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Musical notes...

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  Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-27 08:20 +0000
    Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-27 12:58 +0000
      Re: Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-27 18:06 +0000
        Re: Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-28 10:15 +0000
    Re: Musical notes... ram@zedat.fu-berlin.de (Stefan Ram) - 2026-05-27 16:25 +0000
      Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-27 17:10 +0000
    Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-28 12:30 +0000
      Re: Musical notes... HenHanna@NewsGrouper <user4055@newsgrouper.org.invalid> - 2026-05-28 18:21 +0000
      Re: Musical notes... msb@vex.net (Mark Brader) - 2026-05-28 18:50 +0000
      Re: Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-29 17:19 +0000
      Re: Musical notes... Charlie Roberts <croberts@gmail.com> - 2026-05-29 15:48 -0400
        Re: Musical notes... ram@zedat.fu-berlin.de (Stefan Ram) - 2026-05-29 20:59 +0000
          Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-30 12:07 +0000
            Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-30 12:25 +0000
              Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-05-30 12:33 +0000
              Re: Musical notes... ram@zedat.fu-berlin.de (Stefan Ram) - 2026-05-30 12:56 +0000
      Re: Musical notes... Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-05-29 22:38 +0100
        Re: Musical notes... David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-05-30 06:38 +0000
      Re: Musical notes... Phil Carmody <pc+usenet@asdf.org> - 2026-06-05 16:41 +0300
        Re: Musical notes... James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-06-05 14:54 +0000

#27857 — Musical notes...

I'm curious about the frequency (and wavelength) relationship between the 
notes of the musical scale, but don't know the subject well enough to 
write a coherent question. It would go something along the lines of "What 
sort of Lissajous curve would the following notes generate when applied, 
in equal amplitude, to the input of a Lissajous curve generating device?"

I appreciate the phase-relationship may cause a complication, so it may be 
better to stick to the numerical relationship of the pitch frequencies 
within the question.

Does anyone know this subject well enough to draft a meaningful question? 
If you do know it, can you comment on Western and other pitch 
relationships and build that into a follow-up question?

If there isn't anyone able to do this, I'll try and read up on the subject 
and put something together.

Best wishes from a warm and sunny UK.

-- 
David Entwistle

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#27858

David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> posted:

> I'm curious about the frequency (and wavelength) relationship between the 
> notes of the musical scale, but don't know the subject well enough to 
> write a coherent question. It would go something along the lines of "What 
> sort of Lissajous curve would the following notes generate when applied, 
> in equal amplitude, to the input of a Lissajous curve generating device?"

From this page:
   https://www.physics.wisc.edu/ingersollmuseum/exhibits/waves/lissajous/
Click on Lissajous Curve Plotter and select two notes (at upper right
and lower left) to see the resultant curve.  Select one of three scale
temperaments  at center right.

> Best wishes from a warm and sunny UK.

Weather is also good today here in Chiang Mai.

-----
James

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#27861

On Wed, 27 May 2026 12:58:15 GMT, James Dow Allen wrote:

> From this page:
>    https://www.physics.wisc.edu/ingersollmuseum/exhibits/waves/
lissajous/
> Click on Lissajous Curve Plotter and select two notes (at upper right
> and lower left) to see the resultant curve.  Select one of three scale
> temperaments  at center right.
> 

Thanks, that's an interesting presentation. From the landing page, with 
the default settings:

https://www.physics.wisc.edu/ingersollmuseum/exhibits/waves/lissajous/
plotter/

If I click on audio, to listen to the two inputs, then the y frequency: 
392 Hz (G4) alone sounds like a pure fundamental tone. Muting that and 
listening to x frequency: 262 Hz (C4), that sounds like two tones, I can 
hear a beat. I'm not musical enough to be able identify the tones, but it 
doesn't sound like a fundamental to me.

Does anyone else hear that?

>> Best wishes from a warm and sunny UK.
> 
> Weather is also good today here in Chiang Mai.

Nice. i keep thinking I should move away from the Uk. The winter weather 
is just terrible.

-- 
David Entwistle

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#27862

On Wed, 27 May 2026 18:06:04 -0000 (UTC), David Entwistle wrote:

> doesn't sound like a fundamental to me.
> 
> Does anyone else hear that?

Having spent a little more time listening to the audio, it may just be an 
effect of sinusoidal series sequence length and or quantization. It isn't 
terrible.


-- 
David Entwistle

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#27859

David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote or quoted:
>I'm curious about the frequency (and wavelength) relationship between the 
>notes of the musical scale,

  Here's a generated list of infos, I also added something about birds.

