Harmoni

Written by instrument designer Nick Collier of Nick’s World of Synths. Nick is a British instrument designer who has created a ton of instruments — besides the Harmonicon on which the Harmoni is based — you’ve never even heard of before. Make sure to stop by his site (click here) and check them all out. 

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The Harmoni is a digital version of a real homemade synthesizer. The reason for the strange keyboard is that it is tuned in “just intonation.”

Here are a few notes to help you navigate the keyboard:

  • Octaves are stacked vertically
  • White is the root note
  • Yellow is the 5th
  • Purple is the 4th
  • Red and blue are more major
  • Orange and green are more minor

Follow the colors without mixing major and minor and it should sound pretty harmonious.

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MY EXPLANATION OF JUST INTONATION

Just intonation is the system of tuning that western music used to be based on, so it sounds quite familiar and is easy for someone brought up on modern tuning to relate to. It has also been used my most other musical traditions throughout history and is probably the most instinctive way of tuning an instrument as the intervals can be tuned by ear very easily. Modern equal temperament is not very instinctive, ask a piano tuner how long it takes to learn how to tune all the intervals, it is done by listening to the speed of the beats of the out of tune intervals.

Firstly, we need understand musical intervals as the relative frequency of one note to another. We can use two numbers to represent the frequency ratio of two notes — for example the ratio of an octave is 1:2, which means that the higher note is twice the frequency of the lower note e.g. A-440Hz : A-880Hz, or the fifth 2:3, the higher note does three vibrations for every two of the lower note e.g A-440Hz : E-660Hz.

When using ratios we are just concerning ourselves with relative pitch. We could choose any two numbers to make our interval, but our ears prefer ones based on smaller number ratios. Here is the major scale as ratios.

Do 1:1, Re 8:9, Me 4:5, Fa 3:4, So 2:3, La 3:5, Te 8:15, Do 1:2

do re me pitches

Modern equal temperament does not use these perfect ratios. It has re-tuned them, but re-tuned them by small amounts so they are still recognizable as representing the same intervals.

We can take our major scale and fill in the whole tone gaps to make a 12 semitone scale (picture these two tables as one long row).

1:1 15:16 8:9 5:6 4:5 4:3 √2
Root m2nd M2nd m3rd M3rd 4th Tritone
2:3 5:8 3:5 5:9 8:15 1:2
5th m6th M6th m7th M7th Octave

Seen in this way as simple whole number ratios our familiar intervals seem to have a kind of geometric perfection. The only problem with this system is that the notes are not equally spaced.

Some of the semitones are bigger than others. This means that on an instrument tuned in this way a piece of music will sound different if it is transposed to a different key. In modern tuning, equal temperament has re-tuned these intervals so they are equally spaced, and you can play in any key and the music will have the same essential character. The process of equal temperament in Western music happened around the time that keyboard instruments were being invented.

WHAT DOES IT SOUND LIKE?

Compared with equal temperament, the harmonies of just intonation sound purer, brighter and more defined. Equal temperament sounds fuzzy and more uniform. The intervals that have suffered most by equal temperament, namely minor 3rd, major 3rd, minor 6th, major 6th are what the emotional language of Western music is based upon.

Instruments tuned to just intonation I find far more emotional to play. I remember the first time I experimented with just intonation, it was an old electric organ that I re-tuned, I found the sound so captivating and beautiful, I suddenly found I could play away for hours improvising and really enjoying playing music, whereas before I would find it bland and uninspiring. My point is that just intonation could be seen by some as being a nerdy distraction from music making and obscure pointless subject, but to me it is key to my enjoyment of playing a musical instrument and I want other people to experience its beauty.

BUILDING A SCALE

Just intonation is an open system with no limit on the number of notes that can be added to a scale. While the traditional western 12-semitone scale is just one variation, I have chosen to use it as an example because it is easy for most people to relate to. The system I used on the Harmoni is an extension of this 12-semitone system.

