My Approach to Temperament: Minimizing Compound Errors By Ben Cross, RPT

Introduction

Traditional temperament methods often suffer from compound errors, where inaccuracies in early tuning stages propagate through the entire temperament system. This article presents a new approach that prioritizes tuning notes from intervals as directly from the pitch source as possible early on in the sequence, thereby minimizing compound errors and providing the tuner with accurate foundational intervals to build the temperament.

Understanding Degrees of Separation

A "degree of separation" refers to the number of intervals between a given note and the pitch source. In this temperament system, intervals closer to the pitch source (first-degree intervals) are tuned directly, while subsequent intervals (second-degree and third-degree) are derived from those primary relationships later on in the sequence.

Mitigating Compound Errors

By prioritizing first-degree intervals, the risk of compound errors is significantly reduced. When tuning an interval that is several notes away from the pitch source, any initial inaccuracies are less likely to accumulate and affect subsequent intervals.

The Tuning Process

● Pitch Source and Initial Tuning: The process begins by establishing a stable pitch source ( A4). The interval between A3 and A4 is then tuned as a clean sounding ocatve. You can use an estimated a beat rate of 7 beats per second between the major 10th F3 and A4 (and match the major 3rd of F3 to A3 of about 7 bps.

1.) First-Degree Intervals: Intervals directly connected to the pitch source (A3 or A4) are tuned first. These include major thirds, fourths, fifths, and sixths.

First degree intervals are intervals built from first degree notes, notes tuned from A4 or A3

1. 1)  A3 to F#4 as a major 6th slowest 6th in the temperament

2. 2)  F4 to A4 as a major 3rd the fastest major 3rd in the temperament

3. 3)  A3 to E4 as a clean but narrow 5th

4. 4)  A3toD4asacleanbutwide4th

5. 5)  A3 C#4 as a major 3rd. Your slowest major 3rd. 2nd degree check c#4 to f#4 as a clean

   4th

6. 6)  C4 to A4 as a major 6th the fastest 6th

MAt this point we have :

• ●  our fastest and slowest 6tht outlining the perimeter for our 6ths

• ●  Our fastest and slowest 3rd outlining the perimeter for our 3rds

• ●  A first degree fourth in the middle of the temperament

• ●  A first degree fifth in the middle of the temperament

• ●  A second degree 4th check derived from two 1st degree intervals

• ●  Contagious Major 3rds to check balance the macro distribution of major 3rds across the

  temperament (bonus tune f3 as a clean octave to f4 and gain another 3rd to check the contiguous major thirds against.

  2.) Second-Degree Intervals:

  2nd degree intervals tuned to 1 or more 1st degree notes.

  1. 1)  B3toE4asacleanwide4thandtoF#4asacleanfifth

  2. 2)  Bb3 (A#) to F4 as a clean 5th

  3. 3)  G#4toC#4asaclean5thandB3toG#4asa6thchromaticallyinsyncwithC3toA4a

     1st degree major 6th sixth

  4. 4)  G4toC4asaclean5thandtoD4asaclean4thandtoBb3G4asa6thchromaticallyin

     sync with A3 to F#4 a 1st degree sixth

  3.) Tuning the final 3rd degree note

  D#4 (the back of the circle of 5ths) as a contagious clean wide 4ths between A#3(Bb3) and C#4 (both 2nd degree notes) is the final note that needs to be tuned and it can also be checked as contagious major thirds between B3 and G4.