The Fundamentals of Pitch and Ratios

Musical notes are vibrations measured in Hertz (Hz).
The relationship between two notes is called an inter‐
val, which is expressed as a mathematical frequency
ratio.

-  Octaves double frequency: Two notes an octave apart
   always have a 2:1 frequency ratio.

-  Unisons are identical: A 1:1 ratio means two notes
   share the exact same frequency.

-  Integers create consonance: Small, simple integer
   ratios sound pleasant and stable to the human ear.

-  Complexity creates dissonance: Large, complex mathe‐
   matical ratios sound tense, unstable, and harsh.

-  Pitch is logarithmic: Human perception of pitch
   scales logarithmically, not linearly, relative to
   frequency.

The Mathematics of Major Intervals (Just Intonation)
In pure tuning systems based on natural harmonics,
intervals are defined by clean, uncompromised frac‐
tions.

-  Perfect Fifth (3:2): The most stable interval after
   the octave, vibrating three times for every two of
   the base note.

-  Perfect Fourth (4:3): The inversion of a fifth,
   vibrating four times for every three of the base
   note.

-  Major Third (5:4): The bright, foundational interval
   of a major triad, vibrating five times for every
   four.

-  Minor Third (6:5): The dark, foundational interval
   of a minor triad, vibrating six times for every
   five.

-  Major Sixth (5:3): A highly consonant interval,
   vibrating five times for every three of the funda‐
   mental.

-  Major Seventh (15:8): A highly tense, complex inter‐
   val that strongly desires to resolve upward to the
   octave.

Tuning Systems and Scale Construction

Over centuries, musicians developed different systems
to organize these frequency relationships into pre‐
dictable scales.

-  Pythagorean tuning builds fifths: This ancient sys‐
   tem calculates every note by stacking pure 3:2 per‐
   fect fifths.

-  The Pythagorean comma exists: Stacking 12 perfect
   fifths does not perfectly match 7 octaves, creating
   a small mathematical overshoot.

-  Just Intonation uses fractions: This system tunes
   all notes to clean, whole‐number ratios based on a
   single fundamental note.

-  Just Intonation limits modulation: A scale tuned
   perfectly to one key sounds completely out of tune
   in another key.

-  Equal Temperament compromises physics: Modern
   12‐Tone Equal Temperament (12‐TET) mathematically
   divides an octave into 12 identical steps.

-  12‐TET uses exponents: The frequency of each consec‐
   utive semitone is multiplied by the twelfth root of
   two.

-  Only octaves stay pure: In modern 12‐TET, every sin‐
   gle interval except the octave is slightly out of
   tune with natural physics.
   
-  Modern fifths are narrow: Equal‐tempered fifths are
   roughly 2 cents flat compared to a pure 3:2 ratio.
   
-  Modern thirds are sharp: Equal‐tempered major thirds
   are nearly 14 cents sharp compared to a pure 5:4
   ratio.

Harmonics and the Overtone Series

Frequency relationships within scales are derived
directly from the physics of vibrating strings and air
columns.

-  Fundamentals dictate the pitch: The lowest, loudest
   frequency of a sound determines its perceived musi‐
   cal note.
   
-  Overtones are integer multiples: A vibrating string
   simultaneously produces frequencies at 2x, 3x, 4x,
   and 5x the fundamental.
   
-  Scales mirror the series: The major scale naturally
   emerges from the upper partials of the overtone
   series.
   
-  Timbre relies on overtones: The relative volume of
   these different frequency relationships gives
   instruments their unique sound character.

Psychoacoustics and Perception

How human brains interpret the physical interaction of
these frequencies defines musical emotion and tension.

-  Critical bands cause roughness: When two frequencies
   are too close together, the ear cannot separate
   them, creating physical friction.

-  Beating measures close pitches: Two frequencies
   slightly out of tune create a physical pulsing sen‐
   sation called beating.

-  Beat speed equals difference: The number of beats
   heard per second is exactly equal to the difference
   in Hz between the two notes.

-  Combination tones appear naturally: When two loud
   notes are played, the brain naturally perceives a
   third "difference tone" (f2-f1).

The Hidden Scales of Birdsong

To human ears, birds may seem to chirp "between the
cracks" of our musical notes, but deep acoustic analy‐
sis reveals strict underlying patterns:

-  The Harmonic Series: Species like the Hermit Thrush
   explicitly select notes that follow a harmonic
   series governed by small‐integer ratios - the exact
   same mathematical distribution that builds human
   musical scales.

