Theory of harmonica tuning

Örjan Hansson

(Click on a heading to get directly to that section.)

1. Introduction

1.1 Twelve-tone equal temperament

Most musical instruments today are tuned to what is called 12-tone equal temperament (12TET) using a reference frequency (pitch) of A4 = 440 Hz (1 Hz = 1 Hertz = one cycle of oscillation per second). This intonation (tuning scheme) is a common standard that ensures that different instruments sound good together and that each instrument sounds reasonably good independent of which key it is played in.

12TET is characterized by (1) a doubling of the frequency for each octave and (2) a division of the octave in twelve equal steps (semitones). The first property is common to most tuning schemes. The second property means that a semitone step increases the frequency with a factor of

${2}^{1/12}=1.05946...$

that is, close to 6%. Changing the frequency with twelve semitones in succession results in an increase with a factor of
${(2^{1/12})}^{12} = 2,$

that is, a doubling of the original frequency. A scale consisting of all the semitones in sequence is called a chromatic scale.

Note that a musical interval (step) involves two notes whose (fundamental) frequencies stand to each other in a certain ratio, like two for the common octave or 1.059 for a 12TET semitone.

Each semitone is in turn divided in 100 equal steps called cent. Cents are useful when quantifying small frequency differences (or, more correctly, frequency ratios close to one). For example, a 1-Hz difference corresponds approximately to 4 cent at 440 Hz (but only half as much for each doubling of the frequency). Cents can also be used for larger intervals.

With the following formula one can calculate how many cent (n) a certain frequency (fn) deviates from a reference frequency (f0) (ln is the natural logarithm):

$n = \frac{1200}{\ln{2}} \cdot \ln{\frac{f_n}{f_0}}$

12TET came into general use during the latter half of the 18th century [Doty, 2002, p.5]and is currently the most common tuning scheme in the Western culture. However, other intonations were used in the past and are used today for various reasons. One drawback with 12TET is that chords are always somewhat dissonant. This can be heard as a considerable beating, particularly for the thirds, and some musicians find this distracting. With other tuning schemes one can avoid this problem but only for a limited set of keys at a time.

1.2 Just intonation

There are many alternatives to 12TET. When discussing tuning schemes one often distinguishes between just and tempered intonation. Just intonation (JI) “is any system of tuning in which all of the intervals can be represented by whole-number frequency ratios, with a strongly implied preference for the simplest ratios compatible with a given musical purpose” [Doty, 2002, p.1].

A justly tuned instrument is free from dissonances and sounds very smooth when intervals or chords are played. However, in practise it is difficult to construct fixed-pitch instruments (for example non-electronic keyboards, instruments with fretted strings or those with resonant reeds like the harmonica) that can be tuned to JI in all possible keys at the same time. Either, the instrument will only be in JI for a limited set of keys, or else one has to tune it using some sort of compromise.

Compromised tuning schemes with intervals that deviate from the ideal whole-number ratios are called tempered intonations or temperaments. 12TET is an example of such a compromise solution, meantone temperament is another, older, example. See [Hall, 1991], [Berg & Stork, 1995] and [Doty, 2002] for discussions of intonation and temperament.

A diatonic harmonica is primarily designed to be played in a limited set of keys. Therefore, it can very well be tuned to JI, and the purpose with the present article is to describe some alternative tuning schemes for diatonic harmonicas. The information here has been gathered from many sources (see the list of references at the end), in particular Pat Missin’s text Altered States [Missin, 1996]. I highly recommend visiting Pat Missin’s Web site and reading his explanations and advice.

1.3 An example of a seven-tone just major scale

As an example of JI, let us start with a 7-tone major scale that is designed so that the thirds and fifths of three major chords, the tonic (I), the dominant (V) and the subdominant (IV), in some key are tuned true (beatless) [Berg & Stork, 1995, p.248]. This particular scale will be referred to as a 7-tone just major (7TJM) scale in the following. The table below shows the names and frequency ratios for the notes in this scale and their deviation in cent from the corresponding notes in 12TET.

