Triangle / Lambdoma
This instrument uses a basic cartesian plot of the rational numbers, expressed as fractions, to make a wide palette of just intonation intervals quickly accessible.
The harmonic series (1/1, 2/1, 3/1 ...) is expressed by the rows, while the undertone series (1/1, 1/2, 1/3, ...) is expressed by the columns. The resulting intervals are instantly musically meaningful, though they arise out of these simple ratios.
Brightness indicates octave. Color indicates position in the octave, with red being the root or unison interval 1/1.
- The default instrument is a Hokema sansula, a type of kalimba.
- Right-click notes to turn on sine waves matching the interval.
- Drag-and-drop samples into the window to play with your own sounds.
keyboard shortcuts
| ESC | Stop all sound |
| ESC ESC | Return to home position |
| ? | Show this help |
| + - | Change scale |
| up down left right |
Scroll the grid |
| 0-9 a-z | Keyboard mapped to the top-left 8x8 grid, sorted by pitch |
| ? | Toggle this help |
| \ | Detect MIDI device (listening on channel 1) |
| ⇧ + ⇧ - |
Change scale root by +/- 1 hz |
| ⌘⇧ + ⌘⇧ - |
Change scale root by +/- 10 hz |
| ⌘ up ⌘⇧ up ⌘⇧⌃ up |
Change pitch of sampler by +10 / +1 / -0.1 hz |
| ⌘ down ⌘⇧ down ⌘⇧⌃ down |
Change pitch of sampler by -10 / -1 / -0.1 hz |
scales
Alternate scales are accessed by pressing the +/- keys:
| natural | Natural numbers: 1, 2, 3 ... |
| undertone | Subharmonic intervals under the line 1/1 |
| overtone | Harmonic intervals above the line 1/1 |
| primes | Prime numbers only (most dissonant) |
| arithmetic | Multiply all cells by an interval rather than scrolling |
| Collatz | Hailstone numbers of Lothar Collatz |
| Pythagorean |
Pythagorean intervals
where each ratio is a power of 2 |
about this page
This webpage was inspired by Peter Neubäcker, inventor of the Melodyne software. In the short biographical documentary Wie klingt ein Stein? (What does a stone sound like?), Neubäcker describes the basic principles of harmonic intervals. He first demonstrates how one plays harmonics on a monochord. He then shows it next to a grid of whole-number ratios, and demonstrates how one can use these intervals to find specific ratios. I had never seen the concept of just intonation displayed so elegantly, so I made this page to understand the concept more deeply.
I later learned that I had recreated the Lambdoma as described by Barbara Hero. Hero made an electronic lambdoma instrument for sound healing purposes. Hero traces the Lambdoma back to the Introduction to Arithmetic by Nichomachus of Gerasa (ca. 100 CE), and suggests it was rediscovered by Albert von Thimus who depicts it in his Die harmonikale Symbolik des Alterthums (1876). In the Lambdoma, Hero also sees the image of Georg Cantor's transfinite set of rational numbers, which are shown to be countable through use of a Cartesian plot. More can be read in Hero's article, The Lambdoma Matrix and Harmonic Intervals (1999).
With the root, fifth, and fourth in the top-left corner, the Lambdoma shows how the 3/2 proportion is essential to the perception of consonance. The musical circle of fifths, derived from these simple proportions, can be studied in more detail in "Pythagorean" mode, where related intervals can be found by color and compared.
thank you!
Sansula samples by Freesound user cabled_mess. Thanks to Dave Noyze for telling me about Barbara Hero. Thanks to Hems for the support!
Jules LaPlace / asdf.us / 2018-2025