Unison
Ratio to Unison: 2187:2048
Frequency: Awaiting selection
Diminished Octave
Ratio to Octave: 2187:4096
Frequency: Awaiting selection
Map Guides
Selected reference tone
Primary axes x=0 and y=0
Reference Unison/Octave: Awaiting selection
Unison Coordinate (y, x): Awaiting selection
Transformation Chain: No transforms in force
The template lattice is generated from unity, so (0, 0) = 1 Hz.
Pitch class | F#/Gb | C#/Db | G#/Ab | D#/Eb | A#/Bb | F | C | G | D | A | E | B | F#/Gb |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
y \ x | x=-6 | x=-5 | x=-4 | x=-3 | x=-2 | x=-1 | x=0 | x=1 | x=2 | x=3 | x=4 | x=5 | x=6 |
| y=10 | 1024
────
729 (-6, 10) | 1024
────
243 (-5, 10) | 1024
────
81 (-4, 10) | 1024
────
27 (-3, 10) | 1024
────
9 (-2, 10) | 1024
────
3 (-1, 10) | 1024 (0, 10) | 3072 (1, 10) | 9216 (2, 10) | 27648 (3, 10) | 82944 (4, 10) | 248832 (5, 10) | 746496 (6, 10) |
| y=9 | 512
───
729 (-6, 9) | 512
───
243 (-5, 9) | 512
───
81 (-4, 9) | 512
───
27 (-3, 9) | 512
───
9 (-2, 9) | 512
───
3 (-1, 9) | 512 (0, 9) | 1536 (1, 9) | 4608 (2, 9) | 13824 (3, 9) | 41472 (4, 9) | 124416 (5, 9) | 373248 (6, 9) |
| y=8 | 256
───
729 (-6, 8) | 256
───
243 (-5, 8) | 256
───
81 (-4, 8) | 256
───
27 (-3, 8) | 256
───
9 (-2, 8) | 256
───
3 (-1, 8) | 256 (0, 8) | 768 (1, 8) | 2304 (2, 8) | 6912 (3, 8) | 20736 (4, 8) | 62208 (5, 8) | 186624 (6, 8) |
| y=7 | 128
───
729 (-6, 7) | 128
───
243 (-5, 7) | 128
───
81 (-4, 7) | 128
───
27 (-3, 7) | 128
───
9 (-2, 7) | 128
───
3 (-1, 7) | 128 (0, 7) | 384 (1, 7) | 1152 (2, 7) | 3456 (3, 7) | 10368 (4, 7) | 31104 (5, 7) | 93312 (6, 7) |
| y=6 | 64
───
729 (-6, 6) | 64
───
243 (-5, 6) | 64
───
81 (-4, 6) | 64
───
27 (-3, 6) | 64
───
9 (-2, 6) | 64
───
3 (-1, 6) | 64 (0, 6) | 192 (1, 6) | 576 (2, 6) | 1728 (3, 6) | 5184 (4, 6) | 15552 (5, 6) | 46656 (6, 6) |
| y=5 | 32
───
729 (-6, 5) | 32
───
243 (-5, 5) | 32
───
81 (-4, 5) | 32
───
27 (-3, 5) | 32
───
9 (-2, 5) | 32
───
3 (-1, 5) | 32 (0, 5) | 96 (1, 5) | 288 (2, 5) | 864 (3, 5) | 2592 (4, 5) | 7776 (5, 5) | 23328 (6, 5) |
| y=4 | 16
───
729 (-6, 4) | 16
───
243 (-5, 4) | 16
───
81 (-4, 4) | 16
───
27 (-3, 4) | 16
───
9 (-2, 4) | 16
───
3 (-1, 4) | 16 (0, 4) | 48 (1, 4) | 144 (2, 4) | 432 (3, 4) | 1296 (4, 4) | 3888 (5, 4) | 11664 (6, 4) |
| y=3 | 8
───
729 (-6, 3) | 8
───
243 (-5, 3) | 8
───
81 (-4, 3) | 8
───
27 (-3, 3) | 8
───
9 (-2, 3) | 8
───
3 (-1, 3) | 8 (0, 3) | 24 (1, 3) | 72 (2, 3) | 216 (3, 3) | 648 (4, 3) | 1944 (5, 3) | 5832 (6, 3) |
| y=2 | 4
───
729 (-6, 2) | 4
───
243 (-5, 2) | 4
───
81 (-4, 2) | 4
───
27 (-3, 2) | 4
───
9 (-2, 2) | 4
───
3 (-1, 2) | 4 (0, 2) | 12 (1, 2) | 36 (2, 2) | 108 (3, 2) | 324 (4, 2) | 972 (5, 2) | 2916 (6, 2) |
| y=1 | 2
───
729 (-6, 1) | 2
───
243 (-5, 1) | 2
───
