17 and 19 limit meantone

One must eventually come into conclusion that not only are there conflicting meantone and meanpop in 11-limit and 13-limit (see Meantone notation) but also in turn within grosstone there are in turn two conflicting systems each for 17-limit and 19-limit and one of them, uiffktone, extends to 29-limit, but meanpop extends to 19-limit.

# Meantone

```
3 — +1 fifth
5 — +4 fifths
```

This is the core concept of a meantone, with 81/80 tempered out. It's not like there weren't conflicting systems here already, because there are alternatives like schismatic family with 5 in -8 fifths and superpyth with 5 in 9 fifths. But this is a meantone article.

Edos: 5 7 12 19 24 26 31 36 38 43 45 50 55 57 62 67 69 74 76 81 86 88 93 98 100 105 117 129

## Septimal meantone

```
3 — +1 fifth
5 — +4 fifths
7 — +10 fifths
```

This is a septimal temperament many meantone systems use. Tempers out not only 81/80 but also 126/125 and 225/224.

Edos: 12 19 31 43 50 62 74 81 93 105

### Undecimal meantone

```
3 — +1 fifth
5 — +4 fifths
7 — +10 fifths
11 — +18 fifths
```

In 11-limit, there are two conflicting systems. 31edo matches both meantone and meanpop, and so does the contorted 62edo, but 62edo does not match either 13-limit system as 13 falls outside of the circle of fifths. Tempers also 99/98 and 441/440.

Edos: 12 31 43 62 74 105

### Undecimal meanpop

```
3 — +1 fifth
5 — +4 fifths
7 — +10 fifths
11 — -13 fifths
```

This is the other 11-limit conflicting system. In general sharper fifths than 31edo will follow meantone, while flatter fifths will follow meanpop. Tempers also 385/384.

Edos: 19 31 50 62 81

### Tridecimal grosstone

```
3 — +1 fifth
5 — +4 fifths
7 — +10 fifths
11 — +18 fifths
13 — -16 fifths
```

In tridecimal grosstone we start seeing 13-limit. Tempered out 144/143.

Edos: 12 31 43 74 105

#### septendecimal uiffktone

```
3 — +1 fifth
5 — +4 fifths
7 — +10 fifths
11 — +18 fifths
13 — -16 fifths
17 — -36 fifths
```

This is it. Like how meantone has 31edo split the system to grosstone and meanpop, 74edo splits grosstone. 170/169. Uiffktone is the flatter fifth than 74edo but sharper than 31edo mapping.

Edos: 31 74 105

#### septendecimal vysthtone

```
3 — +1 fifth
5 — +4 fifths
7 — +10 fifths
11 — +18 fifths
13 — -16 fifths
17 — +38 fifths
```

And vysthtone is the alternative mapping. 221/220, 442/441.

Edos: 43 74

#### enneadecimal uiffktone

```
3 — +1 fifth
5 — +4 fifths
7 — +10 fifths
11 — +18 fifths
13 — -16 fifths
17 — -36 fifths
19 — -34 fifths
```

153/152 and 171/170 obviously

Edos: 31 74 105

#### enneadecimal vysthtone

```
3 — +1 fifth
5 — +4 fifths
7 — +10 fifths
11 — +18 fifths
13 — -16 fifths
17 — +38 fifths
19 — +40 fifths
```

153/152 and 171/170 obviously

Edos: 43 74

#### trivigesimal uiffktone

```
3 — +1 fifth
5 — +4 fifths
7 — +10 fifths
11 — +18 fifths
13 — -16 fifths
17 — -36 fifths
19 — -34 fifths
23 — +25 fifths
```

300/299.

Edos: 31 74 105

#### enneavigesimal uiffktone

```
3 — +1 fifth
5 — +4 fifths
7 — +10 fifths
11 — +18 fifths
13 — -16 fifths
17 — -36 fifths
19 — -34 fifths
23 — +25 fifths
29 — -45 fifths
```

204/203.

Edos: 31 74 105

#### untrigesimal uiffktone

```
3 — +1 fifth
5 — +4 fifths
7 — +10 fifths
11 — +18 fifths
13 — -16 fifths
17 — -36 fifths
19 — -34 fifths
23 — +25 fifths
29 — -45 fifths
31 — +55 fifths
```

528/527.

Edos: 74 105

#### untrigesimal uiauntone

```
3 — +1 fifth
5 — +4 fifths
7 — +10 fifths
11 — +18 fifths
13 — -16 fifths
17 — -36 fifths
19 — -34 fifths
23 — +25 fifths
29 — -45 fifths
31 — -50 fifths
```

465/464.

Edos: 31 105

### Tridecimal meanpop

```
3 — +1 fifth
5 — +4 fifths
7 — +10 fifths
11 — -13 fifths
13 — +15 fifths
```

In tridecimal meanpop we start seeing 13-limit as well. Tempered out 144/143 too, but also 105/104.

Edos: 19 31 50 81

#### septendecimal meanpop

```
3 — +1 fifth
5 — +4 fifths
7 — +10 fifths
11 — -13 fifths
13 — +15 fifths
17 — +26 fifths
```

The work in progress.

273/272. 170/169.

Edos: 31 50 81

#### enneadecimal meanpop

```
3 — +1 fifth
5 — +4 fifths
7 — +10 fifths
11 — -13 fifths
13 — +15 fifths
17 — +26 fifths
19 — +28 fifths
```

153/152. 171/170.

Edos: 31 50 81