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# The scales in the velocity keyswitches retuning Kontakt script

### From Tune Smithy

This page is work in progress, will describe all the scales in the examples list for Velocity keyswitches retuning## Contents |

## Screen shot

The menu here has a droplist of example scales to get you started right away

Also a short list of scales you can put on the black keys

## 12 tone tunings

The droplist shows various twelve tone tunings starting with just intonation, and then a historical progression of tunings and well temperaments.

For an introduction to this subject, see Kyle Gann's An Introduction to Historical Tunings

### Just intonation

In just intonation the aim is to make as many as possible intervals pure. It is impossible to make them all pure except in harmonic fragment scales (Dan Schmidt's "Modern Pelog" is an example of a harmonic fragment 7 tone scale).

For instance, of the major thirds C - E, E - G# and G# to C', you can only make two of those three intervals pure. The remaining one will be out by about 41 cents, nearly a quarter tone. Pure here means using the interval you get on a natural horn, or when playing harmonics on a string instrument, it's the interval between the 4th and 5th harmonics

Even if you stick to the seven notes of the white notes on a piano, then you can't get all the intervals pure, not if you want major and mionr thirds to be pure. If you have all the fifths pure and go C G D A E as pure fifths, then the E there is at a frequency ratio of 81/64 to the C, instead of the desired 5/4.

This means that you can't make all the triads in a seven note scale pure. You can make 1, II and V pure, or I, IV and V pure but you can't make all the triads in a diatonic scale pure. In the I, IV and V tuning then the minor triads on iii and vi are pure but the one on vii is impure.

The examples are Ptolemy tunings with some of the triads pure. For more about this see the wikipedia page on Five-limit tuning.

#### Various ways of working with just intonation scales

You can approach these tunings in many different ways.

One approach is that you only use low numbered pure intervals such as 5/4, 3/2, 7/6 etc. You get to know the scale very well and know which triads and other intervals you can play and which you can't play. You only play triads if all the intervals in the triad are pure.

Another approach is to use all of the intervals. This is a type of "just intonation tempering" or "rational intonation". Some of the intervals you get, though not conventionally "just intonation" still may be musically interesting and useful and can contrast, produce moment of dissonance or beats. You go by your ear, what sounds good to you in the tuning.

Some composers like to use "rational intonation" where you can use numbers as large as you like even into the thousands. You can use rational intonation to approximate twelve equal as used for the Hammond organ, but this can also be used creatively.

Jacky Ligon has explored the idea of making tunings entirely based on ratios of large primes. Here is an interesting post by Margo Schulter on the topic in a long discussion we had about it some years ago in the Tuning list at yahoo groups.

Then another approach is to go high up the harmonic seris. If you go high enough you can get any interval size you like at least approximately. The aim is no longer to play beatless intervals, but you feel the notes are all related together by the harmonic series.

David Beardsley uses the 128 notes of the harmonic series for his compositions, with numbers as high as that you don't expect to get beatless chords on all except a few intervals. See for instance his "as beautiful as a crescent of a new moon on a cloudless spring evening" which uses a very large 128 note "harmonic fragment" tuning.

To try out David Beardsley's tuning, choose "harmonics and subharmonics" and set "starts" to 128 to make the harmonic series 128/128, 129/128, ... , 256/128. Then the 5/4 is played at 160/128. With middle C as the root of the tuning, then this is on the G# key two "octaves" to the right on the keyboard. The 3/2 is on the E key 5 "octaves" to the right. You can't go much higher, so to get the full range of one octave on Beardsley's tuning, you need to set the 1/1 to 0 (this will need a "Transpose" feature in the script to get it to work).

So there are lots of ways you can use ratios in tuning.

### Pythagorean

All the notes are based on the pure fifth, a frequency ratio of 3/2. When you add intervals you multiply ratios. So after twelve fifths you get to (3/2)^12 = 129.746337891. This is quite close to 2^7 (128) but not exactly the same, the notes differ by a Pythagorean comma - roughly an eighth tone, 23.46 cents.

