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Scales, arpeggios, and intonation

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Scales, arpeggios, and intonation


The modern tuning used for pianos is an interesting one with a rich sound world. However there are many other tunings such as 19 tone, 31 tone etc, the wealth of just temperament scales, and well tempered scales, which all have their own characteristics.

In early middle ages there was much invention and exploration of various ways of tuning scales. The twelve tone scale gradually evolved from early scales of seven and eight notes. Along the way, musicians also often explored scales of more than twelve notes. See Margo Schulter's article for the Tuning FAQ.: Twelve notes as _one_ attractive arrangement.

The twelve equal system ruled supreme in Western music in the twentieth century - it lets one modulate extremely easily as all the keys are tuned identically, and this is what many Western composers want to be able to do. It means they can go suddenly to the remotest keys, or indeed, drop notions of a tonal centre altogether. It has sharp bright major thirds. Some hardy pioneers continued to explore other tunings.

Now however, there is a renewal of interest in other tunings in Western music. Some of this is inspired by the music of other great traditions such as Indian music, Gamelan music (Indonesia / Java), African music, Japanese traditional music, .... To take an example, Indian musicians play ragas with a drone, and use melodic invention, so a just intonation twelve tone scale is far more suitable than twelve equal for Indian music (though twelve equal is sometimes used for it nowadays).

Another source is from the authentic performance movement, and the rediscovery of the "key colours" of nineteenth century and earlier music - it is only in the twentieth century that Western music became identically tuned in all the scales. Before then, you had a great deal of variation as you moved to the more remote scales, so authentic insrument performers will retune their harpsichords etc. accordingly.

Also, composers are inspired by the exciting possibilities of entirely new sound worlds, and the tuning systems to explore, including newly invented ones.

A lot of music is inspired by natural sounds. Listen to bird song and you will find that they don't sing in twelve tone equal temperament! [#transcribe_robin_intervals Bird song as Tune Smithy files ].

Try listening to some of the music on John de Laubenfel's basement project to hear how music can be transformed by using pure ratios for the intervals - gives wonderfully pure and rich sonorities. He re-tunes music performances (on MIDI) in adaptive just intonation!

Amazingly refreshing to hear a piano played like that, no longer tied to fixed tunings for all the notes.

For a history of tuning systems in European music see Kyle Gann's An introduction to historical tunings .

Scales and arpeggios

If you are a newbie to the idea of musical pitch intervals, you may like to read [Scales_and_Fractal_Tunes.htm#Some_basic_concepts Absolute beginners] at this point.

Some scales, like the Japanese Koto scale for instance, can just be played as they are, and sound nice.

However other scales are not really meant to be used like that; the idea is that you only use a particular selection of pitch intervals from it at any one time, then you can vary the selection for variety during the piece. Sometimes the selection of notes is used only for a few notes, after which one changes to another selection, in which case it is usually called a chord, or arpeggio.

Other selections are used for extended passages, or sometimes the entire piece, and these are the Arpeggio in FTS. They are usually called modes in tuning circles. However, it risks some confusion, as another use of the word mode is to mean a particular sequence of whole and half tone steps, rather than as a selection of notes from a larger scale - this is how it is often understood in modal music and Scottish and irish folk music etc. See the FAQ [faq_scales.htm#why_call_arp Why call them Arpeggios? Mode sounds much better.]

So an Arpeggio is a selection of notes from some other scale. It is a relative term - having selected an arpeggio from a larger scale to play a piece, you can then refer to that arpeggio as a scale in its own right, and select smaller selections of notes from it as smaller arpeggios. For instance, a major scale can be a mode of a twelve tone scale which itself can then be a mode of, say, 31 tone equal temperament or some other larger tuning system..Then within the major scale indeed, you might choose say, one of its major chord triads as an arpeggio. Chords and arpeggios get used for short passages in common practice (ie.. modern western) music - maybe an entire bar will be in a particular chord, or half a bar, or in cases of more rapid chord progressions, every note in the bar may need a new chord to play it. Occasionally a chord may get extended over several bars too. But as far as we are considering it here, all of these get called Arpeggios.

