Musical chords and their Lissajous Patterns

with Fractal Tune Smithy and Lissajous 3D

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Musical connection , Lissajous patterns for pure harmonies , Pure harmony and tempered music , Listening to beats , Transforming tempered music to pure harmonies , Pure harmonies with pitch shifts , Why pitch shifts occur if all harmonies are pure

Musical connection

The musical connection is to do with the frequencies that make up a chord, and the Lissajous patterns that they form. When you play pure harmonic series based chords, the curve joins back to its starting point after only a few turns, so pure harmonies give simpler Lissajous patterns. When you play chords that use somewhat impure harmonies then the curve may not join back at all, or may do so only after many many loops around, and the patterns are far more complex.

The reason this works is that Lissajous patterns join up quickly if the waves per cycle (frequencies) are related to each other by pure ratios. It so happens that pitches related to each other in this fashion also form the most harmonious chords of all on most musical instruments. This observation about musical harmony is originally attributed to Pythagorous, though as his work survives only as fragmentary quotes by later authors, this is more of a matter of legend than established fact.. At any rate it was well known to the early Greeks.

For instance the warmest and most melodious major chord on most instruments is played using the intervals 1/1 5/4 3/2. That's the same as 4/4 5/4 6/4, so if you enter the numbers 4, 5 and 6 in Lissajous 3D you get the curve for this chord.

Add a 7 as a fourth wave : 4 5 6 7, and the result is once more a very harmonious chord (known as the harmonic dominant seventh). Other numbers in small ratios to each other such as 7/6 are also particularly harmonious (the 7/6 one is the so called septimal minor - a dark rich mysterious minor chord much loved by microtonal composers).

The reason these chords are so harmonious is that most instruments have a timbre made out of an overtone series of frequencies based on whole numbers. So if you play pitches in small number ratios with each other, many overtones meet and re-inforce each other and so the notes sound nice together.

Note - there are exceptions to this rule including bells, and gongs. These have "inharmonic timbres", and they are sometimes used by microtonal composers exploring exotic tunnings for that reason.

But most instruments like voice, strings, wind instruments etc have "harmonic timbres" in which the constituent frequencies that make up the note are all whole number multiples of the fundamental. The piano is a special case - it has a basically harmonic timbre, so generally small number ratios sound pretty good on piano too - but the frequencies of the overtone series are somewhat flatter than you would expect - and get flatter the higher up you go up the series - a consequence of the extrremely high tension applied to piano wires. Older pianos from the nineteenth century and earlier have less tension so have more harmonic timbres than a modern piano.

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Lissajous patterns for pure harmonies

To explore the widest range of possibilities for pure harmony chords in FTS, lookin your purple FTS Extra Shortcuts folder for the Lambdoma shortcut. You will find this folder on your desktop after you install FTS.

Then the coloured squares button in the Lambdoma window, will bring up the Lambdoma keyboard. This is actually just a small section of an infinite keyboard with the pitches arranged to show all possible pure ratios organised in a musically meaningful fashion.

Any notes all in the same row will sound together to play particularly harmonious chords. Also any notes in the same column will also play harmonious chords - of another type. So it is particularly easy to find nice sounding harmonic series based chords with this layout.

You can play the Lambdoma keyboard in Fractal Tune Smithy using the mouse. Click on notes to play, and use the space bar as a pedal to sustain overlapping notes. The preset eight by eight board gives plenty to explore as it is, however you can also move it around in the infinite ratios array if you so wish as explained in the help for that window in FTS.

You can also use the PC keyboard to play it - you will find that it will play the top four rows of the Lambdoma. The bottom four rows of an eight row keyboard will be inaccessible from the PC keyboard, which is a bit limiting, so go to Keyboard Picture Options and select Moveable P.c. keyboard region , and you will find that you can move the playing area up and down using the num pad arrow keys (with num lock switched on). You will also find options to show larger regions of the Lambdoma, for instance the 16 by 16 Lambdoma, and then you will find that the left and right arrow keys are also useful for moving the keyboard playing area about.

Note that the PC keyboard isn't particularly designed as a musical instrument and may drop notes when you play some chords. It is suprisingly good and normally all single notes work fine, also most often, two note chords work fine but some three note chords may not work. The details depend on your hardware, it seems (probably the physical keyboard itself). Anyway if you can't get the chord you want by holding down the keys simultaneously on the keyboard, play its notes one at a time, and use the space bar to sustain notes to make chords.