Table 1: A 7-tone just major scale
Note Name of intervall Frequency ratio Deviation from 12TET
rational decimal (cent)
1 Unison 1/1 1 0
2 Major whole tone 9/8 1.125 4
3 Major third 5/4 1.25 -14
4 Perfect fourth 4/3 1.333 -2
5 Perfect fifth 3/2 1.5 2
6 Major sixth 5/3 1.667 -16
7 Major seventh 15/8 1.875 -12
8 Octave 2/1 2 0

Note that all frequency ratios are built up by small integers (whole numbers). This is due to a close relation to the harmonic (overtone) series, but will not be dealt with further here (see [Berg & Stork, 1995, section 3.2] for a detailed discussion). Note also that all integers consist of the numbers 1, 2, 3 or 5, or products between these. These integers are the first few of the so-called prime numbers. Since the highest prime number in the above scale is 5, it is called a 5-limit major scale.

1.4 A numerical example

Before proceeding, let us work through an example to clarify the meaning of the numbers in Table 1.

Consider an A-major chord consisting of the notes A, C# and E. If we take A4 = 440 Hz as the root note, then, for the above 7TJM scale, C# (the major third) should have a (fundamental) frequency of (5/4)x(440 Hz) = 550 Hz. The frequency of E (the perfect fifth) should be (3/2)x(440 Hz) = 660 Hz.

However, in 12TET, C# is four equal semitones above A and should sound at 440 Hz times the factor

${(2^{1/12})}^4 = 2^{1/3} = 1.25992...$

(see section 1.1). This is 554.3 Hz, or approximately 4 Hz higher than the 7TJM major third. This deviation for the 7TJM value can be expressed in cents using the formula given in Section 1.1:
$n = \frac{1200}{\ln{2}} \cdot \ln{\frac{550}{554.3}} \approx -14$

Thus, the deviation corresponds to -14 cent, as given in the last column of Table 1. The minus sign means that the major third in the 7TJM scale is flatter (lower in frequency) than that in 12TET.

Doing a similar calculation for E (the fifth, which is seven semitones above the root) one finds that its frequency should be 659.2 Hz in 12TET, which is 2 cent below the 7TJM value. Or, in other words, the 7TJM perfect fifth is 2 cent sharper (higher in frequency) than the 12TET value, as also given in Table 1.

This example shows that there is a rather large discrepancy between the above 7TJM scale and 12TET when it comes to the major third (-14 cent). However, for the perfect fifth the deviation is not that large (2 cent). The deviations for the other notes are shown in Table 1.

2. Just intonation for major diatonic harmonicas

Consider the layout of a 10-hole major diatonic (Richter) harmonica (that is, a blues harp) played in 1st position (straight harp):

Table 2: Layout of Richter harmonica (1st position)
Channel 1 2 3 4 5 6 7 8 9 10
Blow note 1 3 5 1 3 5 1 3 5 1
Draw note 2 5 7 2 4 6 7 2 4 6

The numbers for the blow and draw notes refer to the intervals in Table 1 (first column). As can be seen, only the notes of the 7-tone major scale appear. Thus, if this were a C-harp, the only notes are those that correspond to the white keys of a piano. This means that these notes could very well be tuned to the 7TJM scale in Table 1 and the tonic blow chord would then sound very smooth (beatless).

Many harmonicas are factory-tuned to an intonation which has some resemblance to the above 7TJM scale, for example Hohner’s Marine Band and MS-series (see Section 3 for actual tuning schemes). Others are tuned to 12TET, for example Hohner’s Golden Melody and Lee Oskar’s major diatonic harmonicas.

Several blues-harp players claim that a JI tuning is required in order to get a big, fat tone. One argument in favor of this is that chord playing on a justly tuned harp often generates a difference tone equal to the root key of the harmonica, but two octaves down. And if one listens carefully to a blow chord on a Golden melody or Lee Oskar, there is a considerable dissonance (beating) which can not be heard from a (well-tuned) Marine Band or MS harp.

