(34C) Musical Notes (Original by J.M. Bowsher)

[Hans Brueggemann]

one of my favourite programs for the HP-34C is the "Calculator Program For Musical Notes", originally written for the HP-29C by J.M. Bowsher, published in the "Journal of The Audio Engineering Society", 1980 June, Volume 28, Number 6, Page 437.
the program lets you convert tonal frequencies into their corresponding musical notation and vice versa, by simply entering either the frequency in Hz or a musical notation in the form of an integer number that represents the notation in the following way:

Code:


C    0
C#    1
D    2
D#    3
E    4
F    5
F#    6
G    7
Ab    8
A    9
Bb    10
B    11

so that for example 440 Hz gets converted to 9.400, which is exactly the 4th octave of A. in the same way, slightly "off" tones like 441 Hz are converted to 9.404, which is again the 4th octave of A, but high by 4c (or "cents").

this is the transcript of Bowsher's program for the HP-34C:

Code:


001    LBL    0
002    R/S    
003    LBL    A
004    1    
005    5    
006    .    
007    8    
008    8    
009    6    
010    0    
011    9    
012    9    
013    5    
014    8    
015    STO    8
016    X<>Y    
017    EEX    
018    1    
019    5    
020    CHS    
021    +    
022    STO    1
023    ABS    
024    STO    2
025    1    
026    6    
027    2    
028    6    
029    4    
030    X<>Y    
031    X>Y    
032    GTO    9
033    1    
034    1    
035    .    
036    9    
037    4    
038    9    
039    5    
040    X<>Y    
041    X>Y    
042    GTO    1
043    FIX    1
044    INT    
045    EEX    
046    2    
047    *    
048    STO    0
049    RCL    2
050    FRAC    
051    EEX    
052    1    
053    *    
054    ENTER    
055    INT    
056    1    
057    2    
058    0    
059    0    
060    *    
061    STO+    0
062    X<>Y    
063    FRAC    
064    EEX    
065    2    
066    *    
067    5    
068    0    
069    STO+    0
070    X<>Y    
071    X>Y    
072    GTO    9
073    RCL    1
074    RCL    2
075    /    
076    *    
077    STO+    0
078    RCL    0
079    1    
080    2    
081    0    
082    0    
083    /    
084    2    
085    LN    
086    *    
087    e^x    
088    RCL    8
089    *    
090    GTO    0
091    LBL    1
092    FIX    3
093    RCL    1
094    RCL    8
095    /    
096    LN    
097    X<0    
098    GTO    9
099    2    
100    LN    
101    /    
102    ENTER    
103    INT    
104    1    
105    0    
106    /    
107    STO    0
108    X<>Y    
109    FRAC    
110    1    
111    2    
112    *    
113    ENTER    
114    INT    
115    STO+    0
116    X<>Y    
117    FRAC    
118    EEX    
119    2    
120    *    
121    5    
122    0    
123    -    
124    EEX    
125    3    
126    /    
127    ENTER    
128    ABS    
129    STO+    0
130    /    
131    RCL    0
132    5    
133    EEX    
134    CHS    
135    4    
136    X>Y    
137    GTO    2
138    RDN    
139    *    
140    GTO    0
141    LBL    2
142    CLX

enjoy!

[Dieter]

(02-09-2019 12:11 PM)Hans Brueggemann Wrote:  the program lets you convert tonal frequencies into their corresponding musical notation and vice versa, by simply entering either the frequency in Hz or a musical notation in the form of an integer number that represents the notation in the following way:

Fine. But how does it work? There are no instructions. How does the program determine which way the conversion between tone and frequency is done?
Could you please explain how the program is supposed to be used, and maybe you can also give some examples.

I also wonder what all the constants in the program are for. Can you say a bit about the way the program works? I have tried a similar conversion in Excel, and here only a single reference frequency is required.

Dieter

[Thomas Klemm]

(02-10-2019 01:26 PM)Dieter Wrote:  Fine. But how does it work? There are no instructions. How does the program determine which way the conversion between tone and frequency is done?

Input smaller than 11.9495 is considered a musical notation.
A frequency higher than 16264 is considered invalid.