So, play a chord in the Lambdoma, and you will see curves appear in Lissajous 3D which are based on small number ratios. The curves will close around back to the starting point, sooner if you play simpler ratios like 3/2, 5/4 etc, or after many twists and turns if you play more complex ratios.

You will find you get a drone for the Lambdoma view in FTS, because it is needed for music therapy work - it may be quite useful for making the chords too (one less note to sound) - anyway when you wish to, you can switch this off using the Drone dialog.

The idea of exploring this musical connection originates with Barbara Hero, creator of the Lambdoma keyboard , used in music therapy. Much thanks to her :-) - if it weren't for her I would never have written this program. She has been using actual recordings of chords to make her Lissajous curves in conjunction with a laser scanner,

What's really nice about this musical connection is that whenever you play the same chord on the Lambdoma, you will see the same Lissajous pattern - this is because we hear musical intervals in terms of the ratios between the frequencies of the pitches.

This means that you see the same pattern for the same chord whatever its pitch. For instance all minor chords 1/1 6/5 3/2 (e.g. C Eb G) have the same pattern, whether it is C minor or A minor or G minor or F# minor or whatever and whether it is near middle c or several octaves above or below. They all have exactly the same Lissajous pattern in 3D.

The ratio needed for a pure harmony can be read off from the harmonic series once you know how to do it. Starting from C, the harmonic series goes

C1   C2  G2  C3  E3  G3 (Ab3) C4  D4  E4
1    2   3   4   5   6   7    8   9   10

where I have numbered the notes according to the octave so c1 is in the first octave, c2 in the second and so on. Most notes here are fairly close to the pitches on a modern piano, but the Ab3 here is a fair amount flatter. The E is a bit flatter too, not so noticeably. The G is sharper, by a minute smidgen of pitch shift.

The pure version of the minor third such as say E3 to G3 (as it occurs for instance in the E minor chord E G A) corresponds to the interval between the fifth and sixth pitches of this harmonic series, so the pure minor third uses the ratio 6/5. The pure major third such as C3 to E3 corresponds to the interval between the fourth and fifth pitches, so it uses the ratio 5/4. You can also read off the fifth C2 to G2 as 3/2 and various other ratios - the G3 to Ab3 is that mysterious sounding septimal minor third 7/6 that many microtonalists are so fond of.

You will find that 6/5 has one pattern, the pure major third 5/4 (C3 to E3) has another, the pure fifth 3/2 (C2 to G2) has another, the septimal minor third between the sixth and seventh harmonics (G3 to Ab3, which uses 7/6) has another, and so on.

The resulting pattern does vary depending on the place in the cycle where you start each oscillation - the phases of the waves. The phase of a wave is something one doesn't hear readily in ordinary circumstances, so all the patterns you get as you vary the wave shifts correspond to the same chord. So there is something to be said for trying these out with the phase of the waves set to drift in Lissajous 3D (from Shape | Let the waves drift by ), as those patterns that the shape transforms to all correspond to the same chord.


Pure harmony and tempered music

Pure harmony (just intonation) music is traditionally most often encountered in music with a drone such as Indian music, because with a long sustained drones rich in harmonics, any departure from pure harmony will be much more noticeable. These typically don't have chord progressions or key changes - they depend instead on rhythmic and melodic subtleties..

"Western Music" is often played in tunings tempered to enable chord progressions to occur smoothly, and to allow transitions to remote keys to occur easily, with the harmonies not quite so pure. Originally it was tuned in systems with the semitones unequal in size. Some tunings had a distinction between two sizes of semitone only - the diatonic semitone was used for the narrow steps in the major scale, E to F or B to C in C major, and the chromatic semitone for the sharps and flats, such as C to C#. Other tunings had a great variety of semitone sizes.

The semitones were at their most uneven in the seventeenth century with the dawn of the use of Western harmonies based on thirds rather than fifths. From then onwards, the semitones have become more and more equal in size, perhaps because of the need to modulate more and more rapidly to new keys. At the beginning of the twentieth century tunings with slightly unequal semitones were still popular. The tuning system most often used now however is the twelve equal temperament which simply spreads all the notes out evenly so that each semitone in the twelve tone scale is the same size, and all keys are tuned identically. So we have reached the ultimate end point of this particular trend :-). Early music has a different harmonic basis, with emphasis on fifths rather than thirds, with thirds thought of as dissonances. This makes it possible to use a system (Pythagorean tuning) with much more even semitones but still with two sizes of semitones and a small distinction between the chromatic and diatonic semitone.