Before turning to actual tuning schemes, a word should be said about the reference frequency. As mentioned in the beginning, A4 = 440 Hz is a common standard. However, harmonicas are often tuned somewhat higher, for example to A4 = 443 Hz. This is because blowing (or drawing) hard tends to lower the frequency of all notes. Also, a slightly sharper tuning of one instrument will help it stand out over the accompanying instruments in a band [Baker, 1991, p.58; Oskar, 1996, p.13; Missin, 1996].

2.1 JI for 1st position only

Mainly as an example, we will now consider a tuning scheme that works good if one is playing in 1st position only. The following table combines the information in Tables 1 and 2 and prescribes how one should tune each note in order to get a 7TJM scale in 1st position on the harmonica.

Table 3: 5-limit JI for 1st position (deviations in cent from 12TET)
Channel 1 2 3 4 5 6 7 8 9 10
Blow note 0 -14 2 0 -14 2 0 -14 2 0
Draw note 4 2 -12 4 -2 -16 -12 4 -2 -16

For example, if this were a C-harp, one should use a tuner to set 1B (the blow note in channel 1) exactly to C4 (middle C), 2B to 14 cent below E4, 3B to 2 cent above G4, and so on. (The master tuning of the tuner should first be set to the desired reference frequency, for example A4 = 443 Hz.)

Since this tuning scheme is based on the 5-limit major scale in Table 1, a harmonica tuned accordingly can be said to be in 5-limit JI. As will be explained in the next subsections, this tuning is not good for playing chords in 2nd and 3rd position.

2.2 JI for 2nd position only

The following Table 4 shows the note layout for a blues harp played in 2nd position (cross harp). If the harmonica is tuned to 5-limit JI in 1st position, then some of the chords in 2nd position are not good: There are problems with the major whole tone (in 6D and 10D) and the flattened seventh (in 5D and 9D). However, all other intervals are in perfect JI in 2nd position [Missin, 1996].

Table 4: Layout of Richter harmonica (2nd position)
Channel 1 2 3 4 5 6 7 8 9 10
Blow note 4 6 1 4 6 1 4 6 1 4
Draw note 5 1 3 5 6# 2 3 5 6# 2

The note in 6D (and 10D) should be a major whole tone (9/8) above the root in 2nd position. Thus, it should be tuned at 27/16 = (3/2)x(9/8) above the 1st position root. This is 6 cents above 12TET. In 1st position the note in 6D (and 10D) is a major sixth (5/3) which is -16 cents from 12TET. Thus, if the harmonica is tuned to 5-limit JI in 1st position, this note will sound 22 cent flat in 2nd position. The workaround is to increase the major sixth from 5/3 to 27/16. This will make it sound good in 2nd position but 22 cent sharp in 1st position.

The note in 5D (and 9D) should be a flattened seventh in 2nd position. In order to get a chord (2-3-4-5 draw) without beating (dissonance), this note should be tuned to what is called 7-limit JI at 7/4 above the 2nd position root [Missin, 2003], which is also called an augmented sixth (6#) (or a subminor seventh). Thus, it should be tuned at 21/16 = (3/2)x(7/4)x(1/2) above the 1st position root. This is -29 cents from 12TET. In 1st position the note in 5D (and 9D) is a perfect fourth (4/3) which is -2 cents from 12TET. Thus, if the harmonica is tuned to 5-limit JI in 1st position, this note will correspond to a diminished seventh (bb7, 16/9) in 2nd position and will sound 27 cent sharper than the beatless augmented sixth. The workaround is to decrease the perfect fourth from 4/3 to 21/16. This will make it sound good in 2nd position but 27 cent flat in 1st position.

The above adjustments will result in the following tuning scheme (differences from Table 3 are highlighted in dark grey).

Table 5: 7-limit JI for 2nd position (deviations in cent from 12TET)
Channel 1 2 3 4 5 6 7 8 9 10
Blow note 0 -14 2 0 -14 2 0 -14 2 0
Draw note 4 2 -12 4 -29 6 -12 4 -29 6

The term 7-limit JI used above simply means that 7 is the highest prime number in the sequence of frequency ratios 4:5:6:7 that makes up the flattened seventh chord [Missin, 2003].

