# Neutral third scales

Neutral third scales are characterized by the division of the 3/2 "perfect fifth" into two equal "neutral thirds". It usually refers to a MOS or MODMOS generated by a neutral third (usually from 342.9c in 7edo to 360c in 10edo).

One possible temperament interpretation is that the neutral third generator can be taken to represent either 11/9 or 27/22. This means the comma 243/242 is tempered out, resulting in the neutral temperament and its extensions (described below).

If the fifth harmonic is used at all, it makes sense to map it to 8 generators by tempering out 81/80, making it a meantone system, sometimes called mohaha. If 7 is also mapped a particular way it is called mohajira. Alternatively, 64/63 could be tempered out, leading to a 2.3.7.11 version of neutrominant temperament. But neither of these adjustments to the farther-out intervals affects its basic nature as a "neutral thirds" scale, which equally subdivides the 3/2 into two equal parts.

Any temperament tempering out 243/242 lends itself to neutral thirds; this becomes more significant when fifths are of low complexity, such as hemififths or the rank three temperament jove, but also includes miracle, harry and so forth. Nor does a scale need to be a MOS to qualify as a neutral thirds scale; that name could reasonably be given to Oktone, for example.

## Interval chains

### Basic 2.3.11 neutral thirds scale

 296.85 647.37 997.90 148.42 498.95 849.47 0. 350.53 701.05 1051.58 202.10 552.63 903.15 32/27 16/11 16/9 12/11 4/3 18/11 1/1 11/9 3/2 11/6 9/8 11/8 27/16

### 2.3.5.11 mohaha

 815.25 1163.34 311.44 659.53 1007.63 155.72 503.81 851.91 0. 348.09 696.19 1044.28 192.37 540.47 888.56 36.66 384.75 8/5 6/5 16/11 16/9~9/5 12/11~11/10 4/3 18/11 1/1 11/9 3/2 11/6 10/9~9/8 11/8 5/3 5/4

### 2.3.7.11 suhajira

 414.38 768.34 1122.29 276.25 630.21 984.17 138.13 492.08 846.04 0. 353.96 707.92 1061.87 215.83 569.79 923.75 77.71 431.66 785.62 14/11 14/9 7/6 16/11 7/4~16/9 12/11 4/3 18/11 1/1 11/9 3/2 11/6 9/8~8/7 11/8 12/7 9/7 11/7

## MOSes

Neutral third scales are associated with the MOS patterns: 3L 4s and 7L 3s.

In the 3L4s MOS, the large step is a 9/8 and the small step is a 12/11 neutral second. In the 7L3s MOS, the 12/11 is now the large step and the small step is a 33/32 quarter tone.

## Munits

Though they differ in the specifics of their temperaments, all neutral third scales have certain munits in common.

• 4/3 Lss, 4/3 sLs: Rather than being divided into two large steps and one small step as in the diatonic scale, 4/3 can always be divided into one large step and two small steps. The large step is 9/8, and the two small steps are "neutral seconds" representing the sub-munit "32/27 ss".
• 11/8 LsL: In contrast, two large steps and a small step make a much less consonant and more colorful interval, 11/8.

## Neutral thirds scales by EDO

Not an exhaustive list.

1 1 1 1 1 1 1

7edo

### 10edo

2 1 2 1 2 1 1

Neutral Hypolydian

2 1 2 1 1 2 1

Neutral Hypodorian

2 1 1 2 1 2 1

Neutral Lydian

1 2 1 2 1 2 1

Neutral Dorian

1 2 1 2 1 1 2

Neutral Hypophrygian

Any other scales which contain the 3rd (360c) and 6th (720c) degrees of 10edo are also considered neutral thirds scales.

### 17edo

3 2 3 2 3 2 2

Neutral Hypolydian

3 2 3 2 2 3 2

Neutral Hypodorian

3 2 2 3 2 3 2

Neutral Lydian

2 3 2 3 2 3 2

Neutral Dorian, Misaelides 2nd Byzantine mode, Maqam Sikah Baladi

2 3 2 3 2 2 3

Neutral Hypophrygian

Any other scales which contain the 5th (~353c) and 10th (~706c) degrees of 10edo are also considered neutral thirds scales.

### 24edo

4 3 4 3 4 3 3

Neutral Hypolydian

4 3 4 3 3 4 3

Neutral Hypodorian

4 3 3 4 3 4 3

Neutral Lydian

3 4 3 4 3 4 3

Neutral Dorian, Misaelides 2nd Byzantine mode, Maqam Sikah Baladi

3 4 3 4 3 3 4

Neutral Hypophrygian

Any other scales which contain the 7th (350c) and 14th (700c) degrees of 10edo are also considered neutral thirds scales.