The so called "equal temperaments" used by Bach that you often see mentioned as tunings for the Well tempered clavier were actually what we nowadays call well temperaments and not equal temperaments in the modern sense. You can play equally well in any of the keys, however, some keys (C major for instance) are favoured with purer harmonies, and each key has its own colour. The tuning still has a fair amount of intentional variation in size from semitone to semitone. At least, certainly that is true of typical tunings that other musicians at the time would have played the pieces on. The WTC is a showcase which demonstrates the newly found ease of playing in any of the keys, something not possible in earlier tunings.

The idea that Bach himself might have experimented with playing the WTC in equal temperament gets discussed from time to time. He was apparently able to tune a piano very quickly so could change the tuning rather easily, but those who discuss this don't seem to have found any evidence to show that he did, only evidence to suggest that it is something he could have done if he had wanted to. So it is a matter open to endless debate.

Composers and musicians knew about the twelve equal tuning in those days - some people are under the impression that it is a new tuning of the twentieth and late nineteenth centuries but that isn't so - it was used early on for lutes . It doesn't require particularly advanced maths to place the frets for fretted instruments - you just need to be able to take the square root of the square root of a cube root to find the theoretical fret positions. Even the ancient Sumerians and Egyptians would have been able to do that, if they had thought of the idea.

It is particularly practical for fretted instruments as it means you can use the same pattern of frets for all the strings. Equal temperaments with various numbers of notes to an octave are particularly favoured by many microtonal guitarists today as well for the same reason. Partly this would have been for practical reasons, however another possible factor is that the beats are much less apparent on lute than they are on harpsichord with its rich timbre and bright third partials, or the long sustained notes of the church organ.

It would have been hard for a tuner at that time to tune a keyboard to equal temperament with any exactness, but in the absence of any other method, a tuner could perhaps have tuned a lute to equal temperament and then used the lute to help tune the keyboard to get a reasonable approximation to equal temperament. So the use of well temperaments for keyboards at the time, and in the case of the Organ, mean-tone temperaments too (highly uneven tunings that favour pure major chords) is surely a matter of musical taste rather than the capabilities of the tuners. Apparently they enjoyed the variety of colour you get from the variation in tuning from one key to another, and were quite happy with that, and had no wish to even it out.

Here is a nice introductory page about it on-line, introducing some of the variety of historical temperaments in more detail:

A Beginner's guide to Temperament (on-line) by Stephen Bucknell, Organ historian .

Tempered chords make Lissajous patterns that don't join up at all, but go on endlessly - obviously one can't draw such a pattern in its entirety, so in Lissajous 3D the ribbon will go on for a few cycles, and you set the number of cycles from the Ribbons window. When chords like this are listened to carefully on intruments with harmonic timbres then you hear patterns of beating intervals in the chords - a wah wah effect which indicates that the notes aren't in perfect just intonation harmony. For instance, play a fifth on a modern piano and you will hear slow beats if you listen carefully (well, an out of tune piano may have beats between the strings of a single note but this is if it is well tuned). Play a major chord and you will hear a complex pattern of superimposed beats which may sound more like a kind of roughness or raw quality, most noticeable if you are used to the mellow warm sound of pure major chords.


Listening to beats

Unfortunately, many of the instruments on your p.c.'s midi synth may be far from ideal to use to hear beats as they often have other beating type effects, which obscure the ones we are listening out for. For instance, sound cards nowadays often use actual samples of recorded sound for the notes, which are great for improved realism, but can cause beats by the looping of the sound clips used to generate the notes - here the noticeable thing is that you get them even with a single note played on its own, while beating normally only happens in chords. A single note may also beat with sounds made by the PC fan or any loud background hum in your room. Strong beating type effects can also be caused by instruments on your soundcard if they have a little residual vibrato in them. They are often recorded with some vibrato - this is done intentionally indeed and with great care to get the vibrato and the underlying wave to both join together smoothly simultaneously at the end of the loop. The result sounds "more natural", but unfortunately when such notes are played together in a chord, the vibratos for the various notes may not be synchronised. This can cause a jarring wah wah type effect from the way the different vibratos interact, which is extremely noticeable in just intonation chords. The noticeable feature this time is that you sometimes get this effect particularly strongly when you play several unison notes (all the same pitch) on the same midi instrument.