Note that this tuning scheme is specifically designed for playing in 2nd position. If it is used in 1st position, the note in 5D and 9D will be too flat (27 cent) and in 3rd position it will be even flatter (49 cent).

2.3 JI for 3rd position only

Tuning a harmonica to 5-limit JI in 1st position is not good for playing chords in 3rd position either (Table 6 shows the note layout). There are problems with the perfect fifth (in 6D and 10D) and the minor third (in 5D and 9D) as well as with the major whole tone (in 2B, 5B and 8B) and the flattened seventh (in 1B, 4B, 7B and 10B) The rest of the intervals are in perfect JI in 3rd position [Missin, 1996].

Table 6: Layout of Richter harmonica (3rd position)
Channel 1 2 3 4 5 6 7 8 9 10
Blow note b7 2 4 b7 2 4 b7 2 4 b7
Draw note 1 4 6 1 b3 5 6 1 b3 5

The note in 6D (and 10D) should be a perfect fifth (3/2) above the root in 3rd position. Thus, it should be tuned at 27/16 = (3/2)x(9/8) above the 1st position root. This is 6 cents above 12TET. In 1st position the note in 6D (and 10D) is a major sixth (5/3) which is -16 cents from 12TET. Thus, if the harmonica is tuned to 5-limit JI in 1st position, this note will sound 22 cent flat in 3rd pos. The workaround is to increase the major sixth from 5/3 to 27/16. This will make it sound good in 3rd position but 22 cent sharp in 1st position.

The note in 5D (and 9D) should be a minor third in 3rd position. In order to get a chord (4-5-6 draw) without beating (dissonance), it should be tuned at 6/5 above the 3rd position root. Thus, it should be tuned at 27/20 = (6/5)x(9/8) above the 1st position root. This is 20 cent above 12TET. In 1st position the note in 5D (and 9D) is a perfect fourth (4/3) which is -2 cents from 12TET. Thus, if the harmonica is tuned to 5-limit JI in 1st position, this note will sound 22 cent flat in 3rd position. The workaround is to increase the perfect fourth from 4/3 to 27/20. This will make it sound good in 3rd position but 22 cent sharp in 1st position.

The major whole tone and the flattened seventh should also be adjusted. Since these are blow notes they are not used in the tonic chord in 3rd position. However, they are part of a dominant minor seventh chord (eg 4-5-6 blow). But if all blow notes are tuned to 5-limit JI in 1st position, this chord will be beatless and sound good in 3rd position playing. We will therefore leave them here at their 1st position values.

The above adjustments will result in the following tuning scheme (differences from Table 3 are highlighted in dark grey).

Table 7: JI for 3rd position (deviations in cent from 12TET)
Channel 1 2 3 4 5 6 7 8 9 10
Blow note 0 -14 2 0 -14 2 0 -14 2 0
Draw note 4 2 -12 4 20 6 -12 4 20 6

Note that this tuning scheme is specifically designed for playing in 3rd position. If it is used in 1st position, the note in 5D and 9D will be too sharp (22 cent) and in 2nd position it will be even sharper (49 cent).

2.4 JI for both 2nd and 3rd position

As explained above, tuning a harmonica to 5-limit JI in 1st position is not good for playing chords in 2nd and 3rd position. The main problems are with the major sixth (in 6D and 10D) and the perfect fourth (in 5D and 9D) in 1st position. How should the harmonica be tuned if one wants it to sound good both in 2nd and 3rd position?

Fortunately, the same adjustment of the major sixth (increase with 22 cent to 6 cent above 12TET) will put this note in perfect JI for both 2nd and 3rd position where it corresponds to a major whole tone and a perfect fifth, respectively.