Quote:Could you please explain how the program is supposed to be used, and maybe you can also give some examples.

Examples:

441 A
9.404
R/S
441.0

Quote:I also wonder what all the constants in the program are for.

\(15.88609958 = \frac{440}{2^\frac{115}{24}}\)

Quote:Can you say a bit about the way the program works?

These two programs for the HP-42S may miss the fancy output but are probably easier to follow.

Frequency → Note

Code:

00 { 30-Byte Prgm }
01▸LBL "F→N"
02 440
03 ÷
04 LN
05 2
06 LN
07 ÷
08 4.75
09 +
10 IP
11 LASTX
12 FP
13 12
14 ×
15 END

Examples:

450
XEQ "F→N"

y: 4.00
x: 9.39

A₄ + 39¢


4698.63
XEQ "F→N"

y: 8.00
x: 2.00

D₈

Note → Frequency

Code:

00 { 27-Byte Prgm }
01▸LBL "N→F"
02 12
03 ÷
04 +
05 4.75
06 -
07 2
08 X<>Y
09 Y↑X
10 440
11 ×
12 END

Example:

G₆

6
ENTER
7
XEQ "N→F"

x: 1567.98

Cheers
Thomas

[Dieter]

(02-10-2019 03:28 PM)Thomas Klemm Wrote:  Input smaller than 11.9495 is considered a musical notation.
A frequency higher than 16264 is considered invalid.

OK, but the 34C has two label keys A and B so that this kind of trick is not required.

(02-10-2019 03:28 PM)Thomas Klemm Wrote:  \(15.88609958 = \frac{440}{2^\frac{115}{24}}\)

I now have a program for the HP67 that is based on a slightly different constant and returns the output in the same way as the original program. Both conversions fit into not much more than 67 steps.

450     [A] => 9,439
4698,63 [A] => 2,800

7,6 [‌B] => 1568,0

Changing the input/output the way you did allows even more compact code, as you have shown. I think it even makes more sense to have the octave and then the tone plus cents. So instead of 9,439 I think that 4 and 9,39 are more intuitive.

Edit:
Here is an HP-34C version of your program. Since there is no two-line display the octave is displayed with "PAUSE". In either case the octave is returned/entered in Y and the note in X.

Code:

001 LBL A
002 FIX 0
003 4
004 4
005 0
006 /
007 LN
008 2
009 LN
010 /
011 4
012 .
013 7
014 5
015 +
016 INT
017 PAUSE
018 LastX
019 FRAC
020 1
021 2
022 *
023 FIX 2
024 RTN
025 LBL B
026 FIX 1
027 1
028 2
029 /
030 +
031 4
032 .
033 7
034 5
035 -
036 2
037 X<>Y
038 Y^X
039 4
040 4
041 0
042 *
043 RTN

450 [A] => "4"  9,39
6 [ENTER] 7 [‌B] => 1568,0

Dieter

[Thomas Klemm]

We could also use 55 with 1.75 instead of 440 with 4.75.
Which makes it a bit shorter.
Or then only use:

\(16.3515978313=\frac{440}{2^{4.75}}\)

Similar to the original program.
I must admit that I still don't really understand why it shifts by an additional 50¢.

Interestingly a search for just 15.88609958 using Google brings up a link to JAES Volume 28 Issue 6; June 1980.

This is in accordance with:

(02-09-2019 12:11 PM)Hans Brueggemann Wrote:  published in the "Journal of The Audio Engineering Society", 1980 June, Volume 28, Number 6, Page 437.

But unfortunately that paper is hidden behind a paywall:

Quote:This paper costs $33 for non-members and is free for AES members and E-Library subscribers.

Cheers
Thomas

[Dieter]

(02-10-2019 06:15 PM)Thomas Klemm Wrote:  We could also use 55 with 1.75 instead of 440 with 4.75.
Which makes it a bit shorter.
Or then only use:

\(16.3515978313=\frac{440}{2^{4.75}}\)

Similar to the original program.

That's what I used for the mentioned HP67 program: 55/4 · √√2.