One thing that can confuse a beginner is that the scale of all the black and white notes is known as a twelve tone scale, when it is actually made out of twelve semitones, or six whole tones. The reason is, the word tone is used in various ways, as a general term for a subdivision of the octave into any number of equal parts, or as a specific term meaning one of the larger steps of the major scale. So for instance, one can also talk about thirty one tone scales, nineteen tone scales, and so forth, and the tones for most of these scales are actually smaller than semitones. To make clear that one intends the other meaning of the word, one can say " whole tone " for a large step of the major scale, rather than, say, one of the steps of the thirty one tone scale. You can then say that the whole tone of thirty one equal consists of so many tones, and it's semitone or half tone consists of so many tones, and so forth. This nomenclature is so prevalent, one just has to get used to it; sorry if it is confusing. That is just how it is, and one soon gets used to it, so it isn't really a handicap at all. There's more about this on the Musical Intervals page: [Scales_and_Fractal_Tunes.htm#Tones_semitones_and_n-et Tones, semitones and n-et]

All the major scales follow the same pattern of large and small steps: W W S W W W S where W is the whole tone, and S is the semitonel. Also always the whole tone is larger than the semitone. In twelve equal it is exactly two semitones, and the pattern is 2 2 1 2 2 2 1 with the numbers showing numbers of semitones (same as the tones in twelve tone scales).

In other scales, its size depends on the tuning used, and also may vary depending on the position in the scale, for instance, just intonatoin major scales have two sizes of whole tone, the 9/8 and the smaller 10/9.

The nineteen tone scale has 19 equally spaced notes to an octave, and the major scale in this system has steps 3 3 2 3 3 3 2. This is the same pattern of large and small intervals as before, but the steps are more similar in size than in the twelve tone system. The semitone for this tuning is two thirds of the size of the whole tone, which makes it rather on the large side. The seventeen equal scale has the smallest of all the equal temperament semitones.

A few of the scales in the drop list

The 12 tone system has all its semitones completely identical. It is called 12 tone Equal Temperament. This is the modern piano tuning that Western musicians are familiar with - much of modern music uses this as its only tuning, often with inflections of pitch for instruments that are capable of it (voice, winds, strings etc).

Other 12 tone systems, such as the historically accurate ones for Mozart's time, Bach's time, or the mean-tone system of Medieval times, have semitones that vary in size, so that some of them are larger than others.

To look at the step sizes, select Show steps - unselect to show the intervals from the note that starts the scale - the 1/1.

Notice how the semitones are unequal for most of the twelve tone scales. Some may be shown in cents - a cent is a hundredth of an equal temperament semitone. If you want to show them all as cents, go to File | scale notation and select Show cents .

The idea of the nineteen tone scale is that you split all the black keys, e.g. C# and Db as distinct notes. Also you add E# = Fb as an extra note between E and F, and B# = Cb between B and C. The nineteen notes are c, c sharp, d flat, d, d sharp, e flat, e, e sharp (same as f flat), f, f sharp, g flat, g, g sharp, a flat, a, a sharp, b flat, b, and b sharp (same as c flat). Now make all the steps the same size.

If you now make all these steps the same size, you find that you have nineteen equal sized notes. The whole tone is three steps in this scale, and the diatonic semitone is two steps.

This scale is favoured because it has some nice pure intervals. It gives better major and minor thirds than the equal temperament twelve tone system. The minor third is the interval c to e flat, and it has another minor third as the interval c to d#. This is a very nice septimal minor third (i.e. bluesy minor third) - using the seventh harmonic. The fifth of the nineteen tone scale is a little flatter than for the twelve tone scale (7 cents flat instead of 2 cents flat). It has wonderfully subtle shades of harmony, when one comes to it from the twelve tone system. For instance, the contrast between the septimal minor chord and the ordinary minor chord is a striking one.