The perfect fourth is more problematic. In order to make a beatless flattened seventh chord in 2nd position it needs to be decreased with 27 cent. However, this will result in a very flat minor third in 3rd position. In fact, to get a beatless minor chord in 3rd position, this note should be adjusted in the other direction, that is, increased by 22 cent from the 1st position value. One alternative suggested by [Missin, 1996] is to leave this note unchanged at the 1st position perfect fourth. This means that the flattened seventh in 2nd position will sound a little too sharp while the minor third in 3rd position will sound a little too flat. The above adjustments will result in the following tuning scheme (differences from Table 3 are highlighted in dark grey).

Table 8A: JI compromise for 2nd and 3rd position
(deviations in cent from 12TET)
Channel 1 2 3 4 5 6 7 8 9 10
Blow note 0 -14 2 0 -14 2 0 -14 2 0
Draw note 4 2 -12 4 -2 6 -12 4 -2 6

Pat Missin’s personal favorite tuning differs slightly from the above for the note in 5D (and 9D). He tunes this to make a so called 19-limit minor triad in 4D, 5D and 6D. One reason for this is that the draw chord will then produce a difference tone equal to the root of the chord but two octaves down [Missin, 2004]. A 19-limit tuning means that the notes should be tuned in the ratios 16:19:24. Thus, the note in 5D should be (19/16)x(9/8) = 171/128 above the 1st position root in 4B. This is 1 cent above 12TET, which is only 3 cent higher than the value given in Table 8A. A similar adjustment of 9D results in the following tuning scheme (differences from Table 3 are highlighted in dark grey).

Table 8B: 19-limit JI for 2nd and 3rd position
(deviations in cent from 12TET)
Channel 1 2 3 4 5 6 7 8 9 10
Blow note 0 -14 2 0 -14 2 0 -14 2 0
Draw note 4 2 -12 4 1 6 -12 4 1 6

This is essentially the tuning scheme suggested by Pat Missin [Missin, 1996] and the one I use in my own tuning. The actual numbers in Table 8B differ slightly from those given by Missin since he includes a stretching of octaves with 2 cent/octave (see Section 5). I also do that but since it is handled automatically by the tuner I am using it is not included in the tuning schemes given here.

3. Other tuning schemes for major diatonic harmonicas

3.1 Hohner Classic Marine Band and Special 20

The Marine Bands were originally tuned to JI for the tonic blow chord as well as for the dominant seventh draw chord, that is, to the 7-limit JI shown in Table 5 [Epping, 1995, 2006]. In Epping’s original posting to Harp-L [Epping, 1995] the relative pitch of the note in 5D (and 9D) was given as -27 cent. This has later been corrected to -29 cent [Epping, 2006].

Over the years, the factory tuning of Marine Bands has drifted closer to 12TET, but due to complaints it has now been brought back closer to the original tuning. The following table shows the current tuning of the Classic Marine Band and Special 20 harmonicas [Epping, 1995, 2006]:

Table 9: Current tuning of Hohner Classic Marine Band
(deviations in cent from 12TET)
Channel 1 2 3 4 5 6 7 8 9 10
Blow note 0 -12 1 0 -12 1 0 -12 1 0
Draw note 2 1 -11 2 -12 3 -11 2 -12 3

3.2 Hohner MS-series

The Hohner MS models are tuned as the Marine Band was up to recently, see the following table [Epping, 1995, 2006].

Table 10: Tuning of Hohner MS Richter harmonicas
(deviations in cent from 12TET)
Channel 1 2 3 4 5 6 7 8 9 10
Blow note 0 -10 1 0 -10 1 0 -10 1 0
Draw note 2 1 -9 2 3 3 -9 2 3 3

Comparing this with the previous tuning schemes one sees that it is most similar to the 19-limit JI scheme in Table 8B. However, the major thirds (2B, 5B and 8B) and the major sevenths (3D and 7D) are somewhat sharper: -10 cent instead of -14 and -9 cent instead of -12, respectively.

3.3 Steve Baker

The following table shows Steve Baker’s tuning as given in the 2nd edition of his Harp Handbook [Baker, 1991, p.59]. Note that he gives deviations in Hz from 12TET in that edition, while Table 11A shows the values in cent [Baker, 2014]. Baker uses octave stretching which is inluded in the scheme. (I do not know if he recommends this tuning in the 3rd edition of his book.)