(02-10-2019 06:15 PM)Thomas Klemm Wrote:  I must admit that I still don't really understand why it shifts by an additional 50¢.

Take a look at line 67 and 121 ff. respectively of the original program. This seems to add or subtract 50 cents.

Other calculations still remain unclear to me. For instance line 017 ff. Here a tiny value of 1E–15 is added. ?!? Then both the sum and its absolute value are stored separately. Can this be negative at all? Later both (073 ff.) values are divided, so this returns the sign of that number. And there is a second occurence of a sign calculation starting at line 127. What is the purpose of that? I also wonder why line 084 ff. calculate e^x · ln 2 instead of a simple 2^x. Maybe Hans can shed some light on this.

Dieter

[Thomas Klemm]

(02-11-2019 07:49 PM)Dieter Wrote:  That's what I used for the mentioned HP67 program: 55/4 · √√2.

With the HP-42S using this expression is one byte smaller:

\(\frac{55}{\sqrt{\sqrt{128}}}\)

Code:

00 { 12-Byte Prgm }
01 55
02 ×
03 4
04 ÷
05 2
06 SQRT
07 SQRT
08 ×
09 END

Code:

00 { 11-Byte Prgm }
01 55
02 ×
03 128
04 SQRT
05 SQRT
06 ÷
07 END

This leads to the following programs:

Code:

00 { 30-Byte Prgm }
01▸LBL "F→N"
02 55
03 ÷
04 128
05 SQRT
06 SQRT
07 ×
08 LN
09 2
10 LN
11 ÷
12 IP
13 LASTX
14 FP
15 12
16 ×
17 END

Code:

00 { 27-Byte Prgm }
01▸LBL "N→F"
02 12
03 ÷
04 +
05 2
06 X<>Y
07 Y↑X
08 55
09 ×
10 128
11 SQRT
12 SQRT
13 ÷
14 END

Or then you simply store that value in a register:

55
ENTER
128
SQRT
SQRT
÷
STO 00
16.3515978313


And end up with:

Code:

00 { 22-Byte Prgm }
01▸LBL "F→N"
02 RCL÷ 00
03 LN
04 2
05 LN
06 ÷
07 IP
08 LASTX
09 FP
10 12
11 ×
12 END

Code:

00 { 19-Byte Prgm }
01▸LBL "N→F"
02 12
03 ÷
04 +
05 2
06 X<>Y
07 Y↑X
08 RCL× 00
09 END

Quote:Take a look at line 67 and 121 ff. respectively of the original program. This seems to add or subtract 50 cents.

This led me to the definition of the constant 15.88609958 in the first place.
Still I don't grok the benefit of doing it.

Quote:Maybe Hans can shed some light on this.

1+


Cheers
Thomas

[Thomas Klemm]

If we don't care about using registers we can use the exponential regression model with:

EXPF
CLΣ
55
ENTER
1.75
Σ+
440
ENTER
4.75
Σ+


And then let the forecast functions do the heavy lifting:

Code:

00 { 16-Byte Prgm }
01▸LBL "F→N"
02 FCSTX
03 IP
04 LASTX
05 FP
06 12
07 ×
08 END

Code:

00 { 14-Byte Prgm }
01▸LBL "N→F"
02 12
03 ÷
04 +
05 FCSTY
06 END

Cheers
Thomas

[Thomas Klemm]

(02-11-2019 10:26 PM)Thomas Klemm Wrote:  Still I don't grok the benefit of doing it.

For those of you who want to give the HP-29C a try here's the original program:

Data

Code:

00: 1200
01: 1731.234049
02: 15.88609958
03: 50
04: 100
05: 11.9495
06: 16264
07: 10
08: 0.0005
09: 1e-15

Program

Code:

