12 Tone Compostion

[Kevin]

Okay so most of us here are tonal writers which is cool, but does anyone have experience in 12 tone writing and would like to share their experience with the members?

[Kevin]

Still hoping for some instruction on 12 tone.Any takers?

[steve]

I've never tried it and wouldn't mind an experienced hand in going thru the process either. Someone out there will respond to the call:)

[Kevin]

I managed to find this Steve, any use?


jan.ucc.nau.edu/~krr2/12tone/12tone1.html

[steve]

I can see what you mean when you call it mathematical - on paper it's ugly to look at! Of course, I suppose many think it's just as ugly to listen to! Thanks for this though - I'll be trying my hand at it one of these days. Maybe soon...

[Kevin]

Is it any wonder music is so technical when it is analized to the nth degree!


Twelve-Tone Musical Scale
Why does our musical scale have twelve notes (counting both the white and black keys on the piano)? Why not ten or fifteen or twenty?


To answer this question, we first need some background information. A note's pitch or frequency is measured in cycles per second; for example, A' is 440 cycles per second. The distance between two notes, measured as the ratio of their pitches, is called an interval. If the interval between two notes is a ratio of small integers, such as 2/1, 3/2, or 4/3, they sound good together — they are consonant rather than dissonant. People prefer musical scales that have many consonant intervals.

There is no absolutely definitive list of consonant intervals because the concept of consonance involves subjective aesthetic judgement. However, the following seven pure intervals, smaller than or equal to an octave (2/1) and larger than unison (1/1), are commonly considered to be consonant.

Basic Consonant Intervals

2/1 octave harmonic inverse of 1/1
3/2 perfect fifth harmonic inverse of 4/3
4/3 perfect fourth harmonic inverse of 3/2
5/3 major sixth harmonic inverse of 6/5
5/4 major third harmonic inverse of 8/5
6/5 minor third harmonic inverse of 5/3
8/5 minor sixth harmonic inverse of 5/4

This list can be constructed mathematically by listing the ratios of the smallest integers and including their harmonic inverses (defined below). First, list the ratios of the integers from 1 to 5, where the ratios are between 1 (unison) and 2 (octave): 1/1, 2/1, 3/2, 4/3, 5/3, and 5/4. Then, include their harmonic inverses [shown in brackets]: 1/1 [2/1], 2/1 [1/1], 3/2 [4/3], 4/3 [3/2], 5/3 [6/5], and 5/4 [8/5]. Remove the duplicates. We can ignore the trivial unison interval. This leaves: 2/1, 3/2, 4/3, 5/3, 5/4, 6/5, and 8/5. If you start with the integers from 1 to 3 or 1 to 4, the result is the top three intervals: 2/1, 3/2, and 4/3. If you start with the integers from 1 to 5 or 1 to 6, the result is this list of seven intervals.

Harmonic inverses: Two intervals are harmonic inverses of each other if they combine to make an octave, in other words, if the ratios multiplied together equals two — for example, 3/2 x 4/3 = 2. Harmonic inverses appear spontaneously when you construct a new musical scale. Imagine making a musical instrument with three strings. Start with two strings making an octave, a low string and a high string with half the length and twice the pitch. Now, add a string somewhere in the middle, for example, 2/3 the length and 3/2 the pitch of the low string. Playing the low and middle strings together makes a 3/2 interval (perfect fifth), and playing the middle and high strings together makes another interval, the harmonic inverse of 3/2, which is 2/(3/2) = 4/3 (perfect fourth). Each time you add a string between the low and high strings (the octave), you always get two intervals that are harmonic inverses.

In the past, people constructed scales based on pure or natural ratios of small integers. For example, the just intonation system uses the exact ratios shown in the table below. However, this method runs into serious problems. Although some of the intervals are perfect, other combinations of notes sound very bad. After the Middle Ages in Europe, music became more complex, with more key changes, and these bad intervals became more common.

The modern equal temperament system was invented (in the 1500s) to solve this problem. The octave is divided into twelve exactly equal intervals. In this system, the smallest interval, the semitone, is not a simple integer ratio, but is the twelfth root of two (21/12) or approximately 1.059. Larger intervals are multiples of the twelfth root of two, as shown in the table below. Although no interval (except the octave) is perfect in this system, the error is "spread around" evenly so there are no very bad intervals.

The table below compares just intonation with equal temperament. The intervals in both systems are never exactly the same (except the octave), but they are very close — always within about one percent or better. For example, the fifth (3/2), obtained by multiplying the twelfth root of two by itself seven times, is 1.498 — very nearly a perfect 1.500. The fourth (4/3), obtained by multiplying the twelfth root of two by itself five times, is 1.335 — very nearly a perfect 1.333.

