The Template - A Unified Mathematical, Geometrical and Musical Instrument

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Unified Harmonic Calculator v0.10

Views / Perspectives

Secondary Tabs (Select a reference / fundamental frequency on the active grid selector to activate the dependent tabs)

Pythagorean Tuning Grid (Template)

Select a Unison on the grid to highlight its 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.

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
y=9
y=8
y=7
y=6
y=5
y=4
y=3
y=2
y=1
y=0
y=-1
y=-2
y=-3
y=-4
y=-5
y=-6
y=-7
y=-8
y=-9
y=-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)

Octave
Ratio to Unison: 2:1
Frequency: Awaiting selection
Unison
Ratio to Unison: 1:1
Frequency: Awaiting selection
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)

Octave
Ratio to Unison: 2:1
Frequency: Awaiting selection
Unison
Ratio to Unison: 1:1
Frequency: Awaiting selection
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