01: 15 13 00    : LBL 0
02: 74          : R/S
03: 24 09       : RCL 9
04: 51          : +
05: 23 .1       : STO .1
06: 15 64       : ABS
07: 23 .2       : STO .2
08: 24 06       : RCL 6
09: 21          : x<>y
10: 14 51       : x>y
11: 13 22       : GTO i
12: 24 05       : RCL 5
13: 21          : x<>y
14: 14 51       : x>y
15: 13 01       : GTO 1
16: 14 11 01    : FIX 1
17: 14 62       : INT
18: 24 04       : RCL 4
19: 61          : ×
20: 23 .0       : STO .0
21: 24 .2       : RCL .2
22: 15 62       : FRAC
23: 24 07       : RCL 7
24: 61          : ×
25: 31          : ENTER
26: 14 62       : INT
27: 24 00       : RCL 0
28: 61          : ×
29: 23 51 .0    : STO + .0
30: 21          : x<>y
31: 15 62       : FRAC
32: 24 04       : RCL 4
33: 61          : ×
34: 24 03       : RCL 3
35: 23 51 .0    : STO + .0
36: 21          : x<>y
37: 14 51       : x>y
38: 13 22       : GTO i
39: 24 .1       : RCL .1
40: 24 .2       : RCL .2
41: 71          : ÷
42: 61          : ×
43: 23 51 .0    : STO + .0
44: 24 .0       : RCL .0
45: 24 01       : RCL 1
46: 71          : ÷
47: 15 42       : eˣ
48: 24 02       : RCL 2
49: 61          : ×
50: 13 00       : GTO 0
51: 15 13 01    : LBL 1
52: 14 11 03    : FIX 3
53: 24 .1       : RCL .1
54: 24 02       : RCL 2
55: 71          : ÷
56: 14 42       : LN
57: 15 41       : x<0
58: 13 22       : GTO i
59: 24 01       : RCL 1
60: 61          : ×
61: 24 00       : RCL 0
62: 71          : ÷
63: 31          : ENTER
64: 14 62       : INT
65: 24 07       : RCL 7
66: 71          : ÷
67: 23 .0       : STO .0
68: 21          : x<>y
69: 15 62       : FRAC
70: 24 00       : RCL 0
71: 61          : ×
72: 24 04       : RCL 4
73: 71          : ÷
74: 31          : ENTER
75: 14 62       : INT
76: 23 51 .0    : STO + .0
77: 21          : x<>y
78: 15 62       : FRAC
79: 24 04       : RCL 4
80: 61          : ×
81: 24 03       : RCL 3
82: 41          : –
83: 33          : EEX
84: 03          : 3
85: 71          : ÷
86: 31          : ENTER
87: 15 64       : ABS
88: 23 51 .0    : STO + .0
89: 71          : ÷
90: 24 .0       : RCL .0
91: 24 08       : RCL 8
92: 14 51       : x>y
93: 13 02       : GTO 2
94: 22          : R↓
95: 61          : ×
96: 13 00       : GTO 0
97: 15 13 02    : LBL 2
98: 34          : CLx

Examples

Code:

   440                            9.400   (A4 exactly)
   441                            9.404   (A4 + 4 ¢)
   439                          – 9.404   (A4 – 4 ¢)
   123                          –11.207   (B2 – 7 ¢)
    50                            7.135   (G1 + 35 ¢)
    60                          –11.149   (B1 – 49 ¢)
 0.400   (C4 exactly)             261.6
10.400   (Bb4 exactly)            466.2
–4.321   (E3 – 21 ¢)              162.8
 4.321   (E3 + 21 ¢)              166.8
14.0     (?)                    Error
 0.451   (C4 + 51 ¢)            Error

Thanks to the examples I understand now that the difference in cents to a pure note must be between –50¢ and 50¢.
A negative value is indicated by a negative sign.

Though I can see that this is a reasonable convention I'm still not convinced to enter the data in this peculiar way.
Furthermore I get -9.400 for the 440Hz of the first example.

Quote:We are left with the problem of displaying the number of cents by which the note is sharper or flatter than the nominal note of the scale. It would be most elegant to use the exponential part of the display to carry this information. The author has studied this problem but has concluded, regretfully, that this is impossible to execute on a reasonably simple calculator, as a great number of steps are required to handle the computations. Thus we are forced to use the next two digits of the display to handle the number of cents.

The real problem is to distinguish for instance B0 + 23¢ from C#1 + 24¢.
Both entries 11.0e23 and 1.1e24 lead to the same number:

1.1        24


Cheers
Thomas