This is an interactive version of a real homemade synthesizer. Each note on the keyboard was recorded separately
So it plays just like the real thing. The reason for the strange keyboard is because it is
tuned in 'just intonation'.
Here are a few tips 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 colours without mixing major and minor and it should sound pretty harmonious
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
2 numbers to represent the frequency ratio of two notes, for example the ratio of an octave is 1:2, which means
that the high note is twice the frequency of the low note e.g A-440Hz:A-880Hz, or the fifth 2:3, the high note does
3 vibrations for every 2 of the low 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
Modern equal temperament does not use these perfect ratios, it has re-tuned them but by small amounts so they are still recognisable as representing the same intervals.
we can take our major scale and fill in the whole tone gaps to make a 12 semitone scale
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. Modern tuning, equal temperament has re-tuned these intervals in so they are equally spaced, 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 is sound like
Sound 3 sad.mp3
Compared to equal temperament the harmonies 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 what 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 and obscure pointless subject, 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. 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 use on my synthesizer (the keyboard at the top of the page) is an extension of this 12 semitone system.
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 2, 3 or 5. The scale has a prime number limit of 5, 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 5 keeps the sound familiar to western ears. If we use 7 as our limit then we get some more exotic sounding intervals. The intervals based on 7 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, 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.
***I have changed around the colours I use since making the keyboard at the top, sorry for the confusion ***
My system is just an extended version of the 12 semitone scale, apart from two notes using ratios of 7, I have stuck to a prime number limit of 5. The reason for using more than 12 notes is to allow for more freedom of movement.
How it was constructed
I made my scale by starting with just 3 fundamental intervals. If we decide to use a 5-limit scale, the the only prime numbers we have to base all our notes on are 2,3,5
The 3 fundamental intervals are made by using combinations of the prime numbers 2:3 2:5 3:5 To keep them all within the same octave 2:5 becomes 4:5. So we have 2:3(fifth) 4:5(major 3rd) 3:5(major 6th)
These intervals are inverted to make 3 new ones 3:4(fourth) 5:8(minor 6th) 5:6(minor 3rd)
Another 6 are made intervals by adding neighbouring intervals. If we try making new intervals by adding other ones together then we just get ones we have already got. So only 6 new ones can be made this way.
Six new intervals are made by adding each interval to itself. These new intervals appear in the web diagram to be only connected to one other note, but the web diagram only shows a small proportion of the harmonies between the notes.
Notice how in the web diagrams above, the scale has symmetry pattern to it. Seeing harmony as a web of interconnections can help us design key boards and other playing interfaces. A web like representation 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. My keyboard uses colours to show a "harmonic web" superimposed on a linear scale of notes that are arranged in order of pitch.
Colour coding the notes
I like to colour code my notes to help me navigate around the keyboard by grouping related notes together with similar colours. My system is loosely based on the colour 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 colours and musical harmony, but there are some similarities. It uses colours 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 3 ideas.
1 warm/cold colours for sad/happy or major/minor notes e.g: orange-M3rd, purple-m3rd.
2 primary/secondary/tertiary colours for the range of dominant to obscure notes e.g: red-5th, purple-m3rd, brown-7:4.
3 complementary colours to show inverted pairs of notes e.g: 5th-red, 4th-green.
Here are 6 fundamental intervals paired up with 6 points on the colour 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 is it a good starting point.
Orange for M3rd warmest colour for the most important major interval
Blue for m6th coldest colour for the saddest interval
Red for 5th
Green appropriate for 4th because calm under-stated qualities
Yellow for M6th to me it is the sunniest interval used a lot in Hawaiian music
Purple for m3rd sad with a touch of warmth unlike m6th which is harsh and despairing
Some more ideas that relate to constructing scales
Overtones and Undertones
An instrument string or pipe has many modes of vibration that produce notes higher than the basic note to which it is tuned 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.
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 2 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 visualise 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 visualisation of a tuning system.