Table 11A: Steve Baker’s previous tuning of Richter harmonicas
(deviations in cent from 12TET)
Channel 1 2 3 4 5 6 7 8 9 10
Blow note 0 -8 0 0 -6 2 2 -4 6 6
Draw note 4 4 -4 4 4 4 -2 8 8 8

Baker moved away from the above tuning scheme about 20 years ago and uses instead the following scheme since then. This is pretty much the same as the intended tuning for Hohner MS and MB Crossover. Octave stretching is not included below, since most digital tuners now do this automatically. [Baker, 2014]

Table 11B: Steve Baker’s present tuning of Richter harmonicas
(deviations in cent from 12TET)
Channel 1 2 3 4 5 6 7 8 9 10
Blow note 0 -6 2 0 -6 2 0 -6 2 0
Draw note 6 4 -2 6 2 8 -2 6 2 8

3.4 Golden Melody, Lee Oskar

The following table shows the tuning scheme for a Richter harmonica tuned to 12TET such as the Hohner Golden Melody or Lee Oskar Major Diatonic. This is indeed a trivial case since none of the notes should deviate from 12TET, and it is only included here for completeness.

Table 12: 12TET for Richter harmonicas (deviations in cent from 12TET)
Channel 1 2 3 4 5 6 7 8 9 10
Blow note 0 0 0 0 0 0 0 0 0 0
Draw note 0 0 0 0 0 0 0 0 0 0

4. Just intonation for natural minor diatonic harmonicas

A 7-tone just minor (7TJm) scale differs from the 7TJM scale in Table 1 in that the third, sixth and seventh are lowered a semitone [Berg & Stork, 1995, p.249], see the following table. This is an example of a 5-limit minor scale since 5 is the highest prime number occurring in the frequency ratios.

Table 13: A 7-tone just minor scale
Note Name of intervall Frequency ratio Deviation from 12TET
rational decimal (cent)
1 Unison 1/1 1 0
2 Major whole tone 9/8 1.125 4
b3 Minor third 6/5 1.2 16
4 Perfect fourth 4/3 1.333 -2
5 Perfect fifth 3/2 1.5 2
b6 Minor sixth 8/5 1.6 14
b7 Minor seventh 9/5 1.8 18
8 Octave 2/1 2 0

4.1 Note layout

A natural minor harp as manufactured by Lee Oskar or Hohner has the following layout with respect to 1st position:

Table 14: Layout of natural minor harmonica (1st position)
Channel 1 2 3 4 5 6 7 8 9 10
Blow note 1 b3 5 1 b3 5 1 b3 5 1
Draw note 2 5 b7 2 4 6 b7 2 4 6

Note that the scale starting on 4B and ending on 7B is not a natural minor scale: The note in 6D (and 10D) should be a flattened sixth. These harps are designed to be played in 2nd position. Then a true natural minor scale is obtained (starting on 6B and ending on 9B) as can be seen from the following table.

Table 15: Layout of natural minor harmonica (2nd position)
Channel 1 2 3 4 5 6 7 8 9 10
Blow note 4 b6 1 4 b6 1 4 b6 1 4
Draw note 5 1 b3 5 b7 2 b3 5 b7 2

4.2 Tuning scheme

Combining the information in Tables 13 and 15 results in the following tuning scheme:

Table 16: 5-limit minor JI for natural minor 2nd position
(deviations in cent from 12TET)
Channel 1 2 3 4 5 6 7 8 9 10
Blow note -2 14 0 -2 14 0 -2 14 0 -2
Draw note 2 0 16 2 18 4 16 2 18 4

Note that Hohner labels their natural minor harps according to the root of the blow chord – usually the blow note in channel 1 [Epping, 1999b], while Lee Oskar labels them according to the root of the 2nd position draw chord – usually the draw note in channel 2.

