Forward Stack

Occupation: Music and Math
Author: Boris Andonov
Initial release: 28 February 2021
Latest update: 26 July 2021
PDF version: Download

Forward Stack


This is the full manual, step by step how to tune all 12 keys following the conception in first article, sub-chapter “Forward Stack” (read it first). It is recommended for beginners to understand this process and master the tuning techniques of just intervals.

This process requires to create 7 perfect fifths and 1 major third as just intervals. Tuning order of the stack is a personal choice.

Images below are an example how I would do it from my perspective. They are for educational purpose to see the whole picture of this procedure. Choose any order that fits best for you. I’m recommending to use sticky notes on every key to track your progress. Otherwise, you might get lost what has been done and what has left.

The range of the referent and jump octave have been chosen on purpose. As already said in first article, the best hearing range for complex sounds is between 80Hz – 500Hz.


Referent octave is the kernel for all other keys. When you finish with referent octave, spread the tunes across all octaves. It is a standard, well-known technique which does not need further explanation. Remember to tune an octave exactly in ratio 2:1. DO NOT STRETCH THE OCTAVE!

Jump octaves is where a single stack is being created. They are called jump because will be constantly detuned.

Keys marked with numbers represent all required steps for a single stack.

X – Black and white “X” symbol have the meaning of already tempered tone. Do not touch this key!

X – Green “X” mark. The tune of this key will be equally tempered within the current stack.

0 – Black or white “0”. The kernel of every stack. This is the initial step. In first picture it’s tone A and assumes that already has the right tune of 440Hz, 432Hz… or whatever you like.

Red numbers – These are perfect fifths, ratio: 3/2

Blue number – This is a major third, ratio: 5/4

Green numbers – shift 1 octave down, ratio: 1/2

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