-  Pentatonic and Consonant Intervals: Many birds
   default to consonant intervals (pleasant combina‐
   tions) and pentatonic structures rather than random
   frequencies.

-  Absolute Pitch vs. Spectral Shape: Humans easily
   recognize a melody if it is transposed to a differ‐
   ent octave (relative pitch).  Songbirds recognize
   melodies by "spectral shape" (timbre and texture)
   rather than moving a melody up or down a scale.

Shared Rhythms at High Speeds

Avian rhythm is incredibly precise, though it can eas‐
ily overwhelm human perception:

-  Too Fast to Track: Birds can sing up to four times
   faster than human music.  When a 2‐second wren call
   is slowed down digitally, it unfolds into a beauti‐
   fully timed, rhythmically distinct composition.

-  Isochronous Beats: Many songbirds utilize
   isochronous rhythms, where the spacing between notes
   is perfectly equidistant - just like a steady
   metronome or a drum beat.

-  Categorical Rhythms: Research published in Current
   Biology shows that songbirds like the Thrush
   Nightingale cluster their notes into rhythmic cate‐
   gories shared by human musicians, using structured
   pacing to pass songs down through generations.

The Biomechanics of the Duet

Birds possess a vocal structure completely different
from our own.  Instead of a larynx, they use a syrinx
located at the base of their trachea.  Because this
organ splits into two bronchi, a songbird can produce
two different notes simultaneously, essentially singing
their own harmony or producing complex chords that
sound like a rapid blur to us.

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#27860

=============================================================
WARNING!  This is a very long, almost useless list.  WARNING!
  (I dump it here lest it be otherwise lost to posterity.)
=============================================================

Octave, perfect fifth, etc. are handy ratios when discussing music,
but the complexities of tuning mean that various esoteric ratios arise.
For example, the "Pythagorean comma" -- 1.0136432 or 531441/524288 --
is the ratio between C and B-sharp in Pythagoras' tuning.

I chose to sort these ratios in ascending numeric order but do
not otherwise recall details of this list's construction.