Number of
Semitones Interval
Name Notes Consonant? Just
Intonation* Equal
Temperament Difference
0 unison C-C Yes 1/1=1.000 20/12=1.000 0.0%
1 semitone C-C# No 16/15=1.067 21/12=1.059 0.7%
2 whole tone C-D No 9/8=1.125 22/12=1.122 0.2%
3 minor third C-Eb Yes 6/5=1.200 23/12=1.189 0.9%
4 major third C-E Yes 5/4=1.250 24/12=1.260 0.8%
5 perfect fourth C-F Yes 4/3=1.333 25/12=1.335 0.1%
6 tritone C-F# No 7/5=1.400 26/12=1.414 1.0%
7 perfect fifth C-G Yes 3/2=1.500 27/12=1.498 0.1%
8 minor sixth C-Ab Yes 8/5=1.600 28/12=1.587 0.8%
9 major sixth C-A Yes 5/3=1.667 29/12=1.682 0.9%
10 minor seventh C-Bb No 9/5=1.800 210/12=1.782 1.0%
11 major seventh C-B No 15/8=1.875 211/12=1.888 0.7%
12 octave C-C' Yes 2/1=2.000 212/12=2.000 0.0%

* This table shows one variation of just intonation.

So, back to the original question: Why does our scale have twelve notes? We have explained that an equal-tempered scale works better in practice than a scale based on pure intervals, but we have not yet explained why we prefer the twelve-tone equal-tempered scale. Why do we not use a ten-tone or twenty-tone equal-tempered scale? Is there something special about twelve?

The answer is: Yes, the twelve-tone equal-tempered scale is remarkable. The nearly perfect intervals seen in the table above are not typical of other equal-tempered scales. Consider the seven basic consonant intervals (described above): 2/1, 3/2, 4/3, 5/3, 5/4, 6/5, and 8/5. We observe:

The twelve-tone equal-tempered scale is the smallest equal-tempered scale that contains all seven of the basic consonant intervals to a good approximation — within one percent.

Furthermore, for the most important intervals, the fifth (3/2) and fourth (4/3), the approximations are better — within about one tenth of one percent.

Let's compare the twelve-tone equal-tempered scale to some other scales.

All equal-tempered scales with 14 notes or fewer, except the twelve-tone equal-tempered scale, contain at most only three of the seven basic intervals (including the octave) within one percent.
Several equal-tempered scales with between 15 and 30 notes (notably the 19-tone and 24-tone scales) contain all seven basic intervals, but in none of these scales are the intervals more nearly pure than in the twelve-tone equal-tempered scale.
The 31-tone equal-tempered scale has all seven basic intervals to a good approximation, some with better accuracy than the twelve-tone scale, but the most important fifth (3/2) interval is less accurate than in the twelve-tone scale (218/31=1.495). Some Indonesian music actually uses a 31-tone equal-tempered scale.
The 41-tone equal-tempered scale is the first with a better fifth (3/2) interval than the twelve-tone scale (224/41=1.5004).
The 53-tone equal-tempered scale has all seven basic intervals with a better accuracy than the twelve-tone scale (the fifth is 231/53=1.49994).
But bigger is not necessarily better. Although scales with many tones have many nearly pure intervals that are consonant (ratios of small integers), they have even more intervals that are dissonant (not ratios of small integers). In contrast, the small twelve-tone equal-tempered scale has more consonant intervals (seven) than dissonant intervals (five). We observe:

The twelve-tone equal-tempered scale is the only equal-tempered scale that contains all seven of the basic consonant intervals to a good approximation — within one percent — and contains more consonant intervals than dissonant intevals.

Also, scales with many tones are too large to be really practical: a keyboard with the same range as a piano would be huge.

In summary, the twelve-tone equal-tempered scale is probably the best compromise of all possible scales, and that is why it is now standard almost everywhere.

Note: These mathematical results are fairly robust. When I perform the same analysis with small variations of the discretionary inputs, the twelve-tone scale still looks remarkable. For example, if I add or remove a few intervals near the end of my list of consonant intervals, the results are similar. Also, if I increase the matching tolerance from 1% to 2% or reduce it to 0.8%, the results are similar. You can try your own variations with the spreadsheet below.


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The Equal Temperament Musical Scales Worksheet (MS Excel spreadsheet or PDF document) shows all the ET scales (up to 100 tones) and shows how well they match the "ideal" intervals. If you don't agree with my ideal intervals, the spreadsheet allows you to enter your own ideal intervals. If you don't agree with my scoring, you can change the score function, if you know basic programming.




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Last updated 2009

Keith Enevoldsen's Think Zone

[Deleted]

Hi, Some time ago I started my 22nd prelude in a dodecaphonic style. I wanted it to be strictly dodecaphonic (the 12 tone row appears again and again in the same order) without losing musicality.
I have finished it and it's now on scoreexchange: Take a look if you want:
www.scoreexchange.com/scores/112081
I write mostly tonal, but I am also attracted to dodecaphonic music (especially in its pure form).
Shoot if you want ... or take it as a good example.

