7+3 scales

This page is about scales with 7 of one size step and 3 of another. Usually this means 7 large and 3 small steps, but I'll try and remember to mention the others as well. There's a brief mention of these scales on my permanently unfinished notation and keyboarding page.

This page is sort of finished, but I'd like to add lots of MIDI examples sometime. It should also link with this page on Dave Keenan's site which covers the same sort of scales with minor rather than neutral thirds.

The 10-note scale
Tuning those 10 notes
The 7-note scale
5-note scales
Tetrachordal structure
Other tunings

The 10-Note Scale

So, we have 7 large and 3 small steps. Typically, the smaller step will be something like a quartertone, so I'll write it as q. The larger step is a neutral second. That is, somewhere between a tone and a semitone. So, I'll call it t for "sort of like a tone" The 10 note scale, looks like this:

C  D\ D  E\ F  G\ G  A\ Bb B\ C
  t  q  t  t  t  q  t  t  q  t

The \ symbol means make the note q lower than it would usually be. The b symbol then means "lower by twice q". This works if you start with a meantone scale.

I'm assuming the scale will be tuned to something like 24-, 31- or 38-equal. Plenty of other tunings are possible, as I'll explain below. However, I need to explain the rule before I cover the exceptions, and these tunings are the rule. In all of them, q is one step. That means q is a true quartertone in 24=. Surprise surprise. More generally, as the notation suggests, q is half of a chromatic semitone. In 38=, that means half a step of 19=. The interval t is 3 steps in 12=, 4 steps in 31= and 5 steps in 38=. In all cases, this is exactly half-way between a tone and a diatonic semitone.

Fitted to a normal 7+5 keyboard, the 10 notes look like this:

C  Db D  E  F  Gb G  A  Bb B  C
  t  q  t  t  t  q  t  t  q  t 

I'll use these names to refer to the notes from now on, because I'm so used to my keyboard taking on different guises it doesn't confuse me. Think of it as being a strange key-signature!

Tuning Those 10 Notes

In my own software, this scale in 31-equal can be specified by a tuning file looking like this:

coords 2
s 0.129032
r 0.032258
notes 12
1 1 0
2 1 1
3 2 0
4 2 1
5 3 1
6 4 1
7 4 2
8 5 1
9 5 2
10 6 2
11 6 3
12 7 3

Because of the way those files are designed, to switch to a different temperament you only need to change the bit at the top. For perfect minor thirds (nearly 38=) it becomes

coords 2
s 0.131480
r 0.026546

for perfect fifths:

coords 2
s 0.122556
r 0.047369

And in 24=:

coords 2
s 0.125
r 0.041667

For tuning in cents, the simplest is 24=:

C      0
Db   150
D    200
E    350
F    500
Gb   650
G    700
A    850
Bb  1000
B   1050
C   1200

For other scales in cents, you can work it out youself. It isn't difficult but I can't be bothered.

The 7-Note Scales

Although I'm defining the scales as having 10 notes to the octave, most of the time I'll talk about 7 note subsets. With these, there are 3 large and 4 small steps to the octave. The smaller step is that neutral second t, and the larger step is a normal tone which I'll write as T.

The white notes then become one of these 7-note scales:

 T t t T t T t 

Or, if you're one of those people who think unmodified letters should be a diatonic scale:

C  D  E\ F  G  A\ B\ C
  T  t  t  T  t  T  t 

The interval T+t is a neutral third. That's either an ambiguous interval between a major and minor third, or an approximation to 11/9, depending on whether you believe in 11-limit harmony or not.

The scale given above can be generated from a single section of the spiral of neutral thirds. That means it is a MOS scale. So, it is also a good approximation to 7-equal. (In fact it's a very good approximation, as MOSs go, as the two steps sizes are unusually close together.) The spiral is:


So, D-F is the "wolf" neutral third, which happens to be a minor third as hinted above. So, the scale could be drawn on an 11-limit lattice:


Here a fifth is two, rather than one, steps to the right. Those fifths are, at least for the white notes, always where you expect them to be on the keyboard. However, D-A isnt a fifth although you might expect it to be. There is another 7-note scale worth note:

C  D  E\ F  G  A\ Bb C
  T  t  t  T  t  t  T 

(I put the \'s back in, but that's only a temporary lapse.) Paul Erlich, on the Tuning List, declared this to be his favourite of the two. He also says it is the characteristic scale of Arabic music. I think it is the stronger melodically, but that that's not always a good thing. One reason for this character is that it is less like 7-equal. It contains two minor thirds, plus the interval 2T, which is a major third. So, on the lattice, it looks like this:

         \ / \         

G-D and Bb-F are fifths. Now, I suppose you want all 10 notes on the lattice. Well, you're going to get them anyway:

     \   \ / \ / \           
      \   \   /   \          
       \ / \ / \   \         

5-Note Scales

I find a lot of good melodies come from pentatonic subsets of the MOS scale. The simplest is a run of 5 neutral thirds:

C   E   F   G   A   C
E   G   A   B   C   E
G   B   C   D   E   G
 t+T  t   T   t  T+t 

Although this leaves the two largest intervals right next to each other, which leaves the scales sounding unbalanced. To correct this, take the subset 1 2 1 2 1 from the original 7-note scale, in the Thai fashion. You get:

D   E   G   A   C   D
B   C   E   F   A   B
  t  t+T  t  T+t  T  

F   G   B   C   E   F
A   B   D   E   G   A
  T  t+T  t  T+t  t  

C   D   F   G   B   C
  T  2t   T  t+T  t  

G   A   C   D   F   G
  t  t+T  T   2t  T  

E   F   A   B   D   E
  t  T+t  T  t+T  t  

That's 5 different scales, 2 of which come in mirror-image pairs. The second one is particularly interesting, reminds me of a CD I have where the tuning is called Pelog. Here's a MIDI file of it.