1.0000000 1/1 unison, perfect prime
1.0000430 23232/23231 lesser harmonisma
1.0000508 19657/19656 greater harmonisma
1.0000939 10648/10647 harmonisma
1.0001020 9801/9800 kalisma, Gauss' comma
1.0001689 450359962737049600/450283905890997363 monzisma
1.0001880 250047/250000 Landscape comma
1.0002286 4375/4374 ragisma
1.0002442 4096/4095 tridecimal schisma, Sagittal schismina
1.0003306 3025/3024 Lehmerisma
1.0004166 2401/2400 Breedsma
1.0004861 2058/2057 xenisma
1.0004979 7629394531250/7625597484987 ennealimmal comma
1.0005777 1732/1731 approximation to 1 cent
1.0007770 1288/1287 triaphonisma
1.0007999 274877906944/274658203125 semithirds comma
1.0008230 1216/1215 Eratosthenes' comma
1.0011291 32805/32768 schisma
1.0012714 1575/1573 Nicola
1.0013351 750/749 ancient Chinese tempering
1.0013580 65625/65536 horwell comma
1.0013654 2200/2197 Parizek comma
1.0014005 715/714 septendecimal bridge comma
1.0014814 676/675 island comma
1.0016276 19073486328125/19042491875328 '19-tone' comma
1.0017530 4000/3993 undecimal schisma
1.0018552 540/539 Swets' comma
1.0019299 6115295232/6103515625 Vishnu comma
1.0019531 513/512 undevicesimal comma, Boethius' comma
1.0020903 19383245667680019896796723/19342813113834066795298816 Mercator's comma
1.0021997 33554432/33480783 Beta 2, septimal schisma
1.0022727 441/440 Werckmeister's undecimal septenarian schisma
1.0026041 385/384 undecimal kleisma
1.0028490 352/351 minthma
1.0030613 1224440064/1220703125 parakleisma
1.0031020 6144/6125 porwell comma
1.0033313 5120/5103 Beta 5, Garibaldi comma
1.0035200 3136/3125 middle second comma
1.0035607 1600000/1594323 kleisma - schisma
1.0037494 10976/10935 hemimage
1.0038164 118098/117649 stearnsma
1.0039215 256/255 septendecimal kleisma
1.0040459 16875/16807 small BP diesis
1.0041322 243/242 neutral third comma
1.0042265 1188/1183 kestrel comma
1.0042754 390625/388962 dimcomp comma
1.0044642 225/224 septimal kleisma
1.0046939 15625/15552 kleisma, semicomma majeur
1.0047095 640/637 huntma
1.0048605 65536/65219 orgonisma
1.0048828 1029/1024 gamelan residue
1.0050551 2187/2176 septendecimal comma
1.0056116 896/891 undecimal semicomma
1.0057142 176/175 valinorsma
1.0058283 2109375/2097152 semicomma, Fokker's comma
1.0066329 393216/390625 W�rschmidt's comma
1.0069444 145/144 29th-partial chroma
1.0075801 1728/1715 Orwell comma
1.0077696 78732/78125 medium semicomma
1.0078105 4000/3969 septimal semicomma
1.0080000 126/125 small septimal comma
1.0082304 245/243 minor BP diesis
1.0083333 121/120 undecimal seconds comma
1.0084200 50421/50000 Trimyna
1.0093688 33075/32768 mirwomo comma
1.0096021 736/729 vicesimotertial comma
1.0096153 105/104 small tridecimal comma
1.0101010 100/99 Ptolemy's comma
1.0102040 99/98 small undecimal comma
1.0102173 67108864/66430125 Misty comma, diaschisma - schisma
1.0105263 96/95 19th-partial chroma
1.0111111 91/90 medium tridecimal comma
1.0113580 2048/2025 diaschisma
1.0115288 36893488147419103232/36472996377170786403 '41-tone' comma
1.0120783 2430/2401 nuwell comma
1.0123096 3125/3087 major BP diesis
1.0125000 81/80 syntonic comma, Didymus comma
1.0127314 875/864 keema
1.0131578 77/76 approximation to 53-tone comma
1.0136432 531441/524288 Pythagorean comma, ditonic comma
1.0156250 65/64 13th-partial chroma
1.0158730 64/63 septimal comma, Archytas' comma
1.0161052 20000/19683 minimal diesis
1.0162962 686/675 senga
1.0172526 3125/3072 small diesis
1.0184012 34171875/33554432 Ampersand's comma
1.0195312 261/256 vicesimononal comma
1.0200000 51/50 17th-partial chroma
1.0203667 1594323/1562500 Unicorn comma
1.0204081 50/49 Erlich's decatonic comma, tritonic diesis
1.0205761 248/243 tricesoprimal comma
1.0206414 15625/15309 great BP diesis
1.0208333 49/48 slendro diesis, septimal 1/6-tone
1.0215737 8192/8019 undecimal minor diesis
1.0220275 48828125/47775744 Sycamore comma
1.0222222 46/45 23rd-partial chroma
1.0227272 45/44 1/5-tone
1.0240000 128/125 minor diesis, diesis
1.0251562 6561/6400 Mathieu superdiesis
1.0253906 525/512 Avicenna enharmonic diesis
1.0256410 40/39 tridecimal minor diesis
1.0283203 1053/1024 tridecimal major diesis
1.0285714 36/35 septimal diesis, 1/4-tone
1.0288065 250/243 maximal diesis
1.0292887 246/239 Meshaqah's 1/4-tone
1.0294117 35/34 septendecimal 1/4-tone
1.0297328 59049/57344 Harrison's comma
1.0299682 16875/16384 double augmentation diesis
1.0312500 33/32 undecimal comma, al-Farabi's 1/4-tone