(table of scale but it is the same as above)

You may be wondering why the particular intervals in the above scale were chosen: Is there any kind of formula for making a scale, or is it just a case of adding more and more intervals and stopping before they start become too obscure?

I think the truth is a bit of both.

We know that intervals based on small numbers are good, so this accounts for 2:3 3:4 4:5 5:6 3:5 there are no simpler intervals than these that can be fitted within our octave. But simplicity is not just about how small the numbers are, it is also about prime numbers. If you look at all the numbers in the scale you will notice that they can all be divided by two, three or five. The scale has a prime number limit of five, and this explains why the next smallest interval is 5:8 and not the smaller 4:7 or 5:7.

Setting a prime number limit of five keeps the sound familiar to Western ears. If we use seven as our limit then we get some more exotic sounding intervals. The intervals based on seven always remind me of traditional West African music.

The last four — 8:9, 5:9, 8:15, 15:16 — are all created by moving a fifth away from the other small number intervals, and they also fit nicely into the spaces at either end of the scale.

The √2 is not a just interval, I use it because it is the halfway point between the root and the octave, it is useful because it is a point of symmetry between the bottom and the top halves of the scale. The top and bottom halves are a mirror image of each other, except for 8:9 and 5:9.

MY SYSTEM

(Note: The colors of the instrument have been changed around since writing this explanation. Sorry for the confusion, but it shouldn’t make much difference in understanding this explanation.)

My system is just an extended version of the 12-semitone scale, apart from two notes using ratios of seven, I have stuck to a prime number limit of five. The reason for using more than 12 notes is to allow for more freedom of movement.

temp image of harmonicon

HOW IT WAS CONSTRUCTED

I made my scale by starting with just three fundamental intervals. If we decide to use a five-limit scale, then the only prime numbers we have to base all our notes on are two, three and five.

two, three and five intervals

The three fundamental intervals are made by using combinations of the prime numbers 2:3, 2:5, and 3:5.

temp image

To keep them all within the same octave 2:5 becomes 4:5. So we have 2:3 (fifth) 4:5 (major 3rd), and 3:5 (major 6th).

inverted intervals

temp image 2

These intervals are inverted to make three new ones 3:4 (fourth) 5:8 (minor 6th) 5:6 (minor 3rd).

Another six intervals

temp image 3

Another six are made intervals by adding neighboring intervals. If we try making new intervals by adding other ones together then we just get ones we already have. So only six new ones can be made this way.

six new intervals

temp image 4

Six new intervals are made by adding each interval to itself. These new intervals appear in the web-pattern diagram to be only connected to one other note, but the web diagram shows a small proportion of the harmonies between the notes.

Notice how in the web diagrams above, the scale has a symmetrical pattern to it. Seeing harmony as a web of interconnections can help us design keyboards and other playing interfaces. This arrangement of notes is very different from the way notes are usually arranged on an instrument, which is in order of pitch in a one-dimensional row.

While it is important to know the order of pitch of the notes from low to high, the real beauty of harmony is its dimensionless quality, the way that notes link not to their immediate neighbors but branch out to various special points. A good keyboard should represent not only the linear order of pitches but also the dimensionless web of harmony. My keyboard uses colors to show a “harmonic web” superimposed on a linear scale of notes that are arranged in order of pitch.

THE TUNING

You may think that my system looks complicated but it is tuned to simple frequency ratios, the same ones that have been used by musicians for thousands of years. They are so natural that I was able to tune them all by ear, and I’ve got a terrible sense of pitch, I can’t even sing Happy Birthday in tune.

Most musicians don’t understand the science of tuning and are unaware of just intonation, but to the ears the difference is obvious. Listen to two more of my sound files. You can hear something special in the sound even though I am not playing anything that special.

 

MAJOR SCALE

major scale

MINOR SCALE

minor scale

COLOR CODING OF NOTES

I like to color code my notes to help me navigate around the keyboard by grouping related notes together with similar colors. My system is loosely based on the color wheel in and attempt to show emotional qualities of the intervals. It is not meant to be taken too literally as there is no literal relationship between colors and musical harmony, but there are some similarities.

