# Music Composition, Teaching, Recording, Instrument Building

# Ben Hjertmann

# Rogue 48-Utonal [Equally-Spaced]

Equally-spaced frets? Is it an optical Illusion? Does it produce random notes?

The Rogue 48-Utonal divides the *scale length* into 48 equal parts.

The result is a 47-prime-limit Utonal (undertone/subharmonic) scale with a mix of familiar (4:3, 3:2, etc.) and unfamiliar (48/47, 48/41, etc.)

More detailed information provided below.

## 48-Utonal Tuning

This fretting uses 48 equal divisions of the scale length. The scale length is the entire length of the string (25.5 inches, 648 mm) from bridge to nut. Each division (fret) is 13.5 mm (648/48). The fretboard runs 70% of the length of the fretboard, containing 34 frets.

The old 12edo fret-marker dots still remain, but should be ignored.

Aesthetically, it is counterintuitive to see frets of identical size. For guitarists, it may even result in an optical illusion, since we are so accustomed to seeing the frets shrink in size along the neck (to produce the standard 12tone Equal-Temperament, a.k.a. 12edo or 12TET). It may look like the frets are getting larger as they move toward the bridge, but they are all the same 13.5 mm size. (Obviously the 12edo fret dots may contribute to the illusion as well.)

Equally-Dividing the scale length produces a "Utonal" scale, from the inverse of the harmonic series (a.k.a. subharmonics or undertones), since wavelength and frequency are inversely proportional. "Utonal" was Harry Partch's term for Undertone-Tonality (U-tonal). The undertone series is most easily described in descending format, since the order of the intervals are the same as the overtone series.

Moving from bridge to nut, the intervals are 48/1, 48/2, 48/3, 48/4, etc. So, the intervals between the successive pitches are 2:1 (octave), 3:2 (fifth), 4:3 (fourth), etc., identical to the overtone series. As the nut is approached, the intervals not only get smaller (from 1200 cents down to 36 cents), but more obscure higher prime intervals are introduced (like in the overtone series), ending in the arcane 48/47 on the lowest fret. You can explore the full fretting plan with measurements below, along with some examples of tuning charts for different songs that use this guitar.