Microtonality with its many equal tempered and non-equal tempered tone systems is a fascinating branch in music that seems to be rather neglected by music education. Especially it might be hard for not specialized interpreters and they may not know where to start. Before dealing with tone systems of high complexity it seems appropriate to start with easy exercises to learn at least some new pitches. For this reason I created a method - a kind of guideline - for (young) players who want to „switch“ to playing microtonal music which I called Ludus microtonalis - in allusion to Hindemith ́s piano circle Ludus tonalis (1942). As I am a composer and guitarist this guideline should primarily appeal to guitarists, but most of the ideas can easily be transferred to string instruments and even to wind - and brass instruments. It can also be used as a key to my microtonal easy-to-play-compositions which will be dealt with later. Like in Bartók ́s Mikrokosmos - which I arranged for guitar solo (UE 38060) - the methodology is not limited to interpretation on the instrument but also extends into the field of composition.

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1. Some necessary basics

Who plays a chromatic scale in the range of an octave on his instrument, divides the octave in 12 equal parts or semi tone steps. With this material of twelve notes can be played most of the classic and romantic and even more recent repertoire. And though it is permissible to ask this question, whether this chromatic 12-tone-system exhausts all musical possibilities. For sure it does not! In theory, there are tones of infinite number in between every halftone step. These notes are called microtones and the quality of music using microtones - in whatever combination - is called microtonality.

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Fig. 1

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These are notes we have on our instruments, but how can we produce the pitches in between? How can we achieve microtones?

The simplest way to achieve microtones is to use a tuner with a cent-display. Microtones are measured in cents. A halftone step consists of 100 cents, the entire octave of 1200 cents. If you have to tune (e. g.) the G string of your instrument 33 cents higher (or lower) you firstly have to

tune the string to the exact pitch of g and then tune even higher (or lower) according to the cent- display on your tuner. The new pitch is a sixth-of-a-tone higher (or lower) than g.

We use this procedure whenever we have to make a scordatura, which means a tuning different to the normal tuning. Scordatura is most common to the fretted guitar but also possible for any other stringed instrument.

On the fretless string instruments (like the violin, viola, violoncello or double bass) there is a second possibility to produce microtones, that is to play the exact (microtonal) pitches between the chromatic halftone steps. It takes some practice to hit the right pitch that is only reached when the first quarter tone of a semitone is equal to the second quarter tone. This, of course, cannot be done without hearing exercise, much more it requires a good deal of ear training. For producing other microtones - even smaller steps - like sixth-of-a tones or partials from the overtone series (which many might know from the natural harmonics) the training of the musical ear will gain even greater importance.

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2. Quarter tone scale

As the quarter tone scale may be the easiest system in the wide range of microtonality we want to start with this one. The quarter tone system divides the half tone step in two. This means the octave consists of 24 equal quarter tone intervals. Playing a series of consecutive quarter tones is called quarter-tone chromatics.

 

Fig. 2

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It is recommended for string instruments to practice quarter-tone chromatics without scordatura on one string upwards and downwards. The quarter tones must be intoned exactly by ear - right in the middle of e and f, f and f sharp, f sharp and g, and so on.
For the guitar, on the other hand, retuning single strings is an effective means of achieving exact micro-intervals. „Bending“ is particularly widespread in light music and means the increase in pitch by pulling the string. Although effective it is rather inexpedient as a means of

generating exact pitches.

On the concert harp, which has its own string for each diatonic step, microtonality is generated with the help of scordatura anyway. Of course, the harp also allows the manual changing of strings by a quarter tone (or any other micro-interval), but quarter-tone chromatics is not possible on the harp due to its diatonic structure.

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3. The creation of quarter tones using scordatura (guitar, string instruments, harp)

The easiest way to produce quarter tones on the guitar is to tune up one of the middle strings 50 cents higher. As for the guitar the high and low E-strings should remain unchanged for the time being so as not to lose the sense of the basic key. The tuning is done with a centable tuner or an equivalent app. For the following examples we will raise or lower the D string:

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a) D string a quarter tone higher

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To check it out you have to play:

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5A: the 5th fret on the A string is the pure D
0D: now is the quarter tone between D and D sharp 6A: is D sharp

0D has to be right in the middle of the 5th and 6th fret of the A string.

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b) D-string a quarter tone lower

To check it out you have to play:
4A - 0D -5A - 0D - 4A

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As a „rule of thumb“ for the hearing test applies:

If a string is tuned a quarter tone higher, the test must be carried out one string lower at the 5th and 6th fret (on the B string: 4th and 5th fret); if the string is tuned a quarter tone lower, the test must be executed one fret lower.