5. Octave stretching

In addition to applying a certain intonation as exemplified above, it is often recommended that harmonicas should be tuned with octave stretching, for example 2 cent/octave [Baker, 1991, p.58; Missin, 1996]. This means that 2 cent are added (subtracted) for each increase (decrease) of the notes by an octave. One reason for the stretching is that most people prefer intervals to be slightly “wide” (especially as they get older), rather than slightly “narrow” [Missin, 1996].

As a comparison, pianos are usually tuned with an octave stretch of approximately 2 cent/octave, at least in the interval from A2 to C6. Outside this range the stretch is even more pronounced [Fletcher & Rossing, 1998, p. 389].

6. References

• Steve Baker (1991) The Harp Handbook, 2nd edition, Edition Louis, Ludwigsburg, Germany.
• Steve Baker (2014) Personal communication, 14 August 2014.
• Richard E. Berg and David G. Stork (1995) The Physics of Sound, 2nd edition, Prentice Hall, New Jersey.
• David B. Doty (2002) The Just Intonation Primer, 3rd edition, available from http://www.dbdoty.com.
• Rick Epping (1995) E-mail to Harp-L mailing list, 24 & 27 June 1995www.harp-l.org.
• Rick Epping (1999a) E-mail to Harp-L mailing list, 7 June 1999www.harp-l.org.
• Rick Epping (1999b) E-mail to Harp-L mailing list, 16 June 1999www.harp-l.org.
• Rick Epping (2006) Hohner’s Tunings for Major Richter Harmonicas,  www.angelfire.com/music/harmonica/hohnertuningsbyepping.html.
• Neville H. Fletcher and Thomas D. Rossing (1998) The Physics of Musical Instruments, 2nd edition, Springer, New York.
• Donald E. Hall (1991) Musical Acoustics, 2nd edition, Brooks/Cole, Pacific Grove, California.
• Pat Missin (1996) Altered States (updated in March 2004), www.patmissin.com/as23.zip.
• Pat Missin (2003) Ramblings on the general topic of Just Intonation, ratios and prime limitswww.patmissin.com/tunings/tun1.html.
• Pat Missin (2004) Tempered diatonic harmonicas, Picardy Thirds, Diminished Tunings and more ratios and limits, www.patmissin.com/tunings/tun3.html.
• Manuel Op de Coul (2001) Scala scale file format,  www.xs4all.nl/~huygensf/scala/scl_format.html.
• Lee Oskar (1996) The Art of Harmonica Maintenance, Lee Oskar Enterprises, Inc.

7. Revision history

2014-08-16

• Corrected Steve Baker’s tuning schemes in section 3.3 based on personal communication [Baker, 2014].

2012-09-03

• Another relocation of files.
• Rendered formulas with Easy WP LaTeX.

2012-08-18

• Changed the Web site to WordPress-based, which required changes of the layout and relocation of this and several other files.

2006-12-29

• Added Table 8B and revised the text in section 2.4 to put more emphasize on 19-limit JI.
• Corrected the text concerning the original tuning of Hohner Marine Bands (section 3.1) based on reference [Epping, 2006].

2006-11-19

• Updated file to XHTML 1.0 Transitional.

2006-06-26

• Major changes to the text in response to criticism raised by Mr Jonathan Ross on the Harp-L mailing list on 16 May 2006, www.harp-l.org. The criticism mainly involved my incorrect use of the words intonation and temperament, which now has been corrected throughout the text.
• Other mistakes have also been corrected.
• The terminology in the section on 12TET (1.1) has been clarified.
• A section on JI (1.2) has been added.
• The sections 1.2 and 1.3 have been renamed and renumbered to 1.3 and 1.4, respectively.
• The term 5-limit JI has been introduced in section 2.1.

2006-03-27

• Renumbered the sections.
• Added formula for calculating cent and an Example to section 1.
• Added info on octave stretching of pianos.
• Added reference [Fletcher & Rossing, 1998].

2006-01-05

• Changes to the layout.

2004-09-30

• Slight changes after input from readers.

2004-08-15

• First version.