However with care you may find an instrument on your PC that works. Try and find a pure voice with little vibrato - but rich in harmonics. Experiment in FTS and see which sound best. The reed organ for instance may be a good choice. If you have a variety of sound devices, try them all as some may be better for this than others. You can install a soft synth to get yet more variety.

The best ones for hearing pure harmonies are ones that are full and rich sounding as they will be rich in harmonics. Strings are amongst the best in this respect but may well have much residual vibrato depending on your system.

Pure flute like voices may seem a good idea, but they are low in numbers of harmonics. The flute itself has a few, but not many. The ocarina voice is often the purest one you have, and sine waves are lowest of all, no harmonics at all apart from the fundamental.

Even if low in vibrato, as they often are, these pure tones aren't the best choice for learning to hear pure harmonies - because they have few or no overtones. They will sound fairly good together if you play any wide enough interval, not just at pure ratios. There is far less in the way of distinction between pure ratio intervals and tempered ones on pure sine wave like instruments.


Transforming tempered music to pure just intonation harmonies

It is a natural thought, since the pure ratio intervals sound most harmonious of all, to wonder if one can play music in which all chords are in this form. This does indeed work if you have limited harmonic movement, as you have in Indian music for instance. However, it isn't so easy to transfer it to other styles of music. Even if you keep to a five note scale (pentatonic scale), you can't keep the pitches of all the chords you might need in pure ratios with each other. You get most of the chords right maybe, but then you find that whatever you do, the last few chords just don't work out.

In the case of the pentatonic scale on C, you can get all the fifths in tune, ratio 3/2, but if you do then the major third C to E becomes a bit sharper at 81/64 instead of the pure ratio 5/4. It's not badly out of tune, in fact will sound fine to modern ears, but is sharper than the most melodious possible major third. So if you want all the fifths in a pentatonic scale to be pure, and want a pure major third as well, then you are asking for something that is impossible to do. When it comes to seven and twelve note scales then you get many and much larger discrepancies, and some chords will be pushed way out of tune if you make other chords pure. This results in the so called wolf chords which beat rapidly and loudly, an effect that most find unpleasant (though indeed sometimes composers will use even this for effect).

So some compromise is needed in fixed pitch music. One possibility is to restrict ones choice of chords to the pure harmonies possible with the given pitches, so you choose a nice scale first with many harmonies, then compose or improvise accordingly to fit the scale, as in Indian music indeed. The other approach used is to adjust the pitches to play approximations to pure harmony, a compromise which enables a wider range of chords to be played. This is what is meant by a temperament - a tuning tempered to play good approximations to pure harmony in many keys.

However, singers, string players, and many other instrumentalists are able to vary the pitch of a note depending on the harmonic context, and so sometimes they can achieve the best of both worlds. Pure ratio harmonies are possible for such performers even in music with much harmonic movement - it has been found that unaccompanied (a Capella) choirs for instance often gravitate to pure ratio harmonies in long sustained chords. This is known as adaptive tuning.

There's some interest in attempts to do this sort of adjustment automatically on a computer, and so make it available for keyboard players and computer generated music. Various approaches are tried. The most effective is probably John deLaubenfels' leisure time adaptive tuning. This sort of thing is best done after the event with an already recorded piece to retune, as then the program can look ahead any number of bars, and see where the harmony is heading towards, as is sometimes necessary to do. It can be done in real time, but isn't quite so effective in that case, with more compromises needed.

Another approach is to let the keyboard player adjust the tuning themselves manually as they play.This is explored experimentally in FTS with an option to let the player shift the tonic as they play (by playing a note in a key changing octave of the keyboard), though as of writing this, it has some limitations and isn't as flexible for the user as one might perhaps want and could achieve. At present one can chose the tonic but can't choose which note to use to dovetail each tuning to the next.