It uses colors as a quick aid to remembering all the intervals and to help the musician build a mental map of a complex web of tonal possibilities. The symbolism I use is quite simple. A combination of three ideas.

  1. Warm/cold colors for sad/happy or major/minor notes e.g: orange-M3rd, purple-m3rd.
  2. Primary/secondary/tertiary colors for the range of dominant to obscure notes e.g: red-5th, purple-m3rd, brown-7:4.
  3. Complementary colors to show inverted pairs of notes e.g: 5th-red, 4th-green.Here are six fundamental intervals paired up with six points on the color wheel. Of course the associations are subjective, and in the right context they can be made to behave in the opposite way, (Yes, it’s possible to write a sad song using the major scale) but I think it is a good starting point.

Orange for M3rd — warmest color for the most important major interval
Blue for m6th — coldest color for the saddest interval
Red for 5th
Green appropriate for 4th because calm understated qualities
Yellow for M6th to me it is the sunniest interval used in a lot in Hawaiian music
Purple for m3rd sad with a touch of warmth, unlike m6th which is harsh and despairing

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SOME MORE IDEAS THAT RELATE TO CONSTRUCTING SCALES

OVERTONES AND UNDERTONES

An string instrument or pipe has many modes of vibration that produce notes higher than the basic note to which it is tuned, and these notes form a scale of overtones, at multiples of the fundamental frequency. This scale is very natural and can be used to influence man-made scales. The harmonic series is closely related to the major scale and has a bright lively character.

overtones

overtones on piano The overtone scale can be inverted to make an undertone scale:

undertones350 undertones piano

The overtone scale can be inverted to make an undertone scale:

The undertone scale is less common in natural systems and has an opposite character to the overtone scale. It is related to the minor scale, but being a purer form of the minor scale it has a very dark, morbid character. The notes of the undertone scale get closer and closer together in the low end and therefore limit its useful range.

Using the overtone or undertone scale in their natural forms can be a bit limiting, but they are important to keep in mind when making music. I like to think of them as two opposite fundamental forces that define the character of music, like major vs minor, but more perfect. The traditional music theory explanation of major and minor scales is based on the circle of fifths and fourths, but I think that overtones and undertone are a more natural explanation.

INVERSION OF INTERVALS

Every interval has it’s own opposite apart from the octave and the tritone. The way to invert an interval is to play it descending from the root instead of ascending, e.g. play a 5th down from the root and you get a 4th in the octave below. One way to visualize this is to imagine a mirror placed at the root note.

Another way to make inversions is to reflect around the tritone. The tritone is exactly half way between two octaves. This way all our inversions say within the same octave that they came from.

Most intervals’ opposites are truly opposite in character and often don’t get placed in the same scale together, so musically they are not particularly useful and create a very atonal sound when played together, although they do give the whole system a nice symmetry, and the symmetry helps to simplify the visualization of a tuning system.


Notes

Dictionary of musical terms

“Just Intonation” is also referred to as “pure temperament”

Quotes

“Instruments tuned to just intonation I find far more emotional to play.”

“I remember the first time I experimented with just intonation, it was an old electric organ that I re-tuned, I found the sound so captivating and beautiful, I suddenly found I could play away for hours improvising and really enjoying playing music, whereas before I would find it bland and uninspiring.”

“Just intonation could be seen by some as being a nerdy distraction from music making and and obscure pointless subject, but to me it is key to my enjoyment of playing a musical instrument and I want other people to experience its beauty.”

“Compared with equal temperament, the harmonies of just intonation sound purer, brighter and more defined.”

“The reason for using more than 12 notes is to allow for more freedom of movement.”

“This arrangement of notes is very different from the way notes are usually arranged on an instrument, which is in order of pitch in a one dimensional row.

While it is important to know the order of pitch of the notes from low to high, the real beauty of harmony is it’s dimensionless quality, the way that notes link not to their immediate neighbours but branch out to various special points.”

“A good keyboard should represent not only the linear order of pitches but also the dimensionless web of harmony.”

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