Keyboards have been made especially for playing nineteen tone scales. The front part of each key plays the most frequently used notes. See on-line article Everyone in Tune, whatever that means , by Anne Johnson (NYT, March 14th, 1999).

The idea of the thirty one tone scale is that it has two shades of sharp or flat for every note. So it begins: c, c half sharp, c sharp, d flat, d half flat, d, d half sharp, d sharp, e flat, e half flat, e, e half sharp (same as f flat), e sharp (same as f half flat), f, .... The semitone is three notes, and the whole tone is five notes.

Add the numbers up and you will find that you have thirty one notes. It has a major third that is very close to the pure ratio - within a single cent, and good minor and septimal minor thirds. It's fifth is better than for the nineteen tone equal temperament one, but still not as pure as the twelve tone ET one (it is flat by a bit over 5 cents). One of it's modes is very close to the medieval quarter comma mean-tone, which is a twelve tone scale designed to have pure major thirds. To find this mode, select Meantone Chromatic (53/220-comma) from the Modes drop list for the 31 tone scale. Compare with the Quarter comma Mean-tone scale from the Scales drop list..

Keyboards have also been made for the thirty one tone scale, though they are rather unusual. Example of a thirty one tone keyboard (Huygens-Fokker foundation).

The 17 tone system has accidentals split into sharps and flats, but with no extra black notes between the B and C, or E and F. It's fifth is slightly sharp, by about 4 cents, and it has a very sharp (and so bright) major third, and tiny, rather elegant semitones of only a third of a whole tone.

It has a septimal minor third (slightly sharp), no conventional minor third, and a "neutral third" close to 11/9 - in this context, "neutral" means halfway between the minor and major version of the interval, so anything around about 350 cents is a neutral third.

17-tones give you the smallest diatonic semitone in any of these equal tone systems, as the next one, 19 tone equal temperament (or 19-et for short) has two steps to the semitons.

If you want the purest possible fifth in an equal temperament scale, the next really good one after the twelve tone equal temperament is the 41 tone ET one (within half a cent of the pure ratio). You can find this one in the SCALA archives. You can also add all the modes lists from the SCALA archive to FTS, including the 41 tone scale and its lists of modes. All you have to do is to download a file, then browse for it in FTS, and click on a button - to find out how to do this, see [More_scales.htm#SCALA_scales SCALA scales] .

This web site has a good selection of links on Mathematics and music - things like, why does the most prevalent scale we use have twelve notes. For that particular question (from a historical perspective), you can also read Margo Schulter's article in the Tuning FAQ.

See also David Loberg Code's The Groven Piano Project FAQ, and The Piano's Ivory Cage.

You can try playing in the 19 and even the 31 tone scale with the extra sharps, or half sharps etc, from the p.c. keyboard. To do this use Make New Seed | +... | Pitches for keyboard rows | Second and fourth rows arpeggio, others accidentals (p.a. only) or Second row arpeggio, others accidentals (p.a. only) . For details see [User_guide.htm#pc_keyboard_accidentals p.c keyboard accidentals] , under the [User_guide.htm#Pc_keyboard_voice_window P.c. keyboard voice window] .

You can also play in them from the Midi keyboard. For details of how to do this in FTS, see [midi_in.htm#Playing_accidentals Playing fine shades of accidentals from the music keyboard ].

The Javanese Slendro scale has five approximately equally spaced notes to an octave, and the Thai modes are for a scale with seven approximately equally spaced notes to an octave.

For a little more on these with some links, see [Scales_and_Fractal_Tunes.htm|C:\Rob\_fts_source\help\Scales_and_Fractal_Tunes.htm#equal_tone Almost equal tone systems].

The modern Bohlen-Pierce scale repeats at a note between the first and second octaves (at an octave plus a fifth). It has thirteen notes, and there is a version with the 13 notes equally spaced.