To achieve quarter-tone chromatics on the guitar, it is advisable to tune the G string a quarter- tone higher. This in turn happens with a tuner with a cent display or an app. The quarter tone interval can also be tuned by ear. The g on the D string (5th fret), the open G string to be tuned and g sharp on the D string (6th fret) are played alternately. The right tuning is only achieved when both quarter-tone steps are equal.

Of course, this test can also be carried out on a string instrument. To give an example: if we need to tune up the A string of a violin a quarter-tone higher we shall compare it to a and a sharp (= b flat) on the D string. The quarter-tone has to be right in the middle of a and b on D. We play these three notes up and down for a few times and listen exactly to the quarter-tone intervals.

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Figure 4 shows quarter-tone chromatics on G and B strings up- and downwards on the guitar.

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Even if the quarter tones are produced with a digital aid, an audio test should often be carried out on the respective string instrument. For what reason? On the one hand, because string instruments could get out of tune and on the other hand our hearing will be more and more sensitized to microtones. This will help later to distinguish even smaller micro-intervals. Quarter- tone chromatics can also be created on other adjacent strings. On the guitar, however, the fingerings become larger since the other pairs of strings are tuned in fourths and not, like g and b, with a major third apart.

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4. How to create sixth-of-a-tones on stringed instruments

My composition Driving Rondo for violoncello solo was especially dedicated to youth competitions. In this piece I included passages of quarter-tones and sixth-of-a-tones, alternately. This sounds more complicated than it is and I can offer an easy way to succeed, namely to practice quarter-tone scales and sixth-of-a-tone chromatics, respectively. Both microtonal scales have to be executed by accurately listening if all the microtonal steps are equal. In the beginning it is recommendable to compare the pitches by using a tuning device.

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Extract from Driving Rondo for violoncello solo

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As stated before the quarter-tone system (24-equally tempered-division of the octave) is the easiest way to start dealing with microtones. However, using a sixth-of-a-tone-system of 36 equal pitches is more pleasable in respect of its sound possibilities.
If we want to tune a string 33 cents higher or lower than the common pitch we use again our tuner to get the correct pitch. To check it out, we can proceed in the same way as we did by producing quarter-tones: we tune (e.g.) the G string a sixth-of-a-tone higher and compare the note with g on the 5th and 6th fret of D.

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Example for a scordatura with sixth-of-a-tones:

In my composition with pedagogical approach Smaller Steps for guitar solo I used a combination of 3 microtonally-tuned strings, each one in intervals of a sixth-of-a-tone. The exact tuning is as follows:

 

E6, A5, D4 = normal tuning

g = g + 1/6-tone (33 cent)

b = b - 1/6-tone
e = d sharp + 1/6-tone

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This means: to achieve the correct pitch of the higher e-string, we have to tune it down to d sharp and then tune a 33 cents higher. B and G strings are tuned in the same way we did before.

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5. Notation of microtones

Similar to my cello piece Driving Rondo I also used a combination of quarter-tones and sixth-of- a-tones in Jeux de langage for two guitars. The main difference here is the use of a scordatura in this guitar duo and
the way of notation of the microtones. As shown in Fig. 2 and 4 I use the common accidentals plus arrow up- or downwards as the only marks for microtonal notes. In Smaller Steps and Driving Rondo these were used throughout the score, whereas in Jeux de language I only indicated the scordatura and renounced the microtonal signs.

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Fig. 6

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To say it right away, it is hard to obtain all 36 pitches of this 1/6-tone system on one guitar. The same applies to playing a complete quarter-tone scale, as we often have to jump from position to position, like 5th - 1st - 6th - 2nd - 7th - 3rd - 8th - and so on.

It should not be concealed that there is an easier way to obtain microtones on the guitar: this is by using two guitars. Guitar 1 may be tuned like normal and guitar 2 a quarter-tone higher or lower. The result is a perfect framework for playing/composing in the quarter-tone system. By using three guitars - each one tuned a 33 cent higher than the other - we accomplish a perfect sixth-of-a-tone system with all its possible 36 pitches. To realize a tone-system of 72 equally tempered steps we would need 5 guitars, respectively.

The disadvantage of this kind of tuning arrangement is obvious: only the result is microtonal whereas each player has to play in the normal tuning without using microtones. It does not matter if the pitch is a 33 or 66 cents higher or not. By studying his part the musician cannot learn anything about microtonality nor can he approve his aural abilities.

In conclusion, quarter-tone and sixth-of-a-tone systems are best for starting to play microtonal music, as it does not take too much imagination to divide halftone steps in two or three equal parts. Dealing with 48 or 72 steps per octave should belong to every musician ́s daily exercises but might be hard for unexperienced players. -

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All microtonal compositions for youth competitions quoted in this article are published with Universal Edition, Vienna:

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- Driving Rondo for violoncello solo (UES100388-141)

- Jeux de langage for two guitars (UES100298-000)
- Smaller Steps for guitar solo (UES100855-711)

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