Tritone (Part 2)

During the Tritone Part 1 blog post, I reviewed the historic significance of the interval briefly, the basic theory behind it and how composers used it throughout the history of music.

In this blog post, I will be demonstrating how to find the tritone interval of 12ET approximately in the harmonic series (with the help of the formulas previously mentioned). I will also talk about the “third tone” that is heard when a tritone interval is played through heavy distortion or played loudly (the difference tone, which is an example of combination tones).

There can be a number of different augmented 4th and diminished 5th intervals in meantone tunings (chain-of-fifths tuning's with the generating fifth roughly in the range from 692 cents [5/11-comma meantone] to 700 cents [1/11-comma meantone], more or less). The meantone augmented 4th range is from 552 cents to 600 cents and for the diminished 5th it is from 600 cents to 648 cents. However, to make things a little simpler I shall only be mentioning the interval of 600 cents and how some ratios in the harmonic series are close to it.

First, convert cents to ratio-decimal format 2^(600/1200) = 1.41421356 (exactly the number as the square root of 2) (Monzo, 2003). We can use continued fractions to find a decent fraction for this by hand, nevertheless, I will be using an online continued fraction calculator to make it quicker, Ratios such as, 99/70, 140/99, 239/169, 577/408 (A Continued Fraction Calculator, 2016) are some ratios that are extremely close to 600 cents, they are as follows,

• 99/70 = 600.088324c (error of approximation from 600 cents = 0.088324c)

• 140/99 = 599.911676c (error of approximation from 600 cents = 0.088324c)

• 239/169 = 599.984846c (error of approximation from 600 cents = 0.015154c)

• 577/408 = 600.0026c (error of approximation from 600 cents = 0.0026c)

As one can see, they are similar to, but not exactly the same as, 600 cents -- though some are extremely close. It is also important to note that as one uses higher numbers in a ratio it is difficult to perceive it in any psychoacoustic or cognitive sense as anything notably different from any nearby interval. Hence, while we can tune a ratio such as 7:5 by ear so that it "locks in", using beats for example, there is no way to do that for a ratio like 239:169 (Paul Erlich, personal communication, May 27th, 2019).

Monzo (1998) has mentioned numerous tritone cents in meantone temperament and various other temperaments:

https://video.wixstatic.com/video/6a665e_b34aa06d6aba414e83fbf48992f24118/1080p/mp4/file.mp4

What happens when a tritone is played through heavy distortion?

A psychoacoustical phenomenon called Tartini tones or Combination Tones gets introduced when one plays loud, and distortion accentuates this phenomenon considerably even at quiet volumes. The phenomenon can be heard when two tones, for example, 1760Hz and 2489Hz* are played extremely loudly and the difference tone 729Hz is introduced by your ears automatically (Wolfe, 2011). The sum tone can be hard to perceive because our brains can suppress them (Paul Erlich, personal communication, November 17th, 2019).

Why does this happen? Because when the amplitude is loud our ears fail to remain linear and due to non-linearity new tones get introduced. Non-linearity here means the ear membranes donʼt move twice as far when it's twice as loud or, more precisely, pushed twice as hard (Paul Erlich, personal communication, November 17th, 2019).

*1760Hz is 2 octaves above A=440Hz x 4 = 1760Hz (A6) and 2489Hz is a b19 above A4=440 x 5.65685425 = 2489.01587Hz (Eb7).

To calculate the "third tone" (Tartini tone or difference tone) when one would play a tritone (600 cents) loudly is as follows,

The difference between the two cent values will be 600 cents (2125.86396-1525.86397). Although, if one would play A6-Eb7 (600 cents).

What is the combinational tone (or the difference tone) that the person would hear?

We can find out by taking the lower note (A6) and the higher cent value -1525.86397 cents,

From A6

1525.864-1200 (octave reduce)

A5

325.86400-300 (minor third reduce)

F#5 (or Gb5)

25.86400-25 (1/8th of a tone reduce approx.) =

F#5 (or Gb5) -1/8th of a tone (flat)

And later taking the higher note (Eb7) with the lower cent value -2125.86396 cents,

From Eb7

2125.864-1200 (octave reduce)

Eb6

925.86400-900 (major sixth reduce)

F#5 (or Gb5)

25.86400-25 (1/8th of a tone reduce approx.) =

F#5 (or Gb5) -1/8th of a tone (flat)

Thus, the difference tone is a F#5 flattened by 1/8th of a tone (error of approximation 0.864 cents).

One can also convert these intervals to Hz and find any specific pitch by playing a tritone anywhere since cents is just an interval from one note to another.

Recommendations:

Some modern 12ET tritone songs that I personally enjoy are as follows,

• Dig by Mudvayne

• Turn Around by Soften the Glare

• Dante Sonata by Franz Liszt

• The Simpsons Theme

• The Pink Panther Theme Song

• Painkiller by Judas Priest

REFERENCES:

A Continued Fraction Calculator (2016). Retrieved from http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfCALC.html

Monzo, J. L. (2003). Definitions of tuning terms. Retrieved from http://www.tonalsoft.com/sonic-arts/dict/tritone.htm

Wolfe, J. (2011). Tartini tones and temperament: an introduction for musicians. Retrieved from https://newt.phys.unsw.edu.au/jw/tartini-temperament.html

APPENDIX:

12tone. (2017, May 26th). How To Play Notes That Aren't There [Video File]. Retrieved from https://www.youtube.com/watch?v=KWJS_Fzs1j4

Hawking, T. (2014). The Devil’s Music: 10 Songs Based Around the Tritone Interval. Retrieved from https://www.flavorwire.com/467540/the-devils-music-10-songs-based-around-the-tritone-interval

Neely, A. (2017, Oct 30th). Combination Tones [Video File]. Retrieved from https://www.youtube.com/watch?v=73_CiAYX00k

Neely, A. (2019, Sep 9th). YouTubers React to Experimental Music [Video File]. Retrieved from https://www.youtube.com/watch?v=PVvOemu-ZTk

Schoen-Philbert, J. (2015). 25 Songs With The Tritone. Retrieved from https://www.uberchord.com/blog/tritone-songs/

Tritone. (2019). In Wikipedia. Retrieved from October 1st, 2019, from https://en.wikipedia.org/wiki/Tritone

Turner, A. (2014, Dec 12th). Tartini Tones [Video File]. Retrieved from https://www.youtube.com/watch?v=gPGA2pkrab