# User:TromboneBoi9

Hello! My name is Andrew and I like screwing around with xenharmony, especially EDOs, JI, and various notations.

*Here's my website...if you could call it that.*

I also exist on the XA Discord, currently under the alias *Sir Semiflat*.

## Xenharmonic Discography

As of mid-April 2024 (non-comprehensive)

### Completed works

for solo piano in 31edo, 2023**Torn Gamelan**for solo piano in 7-limit just intonation, 2023**Apollo's Broken Piano**for rock band in 19edo, 2023**Chicago Olēka**for string quartet in 24edo, 2024**A Harmonization of a Microtonal Etude**for solo piano in 16edo, 2023**Adagio con demenza**

### Incompleted works

for solo keyboard in 20edo**Keyboard Suite in 20-equal Tuning**for solo piano in 16edo**Treated Four by Four**

## Extended Ups and Downs

See User:TromboneBoi9/Extended Ups and Downs

## Scales n' Stuff

### Website

I have some scales ready in Scala format on my website here.

### "Drewnian" Scale System

I recently developed an eleven-tone scale system, similar to Western theory in construction but by no means similar in sound. I like to think it's what Wstern tonality would look like if the 7/4 was prioritized over the 5/4.

It involves two scales, each built with a seven-based tetrachord. One is built with a 8/7, 8/7, 49/48 tetrachord, generating the "major" intervals; and the other is built with a 49/48, 8/7, 8/7 tetrachord, generating the "minor" intervals. As a result, the scale has very large major seconds and very small minor seconds.

Degree | Name | Ratio | Cents |
---|---|---|---|

0 | P1 | 1/1 | 0.000 |

1 | m2 | 49/48 | 35.697 |

2 | M2 | 8/7 | 231.174 |

3 | m3 | 7/6 | 266.871 |

4 | M3 | 64/49 | 462.348 |

5 | P4 | 4/3 | 498.045 |

6 | P5 | 3/2 | 701.955 |

7 | m6 | 49/32 | 737.652 |

8 | M6 | 12/7 | 933.129 |

9 | m7 | 7/4 | 968.826 |

10 | M7 | 96/49 | 1164.303 |

11 | P8 | 2/1 | 1200.000 |

The intent with these weird superpyth-like constructions is to make 5-EDO-like scales with some extra small intervals for spice.

It's very easily possible to map this scale to a 12-tone keyboard, in fact it is by design a mimicry of Western tonality. However, there is no tritone. For this interval you can either use the eleventh harmonic 11/8, a septimal whole tone below the "minor sixth". or a septimal whole tone above the "major third".

### Enneatonic scale in JI

Supahstar Saga described a scale in 19-EDO in his *Exploring 19-TET* YouTube series called the Enneatonic scale:

In 19-tone, the third harmonic is an even number of steps in 19 (30 steps), splitting evenly into two harmonic (subminor) sevenths. If you take a major pentatonic scale, put a harmonic seventh in between each fifth, and reduce the whole scale into an octave, you get a nine-tone scale somewhat similar to the double harmonic scale in 12.

My thought was, if you use pure 3-limit Just Intonation, you can split the third harmonic into two "ratios" of √3, measuring around 950.978 cents. What would that sound like?

Degree | Ratio | Decimal | Cents |
---|---|---|---|

1 | 1/1 | 1.0000 | 0.000 |

2 | 9/8 | 1.1250 | 203.910 |

3 | 81/64 | 1.2656 | 407.820 |

4 | 3√3/4 | 1.2990 | 452.933 |

5 | 27√3/32 | 1.4614 | 656.843 |

6 | 3/2 | 1.5000 | 701.955 |

7 | 27/16 | 1.6875 | 905.865 |

8 | √3/1 | 1.7321 | 950.978 |

9 | 9√3/8 | 1.9486 | 1154.888 |

10 | 2/1 | 2.0000 | 1200.000 |

But of course Saga wasn't looking for a √3 interval, he meant to use the harmonic (subminor) seventh. The two intervals are rather close though: the seventh is about 17.848 cents sharper.

Using harmonic sevenths of 968.826 cents:

Degree | Ratio | Decimal | Cents |
---|---|---|---|

1 | 1/1 | 1.0000 | 0.000 |

2 | 9/8 | 1.1250 | 203.910 |

3 | 81/64 | 1.2656 | 407.820 |

4 | 21/16 | 1.3125 | 470.781 |

5 | 189/128 | 1.4766 | 674.691 |

6 | 3/2 | 1.5000 | 701.955 |

7 | 27/16 | 1.6875 | 905.865 |

8 | 7/4 | 1.7500 | 968.826 |

9 | 63/32 | 1.9486 | 1172.736 |

10 | 2/1 | 2.0000 | 1200.000 |

Using harmonic sevenths inverted around the third harmonic, or just supermajor sixths (12/7), of 933.129 cents:

Degree | Ratio | Decimal | Cents |
---|---|---|---|

1 | 1/1 | 1.0000 | 0.000 |

2 | 9/8 | 1.1250 | 203.910 |

3 | 81/64 | 1.2656 | 407.820 |

4 | 9/7 | 1.2857 | 435.084 |

5 | 81/56 | 1.4464 | 638.9941 |

6 | 3/2 | 1.5000 | 701.955 |

7 | 27/16 | 1.6875 | 905.865 |

8 | 12/7 | 1.7143 | 933.129 |

9 | 27/14 | 1.9286 | 1137.039 |

10 | 2/1 | 2.0000 | 1200.000 |

## Irrational HEJI Extensions

I've heard phi is somewhat useful in xen areas, as well as other popular irrational numbers, so what would it look like if I extended HEJI (my go-to Just Intonation notation) to support these numbers like factors?

### Commas

#### Golden Ratio

The ratio phi adds up to 833.0903 cents, a sharp minor sixth. The Pythagorean minor sixth is 128/81, about 792.1800 cents. This leaves a comma of 81ϕ/128, about 40.9103 cents. I dub this interval the **Golden quartertone**.

#### Pi

The ratio π/2 adds up to 781.7954 cents, an okay minor sixth. The Pythagorean minor sixth is 128/81, about 792.1800 cents. This leaves a comma of 256/81π, about 10.3846 cents. I dub this interval the **Circular comma**.

#### Euler's constant

The ratio *e*/2 adds up to 531.2340 cents, a pretty sharp fourth. The Pythagorean perfect fourth is, of course, 4/3, 498.0450 cents. This leaves a comma of 3*e*/8, about 33.1890 cents. I dub this interval the **Eulerian comma**.

### Notation

For the golden quartertone, I plan to use the symbol Blackwood used in his microtonal notation, because it already resembles a phi symbol (ϕ). For pi, I designed a symbol similar to the 55-comma symbol in Sagittal, but the "arrowhead" is replaced with a circular cap, making the symbol resemble a *J* with an extra shaft.

I'm yet to design a symbol for e.