If you follow a cycle of pure fifths you get ..., Cbb, Gbb, Dbb, Abb, Ebb, Bbb, Cb, Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#, D#, A#, E#, B#, F##, C##, D##, A##, E##, B##, F##,..., where for instance, Cb, B and A## are different notes

The pythagorean twelve tone cycle was developed in the middle ages, gradually one note at a time, started with the white keys, then added Bb and then gradually added the other notes as time developed as the music got more chromatic.

They eventually stopped at twelve notes in Western music though in other cultures then musicians and theorists continued to more notes, for instance 17, or 53 notes. These numbers are the "moments of symmetry" with two step sizes. You also get a moment of symmetry after 5 notes for the pythagorean pentatonic, and after 7 notes for the diatonic scale with two step sizes whole tone and semitone. In the pythagorean 12 tone system you have two step sizes again, there are two sizes of semitone, the diatonic semitone (from e.g. E to F) of 90 cents, and the chromatic semitone (e.g. C to C#) of 114 cents, but they are similar in size so it is not that noticeable.

It is impossible to have all the fifths pure in Pythagorean tunings. So there is always one "wolf fifth" which is not pure. The wolf fifth in Pythagorean is not as extreme as the mean tone wolf, but has one of the fifths flat by about an eighth tone. It is usually placed in one of the remote keys rarely used.

In this tuning the fourths and fifths are consonances. The major and minor thirds in this tuning were considered dissonances in medieval music.

The only consonances in medieval music theory were the fourths and the fifths (and of course the octave), and music came to a rest on open fifths or fourths. Nowadays a modern listener might find the major and minor thirds in this tuning tolerable as they are only slightly sharp compared with the equal tempered thirds, and our ears are already used to the "bright" rapid beating you get in twelve equal thirds.

To find out more: Pythagorean tuning (wikipedia)

To find out more about how medieval composers used the Pythagorean tuning see Margo Schulter's Pythagorean Tuning and Medieval Polyphony

### Quarter comma meantone

This is a tuning from the sixteenth and seventeenth centuries, also it is still used today for some organs (because the beating thirds of twelve equal are more noticeable in the long sustained notes of an organ).

In this tuning the idea is to adjust the fifths in order to achieve pure 5/4 major thirds. Eight of the major thirds are pure, the remaining four are sharp, a lot sharper than the major thirds of twelve equal. Like the Pythagorean tuning it has a wolf fifth, but this time it is sharp rather than flat (which you may find more noticeable) and it is sharp by more than a third of a tone. In period music this fifth wasn't used at all, you simply avoided that key. This was acceptable so long as modulation was restricted to a small range of keys.

To find out more, Quarter comma meantone on Wikipedia

## The golden horogram tunings

This is a method of tuning developed by Erv Wilson. It uses a cycle of intervals just as for the Pythagorean and Mean tone approaches - but the aim isn't to try to approximate pure intervals. Instead you generate scales with two step sizes, large and small (a bit like our whole tones and semitones) but with the two step sizes in the ratio of the golden ratio.

You start with some formula involving the golden ratio, for instance the 3-5 horogram starts with the interval Phi÷(5Phi+1) where Phi is the golden ratio. You then create a cycle of those intervals just as for the Pythagorean tuning, and you look for the "moments of symmetry" with two step sizes. As you continue the process, if you start with a ratio defined using one of these simple phi based formulae, then when you get to enough notes you find that the intervals are in the golden ratio with each other - and then from then on all the moments of symmetry have two intervals in the golden ratio.

You can hear a couple of examples from David Finnamores golden horograms website. Best way to find out what they are is to read his site.

I thought it was nice to include a couple to give a taste of other kinds of microtonal music, in this case, using the golden ratio to construct intervals.

You can listen to examples on his listening page. The two tunings I included are the one for Ring 3-5, which you can hear here: Ring 3-5, and the one for ring 9-7 which you can hear here: Ring 9-7

## WORK IN PROGRESS

Will expand this page later on.

## Listen to examples of mirotonal music

See the Microtonal Listening List at the Xenharmonic wiki