[Deleted]

And to talk a little about the experience: it's very fun to write in that style. And my trick is: don't begin with your tone-row, but with a sort of improvisation/melody (knowing ofcourse that you can't use a note but one time), and then "finish" that phrase with the notes that you didn't use. You have now a tone row AND the start of your piece. Beginning with the tone row does not work with me.

[Deleted]

Good advice! If a row doesn't make sense as a melody line, or if there is no memorable pattern in it, composing something with the material from it will be harder, and take much more energy and concentration away from the soul and brain of the composer.

[Kevin]

Then can you tell me why when you hear a lot of 12 tone music it always appears to be random and definitely not tonal sounding. For example:

Berio

[Deleted]

Hi Kevin, in my opinion this is not an example of 12-tone music, but of experimental music. Not all 12-tone music is like this. On YouTube there is f.i. the Symphonie, Op; 21 of Webern that one can follow with the score, here is the link:
I like this piece very much. It's not really "at random" but there is a system behind it and at the same time it is (certainly in the first movement) very lyrical, dramatic and colorfull.

[Kevin]

Correct me if I am wrong but it seems like the opening is just the 12 tone row spread out over a set of instruments?

[Deleted]

Yes Kevin. And do you like it ?

[Kevin]

I just wonder what I am supposed to be listening to with this kind of music. It tends to have little or no harmony.

[steve]

This is a funny statement to me because I have never thought to think what should I be listening to. There is to me only listening. It sounds as though you attempt to analyse while listening which for me destroys the effect of listening to the music. It's somewhat akin to a doctor looking at his family members and seeing collections of organs and tissue...

[steve]

Let's have a competition to create a 12 tone composition where the winning piece will be the most enjoyable to listen to, shall we? Of course this will be a purely subjective thing but it will be fun and educational all at the same time. I don't know what a suitable prize would be, but I think it would be fun nonetheless.

[Kevin]

That sounds like an interesting idea Steve. Maybe someone would like to come up with some guidelines?

[Deleted]

Hi All - interesting reading.

I've been keen on 12 tone composition for a long time now but have never really approached it from an 'academic' point of view. Back in the early 90's (yes, I'm that old) I wrote a 12-tone composition for an 11 piece jazz ensemble called Boris.

I'll hunt out the score and post it later. I have a dodgy old recording of it too, so will upload that to my soundclick player (5th one down on the left of SOS site) or on my web site at: www.doriankelly.com

[Kevin]

I am sure that we shall all look forward to looking/listening to it Dorian! Tell us more about Boris.

[Deleted]

Hi - here's a pdf of the score for Boris.

Will upload the studio recording to my soundclick player a little later.

Any feedback would be good as I'm about to start a new 3 movement 12 tone piece for Wind Band in the not to distant future!

Regards - Dorian Kelly

Attachments:

[Kevin]

I shall look forward to the mp3 for Boris!

12 tone piece for wind band..............wow! Obviously you are up for the challenge then?!

[ttw]

This sounds interesting. I'll probably skip it because I have no idea how 12-tone composition works.

[Kevin]

Funny thing is ttw, neither do I! I am hoping for some guidance!

[Deleted]

Hi - the recording of 'Boris' is on my Soundclick player now; please forgive the quality of the recording as it was lifted from a cassette tape (any one out there remember those?).

Unlike 'geert' & 'maria' I did start Boris by working out the row first. I then assigned a chord to each degree of the row. Each chord stays with it's note of the tone row so when you start manipulating the row, the chords change also. There's also quite a bit of improvisation over the chord sequence created by those chords.

Anyway ... enjoy; I hope! Dorian

[Kevin]

So just how do I work out the row and then the harmonisation.?

[Deleted]

Well, as geert said, I think it's important that the row has some melodic quality and can stand on its own if required.

I personally approach the harmonisation of a row the same as I would a conventional composition in that the sequence has to flow and needs to reflect the overall mood / feel of the piece. The only difference I s'spose is that you might be using more complex harmony. Pedals work well too. Which is why the 12 tone approach can sit well in the modern jazz style.

Regards - Dorian

[barrystronge]

I have never understood why a tone row must be dodecaphonic - using all the semitones. Why cannot a tone row consist of less than eleven different notes?

[Kevin]

Mmm? Do you have to use all 12 notes then?

[Deleted]

I suppose serial purists might say that you have to use all 12 notes as it imposes certain parameters but I'm a firm believer in freedom of expression!

However, if you reduce the number of 'tones' in your row too much you're in danger of just creating a melody or motif.

Regards - Dorian

[Kevin]

Is there an easy way to create a matrix?

Just listened to Boris! Great sounds. It reminds me of the Peter Herbolzheimer Rhythm Combination and Brass from the 70's.