Pure harmonies with pitch shifts

FTS explores one particular approach to adaptive tuning, which is of interest here as it provides a way to get as many pure ratio chords as one can in the music, so good Lissajous patterns - though it has musical drawbacks. To try it out, just play some music in FTS, with FTS | Out | Retune to j.i. harmony (with diesis pitch shifts) selected.

One could call it strict just intonation adaptive tuning with pitch shifts. It makes as many of the chords just (pure ratios) as it can, but the drawback, sometimes a major one, is that it may cause some shifting up and down in pitch from time to time . These small pitch shifts can sometimes sound okay, even very beautiful indeed as small melodic steps, but on other occasions may sound strange, a kind of seasickness type lurch in the pitch of the melody, to unaccustomed ears.

That's particuarly so when a whole chord shifts in pitch from one note to the next. The chord is perfectly in tune with itself, but the pitch of the entire chord has lurched by a small amount from the previous chord - to nearly all it will sound strange, maybe "out of tune" until you listen carefully to the individual chords.

Even this can be used to good effect by a composer. Maybe you could build up to the pitch shift as a central feature of a piece of music, do the pitch shift very prominently so that listener can hear that it is intended as something to pay attention to, maybe repeat it to re-inforce the impression, maybe play it with sustained chords before and after so that you can hear that the chords are beautifully in tune with themselves, then maybe even develop it in some way, that sort of thing. But probably not something to over-use.

This is a matter of compromise, either you have perfectly tuned chords, and introduce the potential for occasional pitch shifts; or you temper the chords to remove the pitch shifts. You can also try for a middle road between those extremes - temper the chords just a little, and have either smaller less perceptible pitch shifts, or gradual sliding of the pitch, as John deLaubenfels has done. The FTS option currently plays them all as perfectly tuned chords (where possible), but with occasional pitch shifts.


Why pitch shifts occur if all harmonies are pure

The reason pitch shifts are inevitable with pure ratios is that for instance three major thirds such as c e g# c' will reach c' at (5/4)*(5/4)*(5/4) - because when you combine musical intervals you multiply ratios; here you mulitply by 5/4 for each of the major thirds. This reaches c' at the interval of 125/64 above the c instead of the expected octave 2/1 (or 128/64). That's quite a large discrepancy as these things go, from 125/64 to 128./64. It is smaller than a semitone indeed, but very noticeable for most musicians. So if you go up three pure major thirds then quite a large pitch shift is needed at some point to reach the expected octave.

Progression by minor thirds (6/5) in a pure ratio diminished seventh such as c eb f# a c' also introduces pitch shifts. This is an even larger and very noticeable shift. This is known as the (major) diesis - and the smaller shift that results from a progression by major thirds is known as the minor diesis .

Then there is a very commonly used chord progression of major and minor chords which shifts the pitch of the tune if you use pure harmonies for all the chords in the progression. One example of this progression is C major, E minor, A minor, D minor, G major, C major (I iii vi ii V I if you know the roman numerals notation). It goes up by a major third (5/4) to E to start the progression, then the roots of the chords move down by fifths (3/2) four times in succession. The result is a pitch shift of the final c downwards by the same amount that we found for the E in our pentatonic scale tuned to fifths (which was at 81/64 instead of 5/4).

This is a rather smaller shift, only 80/81. Small though it is, such a shift is easily noticeable to most musicians when you play the two notes one after another. Do this a few times, maybe in a repeating section of the piece, and the music will gradually spiral down in pitch until it is very noticeably flatter. This progression I iii vi ii V I is known as a "Comma pump" because the 81/80 interval that it shifts the music by is known as a comma .

The strong dovetailing options in the FTS Out | Just intonation Opts window work to reduce the number of pitch shifts from a note to the next note or chord in the piece. This is quite effective for some styles of music if they require few of these comma pumps or diesis shifts, and can work well over short sections of the music. However, since all the harmonies are pure, this increases the potential for a comma pump or diesis shifts over larger sections of the music.

The strong dovetailing can be quite effective even for complete pieces if they have limited harmonic movement and don't happen to have comma pump chord sequences in them. So for instance, I ii V I is a common chord progression that doesn't introduce a comma pump, so if you are playing say an early baroque piece that uses this sequence it could work quite well. The more extended sequence I vi ii V I could introduce a comma pump if the vi gets tuned to a pure 5/3 (a pure 4/3 fourth above the 5/4 and a minor third 6/5 below the octave) as it normally would.