Triangle / Lambdoma
This page is a musical instrument made up of a grid of just intonation intervals, all based on a root frequency.
The rows display the overtone series (1/1, 2/1, 3/1 ...). The columns follow its inverse, the undertone series (1/1, 1/2, 1/3, ...). Multiplying everything out, the resulting intervals contain an inherent music arising from simple, whole-number ratios.
Color indicates position in the octave, with red being the root or unison interval 1/1. Brightness indicates octave, with white and black tending toward the extremes of human hearing.
- The default instrument is a Hokema sansula, a type of kalimba.
- Right-click notes to turn on sine waves matching the interval.
- Resonator mode puts white noise through tuned bandpass filters.
- 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 sine and resonator mode |
| \ | 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 |
| hyperbolic | Change stride rather than scrolling. Denominator magnifies along the 1/1 line, numerator emphasizes the hyperbolic extremes. |
| Collatz | Hailstone numbers of Lothar Collatz |
| Pythagorean |
Pythagorean intervals
where each ratio is a power of 2 |
about this page
This webpage was inspired in part 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 ratios to find specific intervals. I had never seen just intonation demonstrated so elegantly, so I made this page to explore the concept.
I later learned that I had constructed the "lambdoid diagram" or Lambdoma, named for its resemblance to the Greek letter Lambda Λ. The synergy of color and tone, linking the octave to the color wheel, seemed intuitive, and revealed a beautiful pattern, both visual and musical. This pattern had previously been uncovered by artist and sound practitioner Barbara Hero, who built an 8x8 electronic Lambdoma instrument for sound healing purposes, using the same pattern of colors.
Hero learned of the Lambdoma from Tone: A Study in Musical Acoustics (1968) by Levarie and Levy, who trace the Lambdoma back to Pythagoras (ca. 500 BCE) by way of the Introduction to Arithmetic by Nicomachus of Gerasa (ca. 100 BCE) and the Theologumena arithmeticae of Iamblichus (ca. 300 CE). The Lambdoma is also mentioned by Plutarch (ca. 100 CE) in his commentary on Plato's Timaeus. It was depicted in the 19th century by Albert von Thimus in the neo-Pythagorean treatise Die harmonikale Symbolik des Alterthums (1876) which connects musical intervals to other harmonic relationships in nature. The Lambdoma was also used by mathematician Georg Cantor in his theory of transfinite sets (see below). More information can be gleaned from Hero's paper, The Lambdoma Matrix and Harmonic Intervals (1999).
the music of whole numbers
The smallest whole numbers have the greatest significance to our understanding of music. With the root, fifth, fourth, and octave in the top-left corner, the Lambdoma shows how the 3:2 proportion is the foundation of tonality. These frequencies sound quite similar to each other, which as any musician knows, can sometimes fool the ear.
The next prime number interval, 5:4, is a just major third, and its inverse, 4:5 is a minor third. Thus the overtone series sounds "major", and the undertone series sounds "minor". The next prime number out, 7, extends the minor to diminished, and major to dominant.
At this point, we have just enough notes to form a fairly melodious just-intonated scale, which you can play using your computer keyboard. Within a just-intonated tuning system, every chord can sound highly distinctive, since the notes are not distributed evenly across the octave.
tuning systems
The musical Circle of Fifths, created by repeatedly multiplying a frequency by 3:2, can be studied in more detail in this program's Pythagorean setting, where each ratio is a power of 2 or 3. Powers of 3 move by fifths, and powers of 2 by octaves. Using this principle, we can transpose any note back down into the same octave and create a scale.
On this instrument, similar notes can be found by color and compared. One can easily hear how stacking fifths does not bring you back to the starting note: find two far-apart red notes of the same brightness, and play both at once. These two frequencies are not quite the same, and they will audibly vibrate or "beat" against each other.
The interval between this pseudo-unison is known as the ditonic comma, and it is one of various "commas" that tuning systems adjust for. Another important comma is the syntonic comma, which is the difference between a just major third (5:4) and the closest equivalent achieved from stacking fifths.
In a sense, 12-tone equal temperament "bends" all of the notes to make the intervals evenly spaced. Fifths in equal temperament are all nearly in tune, making it easy to modulate between keys. By comparison, thirds are quite out of tune compared to a just major third (5:4). An alternative is meantone temperament, which favors harmonious major thirds over out-of-tune fifths, by distributing the comma differently.
tone and transfinite sets
Intervals in just intonation are rational numbers, composed of a
numerator and denominator that are both whole numbers. Notes in equal
temperament, however, are separated by a semitone of 1 to the 12th
root of 2 (1:
In the Lambdoma, Barbara Hero also sees the image of Georg Cantor's transfinite set of rational numbers ℚ, which Cantor proved countably infinite. Consider that although there are infinitely many natural numbers, each of these numbers is by definition finite, and we may count up to it by starting from 1 and adding 1 repeatedly. Similarly, we can count the cells in a Lambdoma by starting from 1:1 and moving outward diagonally in a snake-like pattern, thus mapping the rationals to the natural numbers. Though there are infinitely many rational numbers, by their nature they are discrete, countable, and not completely dense. Between any two rational numbers, there lies an uncountable continuity of irrational numbers in ℝ.
thank you!
Sansula samples by Freesound user cabled_mess. Gradient algorithm via Inigo Quizeles. Thanks to Dave Noyze for telling me about Barbara Hero. Thanks to Hems for the support!
Jules LaPlace / asdf.us / 2017-2025