Temperament Report for 004-01

This is a free analysis generated by recording submitted to me by one of my subscribers.

Contact Me Now if you would like a free analysis of one of your temperaments.

Piano: Unknown Console
Submitted by: 004


We use F2 to create beats with A4 and A440.
We try to tune A4 so that the beats created by F2A4 are the same speed as those created by F2A440.
F2 creates beats because it is flat from pure with A4 and A440.
If the beats are the same speed, then A4 and A440 are the same amount sharp of the fifth partial of F2.
Click the filtered buttons to hear the beating A4 easier.
The faster beating pair indicates the sharper A4 or A440.

4.6 bps
5.4 bps

Frequency of A4: 440.55 (F2A4 beats faster than F2A440)

Cents off: 2.2 (Must be within 1.4 cents)

Mark: 48% (Minimum pass is 80%)

Criteria: No marks lost if 0.9 cents off, or less. Lose 4 marks for every 0.1 cent off after that.

F3F4 and A3A4 Octaves

Criteria: Must be tuned to one of the following sizes.
(Note: Only one size is possible because of inharmonicity.)
Small Scale Octave: Pure 4:2 and Pure 6:3 (Within 3%) Common
Medium Scale Octave: Wide 4:2 and Narrow 6:3 (3% - 5%) Less Common
Large Scale Octave: Pure 4:2 and Very Narrow 6:3 (>5%) Rare
Reversed Octave: Narrow 4:2 and Wide 6:3. Very Rare.
(Click Here for the lesson on Octave Sizes.)


= 7.3bps
= 6.8bps

Size of 4:2 A3A4: Narrow (6%)

= 9.0bps
= 7.6bps

Size of 6:3 A3A4: Narrow (16%)


= 4.6bps
= 4.4bps

Size of 4:2 F3F4: Narrow (5%)

= 8.7bps
= 7.8bps

Size of 6:3 F3F4: Narrow (11%)

Comments on the Octaves F3F4 and A3A4

Both octaves are narrow. The 4:2 and 6:3 are both narrow for each octave.

When you click the F3F4 and A3A4 buttons, you can hear a slight roll in the upper partials.

Notice that the 4:2's are less narrow than the 6:3's. This implies that the piano may have Medium Scaled Octaves, that could be tuned as wide 4:2/narrow 6:3.

It is also possible that they are close enough in size (the 4:2's and the 6:3's) that the octaves may be tuned as pure 4:2/pure 6:3, each within 3%, and then we would identify this piano as having Small Scale Octaves.



I have designed two different grading schemes for measuring temperaments.

One is based on the error of the frequencies relative to a master tuning. This method is similar to the Registered Piano Technicians exam administered by the Piano Technicians Guild. The "master tuning" is determined by those frequencies that would produce beat speeds that increase by the same ratio as the average ratio of the actual beat speeds.

The second method assigns a penalty for each beat speed that does not fall within a certain limit. M3's must increase by 0 to 10%. P4's must beat between 0.5 and 1.5 beats per second.

Below are the marks calculated by each method. The PTG requires a minimum mark of 80% in many of its exams.

Temperament Mark

Average Ratio Method: 84%
(Similar to RPT method)

Beat Speed Limit Method: 79%
(Represents more directly what the tuner can hear. Preferred method.)



M3 intervals should increase speed chromatically as smoothly as possible.

1.Listen to the recordings below and look at the graph.

2.Listen to the filtered recording and try to confirm aurally that speeds are following what the graph says. This improves your Beat Speed Ratio Sensitivity.

3.Listen to the unfiltered recording and try to confirm aurally that speeds are following what the graph says. This improves your Beat Speed Clarity - the ability to hear beat speeds clearly.

Client 004-01 M3 Progressions

M3, filtered:

M3, unfiltered:



P4 intervals should beat close to 1 bps.

1.Listen to the recordings below while looking at the graph.

2.Listen to the filtered recording and try to confirm aurally that speeds are following what the graph says. This improves your Beat Speed Ratio Sensitivity.

3.Listen to the unfiltered recording and try to hear the quality of a tuned P4 (1 bps), a pure P4 (0 bps), and an out of tune P4 (>1.5 bps). Identifying correct and incorrect P4's are a very powerful refinement tool.

Client 004-01 P4 Progression

P4, filtered

P4, unfiltered


You must understand interval sizes in Equal Temperament, and Wide, Narrow, and Pure Intervals, in order to follow the logic explained below.

The logic used is to find a M3 that does not progress smoothly, and then find a P4 that confirms which note needs correction.

Specific Comments

While the M3's seem to progress relatively smoothly, except for the last one, The P4's are all on the fast side, except for the last one.

The obvious places to look for confirmation of corrections needed, are the fast C4E4 and the fast B3E4

C4E4 is too fast. C4 is low and/or E4 is high.
B3E4 is too fast. B3 is low and/or E4 is high.
E4 is high.

C#4F4 is too slow. C#4 is high and/or F4 is low.
C4F4 is too slow. C4 is high and/or F4 is low.
F4 is low


Try to hear the following inconsistencies in the following chromatic ladders. These are predicted by the findings above.

Chromatic M6's:
1. E4 is too high. G3E4 should sound too fast.
2. F4 is too low. G#3F4 should sound too slow.

Chromatic P5's:
1. E4 is high. A3E4 may sound too pure, or be wide.
2. F4 is low. A#3F4 should sound too narrow.

Chromatic m3's:
1. E4 is high. C#4E4 should sound too slow.
2. F4 is low. D4F4 should sound too fast.

Chromatic m6's:
1. E4 is high. G#3E4 should sound too slow.
2. F4 is low. A3F4 should sound too fast.

Compare the above intervals with their neighbours.

Notice how we identify two problems that have the same note, and then look for a way to change that note and fix both problems. This reiterative method helps us improve the temperament bit by bit.

This is why it is imperative that, if you want to improve your aural tuning, you must improve your beat speed recognition. Exercises like these can help greatly.

This is not necessarily a definitive description of the actual beat rates of your temperament. Beat rates within a piano interval can vary over the sustain of the interval.

You are encouraged to read the analysis and confirm aurally that you can hear the findings, and understand how the improvements suggested would improve the beat speed progressions.

In this way you will improve your ear and be able to hear smaller differences in beat speeds, which will ultimately improve your temperament.

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