A twelve tone scale that uses pure ratios is called a just temperament scale. For details of the construction of one of the frequently used just temperament twelve tone scales, see [Scales_and_Fractal_Tunes.htm#Harmonics Harmonics and just temperament] .The Pythagorean scale is based on the idea of using pure major fifths as much as possible. There will always be one of them less than pure - see [Scales_and_Fractal_Tunes.htm#The_circle_of_fifths The_circle_of_fifths] .

The Arabic Pythagorean scale has 17 notes, and uses pure fifths as much as possible too.

The Indian Shruti scale has 22 notes, all pure ratios. These are either small ratios, or related to small ratios by sequences of pure fifths. The Indian ragas use notes from the Indian Shruti scale. The 22 note scale is a synthesis of these - there is some discussion about which ratios to use for some of the notes.

Modern Indian music uses notes from the Modern Indian gamut - a twelve tone scale which you can find in this drop list towards the end. If you want to retune your keyboard for modern Indian music, select this one.

There are many other scales included with FTS, and more than 2000 in the SCALA archive.

Bird song as Tune Smithy files

Here is a short phrase from a robin's song as a Tune Smithy file:

[help_files/robin_phrase.ts robin_phrase.ts]

To hear all the micro-tones, try playing it more slowly by adjusting the Time for one note . It may also help to lower the Pitch... Actually there were many more notes in the song, with some sliding from one to the other, but I've transcribed enough to make it recognisably the same as the original.

For a page of links to other bird song sites worldwide, see Bird Song Links. However, the site with the clip that I transcribed this phrase from isn't listed there any more - it was called the European .

For a site with listing of many sound clips from other

I transcribed this one by ear. You can use FTS to transcribe bird song using the wave counting method, but the methods are not yet very well developed. Possibly for a future release.

This bird song page shows some of the results of my early experiments, which were quite promising as you hear, but I haven't yet been able to get it working well enough without considerable tweaking all the time. Hopefully, more for a later release. Meanwhile, you can send a bird song midi musical card from that site!

This is one way to do it by ear (takes a while to do):

Slow down the recording to a quarter speed in Goldwave or some other audio editor, and loop at each note, then match the pitch in FTS.

You can slow down the recording in Goldwave by customising the Goldwave fast forward button to play at a quarter speed (recommended in the help for Goldwave as a way to do it).

If you can't seem to match a note at all, that probably means it consists of two or more very short notes, fairly close in pitch. So try to find where one begins and the next ends. There may be a short glissando from one note to the next, but often the notes themselves have a clear pitch once you've isolated them.

Look out for very short microtones ending a note (there is one at the end of the second last note in the example) - these can colour the perception of the whole note and make it seem too sharp or too flat on play back if you leave them out.

You can match the pitch in FTS by setting the note time to some large number, the seed to 0, and then varying the pitch using the Pitch window. You can vary the pitch in this way while the note is playing in FTS, and do this until it matches the pitch of the note from the birdsong you have just played in Goldwave.

Note down the frequencies for all the pitches.

After transcribing all the pitches in the phrase, enter them into the Intervals box as frequencies - you can do that by entering the values in hertz, followed by hz: 1053.2 hz.

Then listen to the original, play it in FTS with the same rhythm as best you can, then speed it up four times and raise the pitch by two octaves to the correct value.

You can fine tune the pitches from the New Scale window using the scroll bar to the right of the Interval box. Use Play note to play it, and select the note to play using the scroll bar to the right of the Pos box.

To adjust the pitch of the first note of the scale, use the Pitch window.

I'm sure many of you will be able to do a much better transcription than I did! Only reason for including this one is because it makes nice sounding Tune Smithy files, and I needed an example of bird song as a Tune Smithy file to demonstrate the idea.

In fact some of the tune smithy files use an earlier version, including unchecked intervals beyond the end of the scale, but since they sound nice as they are, decided to leave them like that rather than redo them with the new version of the robin's phrase.

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