Starting from the other kind of 7-note scale, you can get -- shock! horror! -- the classic pentatonic scale:

C   D   F   G   Bb  C
  T  2t   T  2t   T  

But, don't worry, there are lots of odd 5-note scales as well. Let's see:

Bb  C   E   F   A   Bb
  T  t+T  t  T+t  t   

F   G   Bb  C   E   F
  T  2t   T  T+t  t  

A   Bb  D   E   G   A
  t   2T  t  T+t  t  

E   F   A   Bb  D   E
  t  T+t  t   2T  t  

Tetrachordal Structure

"Tetrachord" is a posh word for a set of notes that spans a fourth. Tetrachords come up a lot in scale formation, although not always in the same way. One idea is that each scale can be described by an upper and lower tetrachord, where the upper tetrachord starts a perfect fifth above the lower one. A system of classification of South Indian scales works like this.

The other idea is that a scale should be constructed by the same tetrachord used twice. That leaves a tone which can be placed at the top, bottom or middle of the scale. It happens that all the modes of the diatonic scale can be described like this.

To start off, I'll unify the two and look at scales where the same tetrachord is repeated, with a tone in the middle. Back in diatonic land, this gives us (note names mean what they always used to mean):

Ionian    C D E F   G A B C
           T T s  T  T T s 

Dorian    D E F G   A B C D
           T s T  T  T s T 

Phrygian  E F G A   B C D E
           s T T  T  s T T 

These become transmuted into (note names regain the meaning I adopt for them on this page in order to confuse you):

anti-Ionian    G A Bb C  D E F G
                t t T  T  t t T 

anti-Dorian    E F G A   B C D E
                t T t  T  t T t 

anti-Phrygian  F G A Bb  C D E F
                T t t  T  T t t 

Using the method from Carey and Clampitt's paper in Perspectives of New Music 34/2 we can get anti-Ionian as the average of Dorian and Phrygian, anti-Dorian as the average of Ionian and Phrygian and anti-Phrygian as the average of Ionian and Dorian.

Placing the remainder tones in different places gives:

anti-Hypoionian    D E F G  A Bb C D
                    t t T  t t T  T 

anti-Hypodorian    B C D E  F G A  B
                    t T t  t T t  T 

anti-Hypophrygian  C D E F  G A Bb C
                    T t t  T t t  T 

anti-Hyperionian   C  D E F G  A Bb C
                    T  t t T  t t T  

anti-Hyperdorian   A  B C D E  F G A
                    T  t T t  t T t 

anti-Hyperphrygian Bb C D E F  G A Bb
                    T  T t t  T t t  

As anti-Hypophrygian and anti-Hyperionian are the same, that gives 8 distinct modes with identical tetrachords.

That Paul Erlich says the non-MOS scales are better because they give more modes with identical tetrachords (5 against 3). There may be something in this, but the presence of the classical pentatonic, the major and minor triads, and being further from 7-equal all give melodic distinctiveness to this scale.

Manuel Op De Coul's mode list uses a different naming system for these scales. It's based on a cyclic permutation of the original Greek meanings of the modes. (Thanks to Manuel for explaining that.) It also gives some Arabic names. Maqam Rast is my anti-Phrygian. Maqam Sikah doesn't have identical tetrachords. It's the mode starting on A with Bb in it. Miha'il Musaqa's mode is that anti-Hypophrygian/anti-Hyperionian. Mohajira is my anti-Dorian.


Modulation is a posh word for key changes. I don't know if you can really talk about keys with these scales, but the same principles apply.

Traditionally, modulation is by fifths. With a diatonic scale, this means only one note changes. With 7+3 scales, two notes change. If you want one note to change, you have to modulate by neutral thirds. However, neutral thirds are nowhere near as strong consonances, so the modulation is less definite.

The other option is to switch between the two different 7-note scales, changing one note each time. The result is that there's a transitional phase, followed by a modulation by a fifth. This should work best with the MOS scale as the transition one, as it's more ambiguous. Another idea would be to use the non-MOS scale in the chorus.

I'll try and get some MIDI files together to illustrate all this.

Other Tunings

Oops, nearly forgot this bit. Fatten the neutral third to become a major third and you still get a 7+3 scale. These don't immediately look as good, because the interval sizes are so unequal. However, tuning to 19-equal is really magical. Try it with a celesta sound. More needs to be written about these scales!

home page
my microtonal documents