81 (-4, 1) | 2
───
27 (-3, 1) | 2
───
9 (-2, 1) | 2
───
3 (-1, 1) | 2 (0, 1) | 6 (1, 1) | 18 (2, 1) | 54 (3, 1) | 162 (4, 1) | 486 (5, 1) | 1458 (6, 1) |
| y=0 | 1
───
729 (-6, 0) | 1
───
243 (-5, 0) | 1
───
81 (-4, 0) | 1
───
27 (-3, 0) | 1
───
9 (-2, 0) | 1
───
3 (-1, 0) | 1 (0, 0) | 3 (1, 0) | 9 (2, 0) | 27 (3, 0) | 81 (4, 0) | 243 (5, 0) | 729 (6, 0) |
| y=-1 | 1
────
1458 (-6, -1) | 1
───
486 (-5, -1) | 1
───
162 (-4, -1) | 1
───
54 (-3, -1) | 1
───
18 (-2, -1) | 1
───
6 (-1, -1) | 1
───
2 (0, -1) | 3
───
2 (1, -1) | 9
───
2 (2, -1) | 27
───
2 (3, -1) | 81
───
2 (4, -1) | 243
───
2 (5, -1) | 729
───
2 (6, -1) |
| y=-2 | 1
────
2916 (-6, -2) | 1
───
972 (-5, -2) | 1
───
324 (-4, -2) | 1
───
108 (-3, -2) | 1
───
36 (-2, -2) | 1
───
12 (-1, -2) | 1
───
4 (0, -2) | 3
───
4 (1, -2) | 9
───
4 (2, -2) | 27
───
4 (3, -2) | 81
───
4 (4, -2) | 243
───
4 (5, -2) | 729
───
4 (6, -2) |
| y=-3 | 1
────
5832 (-6, -3) | 1
────
1944 (-5, -3) | 1
───
648 (-4, -3) | 1
───
216 (-3, -3) | 1
───
72 (-2, -3) | 1
───
24 (-1, -3) | 1
───
8 (0, -3) | 3
───
8 (1, -3) | 9
───
8 (2, -3) | 27
───
8 (3, -3) | 81
───
8 (4, -3) | 243
───
8 (5, -3) | 729
───
8 (6, -3) |
| y=-4 | 1
─────
11664 (-6, -4) | 1
────
3888 (-5, -4) | 1
────
1296 (-4, -4) | 1
───
432 (-3, -4) | 1
───
144 (-2, -4) | 1
───
48 (-1, -4) | 1
───
16 (0, -4) | 3
───
16 (1, -4) | 9
───
16 (2, -4) | 27
───
16 (3, -4) | 81
───
16 (4, -4) | 243
───
16 (5, -4) | 729
───
16 (6, -4) |
| y=-5 | 1
─────
23328 (-6, -5) | 1
────
7776 (-5, -5) | 1
────
2592 (-4, -5) | 1
───
864 (-3, -5) | 1
───
288 (-2, -5) | 1
───
96 (-1, -5) | 1
───
32 (0, -5) | 3
───
32 (1, -5) | 9
───
32 (2, -5) | 27
───
32 (3, -5) | 81
───
32 (4, -5) | 243
───
32 (5, -5) | 729
───
32 (6, -5) |
| y=-6 | 1
─────
46656 (-6, -6) | 1
─────
15552 (-5, -6) | 1
────
5184 (-4, -6) | 1
────
1728 (-3, -6) | 1
───
576 (-2, -6) | 1
───
192 (-1, -6) | 1
───
64 (0, -6) | 3
───
64 (1, -6) | 9
───
64 (2, -6) | 27
───
64 (3, -6) | 81
───
64 (4, -6) | 243
───
64 (5, -6) | 729
───
64 (6, -6) |
| y=-7 | 1
─────
93312 (-6, -7) | 1
─────
31104 (-5, -7) | 1
─────
10368 (-4, -7) | 1
────
3456 (-3, -7) | 1
────
1152 (-2, -7) | 1
───
384 (-1, -7) | 1
───
128 (0, -7) | 3
───
128 (1, -7) | 9
───
128 (2, -7) | 27
───
128 (3, -7) | 81
───
128 (4, -7) | 243
───
128 (5, -7) | 729
───
128 (6, -7) |
| y=-8 | 1
──────
186624 (-6, -8) | 1
─────
62208 (-5, -8) | 1
─────
20736 (-4, -8) | 1
────
6912 (-3, -8) | 1
────
2304 (-2, -8) | 1
───
768 (-1, -8) | 1
───
256 (0, -8) | 3
───
256 (1, -8) | 9
───
256 (2, -8) | 27
───
256 (3, -8) | 81
───
256 (4, -8) | 243
───
256 (5, -8) | 729
───
256 (6, -8) |
| y=-9 | 1
──────
373248 (-6, -9) | 1
──────
124416 (-5, -9) | 1
─────
41472 (-4, -9) | 1
─────
13824 (-3, -9) | 1
────
4608 (-2, -9) | 1
────
1536 (-1, -9) | 1
───
512 (0, -9) | 3
───
512 (1, -9) | 9
───
512 (2, -9) | 27
───
512 (3, -9) | 81
───
512 (4, -9) | 243
───
512 (5, -9) | 729
───
512 (6, -9) |
| y=-10 | 1
──────
746496 (-6, -10) | 1
──────
248832 (-5, -10) | 1
─────
82944 (-4, -10) | 1
─────
27648 (-3, -10) | 1
────
9216 (-2, -10) | 1
────
3072 (-1, -10) | 1
────
1024 (0, -10) | 3
────
1024 (1, -10) | 9
────
1024 (2, -10) | 27
────
1024 (3, -10) | 81
────
1024 (4, -10) | 243
────
1024 (5, -10) | 729
────
1024 (6, -10) |
Legend
Legend: Pythagorean Intervals & Ratios
Reference Unison & Octave Anchors
Unison
Ratio to Unison: 1:1
Frequency: Awaiting selection
Octave
Ratio to Unison: 2:1
Frequency: Awaiting selection
Pythagorean Comma Offsets
Diminished 2nd / Unison - Comma
Ratio to Unison: 524288:531441
Frequency: Awaiting selection
Unison + Comma
Ratio to Unison: 531441:524288
Frequency: Awaiting selection
Augmented 7th / Octave + Comma
Ratio to Unison: 531441:262144
Frequency: Awaiting selection
Diminished 9th / Octave - Comma
Ratio to Unison: 1048576:531441
Frequency: Awaiting selection
Positive Intervals - Ascending from Unison (to Octave)
Negative Intervals - Descending from Octave (to Unison)
Perfect 5th
Ratio to Unison: 3:2
Frequency: Awaiting selection
Perfect 4th
Ratio to Octave: 3:4
Frequency: Awaiting selection
Major 2nd
Ratio to Unison: 9:8
Frequency: Awaiting selection
Minor 7th
Ratio to Octave: 9:16
Frequency: Awaiting selection
Major 6th
Ratio to Unison: 27:16
Frequency: Awaiting selection
Minor 3rd
Ratio to Octave: 27:32
Frequency: Awaiting selection
Major 3rd
Ratio to Unison: 81:64
Frequency: Awaiting selection
Minor 6th
Ratio to Octave: 81:128
Frequency: Awaiting selection
Major 7th
Ratio to Unison: 243:128
Frequency: Awaiting selection
Minor 2nd
Ratio to Octave: 243:256
Frequency: Awaiting selection
Augmented 4th
Ratio to Unison: 729:512
Frequency: Awaiting selection
Diminished 5th
Ratio to Octave: 729:1024
Frequency: Awaiting selection
Augmented 5th
Ratio to Unison: 6561:4096
Frequency: Awaiting selection
Diminished 4th
Ratio to Octave: 6561:8192
Frequency: Awaiting selection
Augmented 2nd
Ratio to Unison: 19683:16384
Frequency: Awaiting selection
Diminished 7th
Ratio to Octave: 19683:32768
Frequency: Awaiting selection
Augmented 6th
Ratio to Unison: 59049:32768
Frequency: Awaiting selection
Diminished 3rd
Ratio to Octave: 59049:65536
Frequency: Awaiting selection
Augmented 3rd
Ratio to Unison: 177147:131072
Frequency: Awaiting selection
Diminished 6th
Ratio to Octave: 177147:262144
Frequency: Awaiting selection
Augmented Unison
Ratio to Unison: 2187:2048
Frequency: Awaiting selection
Diminished Octave
Ratio to Octave: 2187:4096
Frequency: Awaiting selection
Map Guides
Selected reference tone
Primary axes x=0 and y=0