1.0322580 32/31 Greek enharmonic 1/4-tone
1.0333333 31/30 31st-partial chroma
1.0355113 729/704 undecimal major diesis
1.0368000 648/625 major diesis
1.0370370 28/27 Archytas' 1/3-tone
1.0384615 27/26 tridecimal comma
1.0393182 134217728/129140163 Pythagorean double diminished third
1.0400000 26/25 tridecimal 1/3-tone
1.0404917 20480/19683 grave minor second
1.0409948 17496/16807 septimal major diesis
1.0416666 25/24 classic chromatic semitone, minor chroma
1.0434782 24/23 vicesimotertial minor semitone
1.0448979 256/245 septimal minor semitone
1.0476190 22/21 undecimal minor semitone
1.0500000 21/20 minor semitone
1.0526315 20/19 small undevicesimal semitone
1.0534979 256/243 limma, Pythagorean minor second
1.0546875 135/128 major chroma, major limma
1.0555555 19/18 undevicesimal semitone
1.0578512 128/121 undecimal semitone
1.0581580 413343/390625 BP small link
1.0588235 18/17 Arabic lute index finger
1.0594600 52973/50000 Mersenne's quasi-equal semitone
1.0595238 89/84 quasi-equal semitone
1.0625000 17/16 17th harmonic
1.0664062 273/256 Ibn Sina's minor second
1.0666666 16/15 minor diatonic semitone
1.0668672 83349/78125 BP minor link
1.0678710 2187/2048 apotome
1.0711836 6561/6125 BP major link
1.0714285 15/14 major diatonic semitone
1.0769230 14/13 2/3-tone
1.0800000 27/25 large limma, BP small semitone
1.0824403 1162261467/1073741824 Pythagorean-19 comma
1.0833333 13/12 tridecimal 2/3-tone
1.0843696 18225/16807 minimal BP chroma
1.0864197 88/81 2nd undecimal neutral second
1.0872483 162/149 Persian neutral second
1.0880755 4608/4235 Arabic neutral second
1.0888888 49/45 BP minor semitone
1.0904977 241/221 Meshaqah's 3/4-tone
1.0909090 12/11 3/4-tone, undecimal neutral second
1.0922666 2048/1875 double diminished third
1.0932944 375/343 BP major semitone, minor BP chroma
1.0937500 35/32 septimal neutral second
1.0973936 800/729 grave whole tone
1.0986328 1125/1024 double augmented prime
1.1000000 11/10 4/5-tone, Ptolemy's second
1.1020408 54/49 Zalzal's mujannab
1.1022927 625/567 BP great semitone, major BP chroma
1.1034482 32/29 29th subharmonic
1.1098579 65536/59049 Pythagorean diminished third
1.1111111 10/9 minor whole tone
1.1113651 21875/19683 maximal BP chroma
1.1160714 125/112 classic augmented semitone
1.1176470 19/17 quasi-meantone
1.1193415 272/243 Persian whole tone
1.1200000 28/25 middle second
1.1224489 55/49 quasi-equal major second
1.1224637 134217728/119574225 whole tone - 2 schismas, 5-limit approximation to ET whole tone
1.1250000 9/8 major whole tone
1.1363636 25/22 undecimal acute whole tone
1.1377777 256/225 diminished third
1.1390625 729/640 acute major second
1.1403486 4782969/4194304 Pythagorean double augmented prime
1.1428571 8/7 septimal whole tone
1.1458333 55/48 undecimal semi-augmented whole tone
1.1520000 144/125 classic diminished third
1.1538461 15/13 tridecimal 5/4-tone
1.1550000 231/200 5/4-tone
1.1562500 37/32 37th harmonic
1.1571428 81/70 Al-Hwarizmi's lute middle finger
1.1574074 125/108 semi-augmented whole tone
1.1666666 7/6 septimal minor third
1.1692330 16777216/14348907 Pythagorean double diminished fourth
1.1705532 2560/2187 grave minor third
1.1718750 75/64 classic augmented second
1.1739130 27/23 vicesimotertial minor third
1.1764705 20/17 septendecimal augmented second
1.1785714 33/28 undecimal minor third
1.1818181 13/11 tridecimal minor third
1.1851851 32/27 Pythagorean minor third
1.1865234 1215/1024 wide augmented second
1.1875000 19/16 19th harmonic
1.1904761 25/21 BP second, quasi-tempered minor third
1.1911764 81/68 Persian wosta
1.2000000 6/5 minor third
1.2013549 19683/16384 Pythagorean augmented second
1.2136296 4096/3375 double diminished fourth
1.2142857 17/14 supraminor third
1.2150000 243/200 acute minor third
1.2187500 39/32 39th harmonic, Zalzal wosta of Ibn Sina
1.2190476 128/105 septimal neutral third
1.2203389 72/59 Ibn Sina's neutral third
1.2222222 11/9 undecimal neutral third
1.2240000 153/125 7/4-tone
1.2244897 60/49 smaller approximation to neutral third
1.2250000 49/40 larger approximation to neutral third
1.2272727 27/22 neutral third, Zalzal wosta of al-Farabi
1.2307692 16/13 tridecimal neutral third
1.2345679 100/81 grave major third
1.2352941 21/17 submajor third
1.2359619 10125/8192 double augmented second
1.2485901 8192/6561 Pythagorean diminished fourth
1.2500000 5/4 major third
1.2592592 34/27 septendecimal major third
1.2600000 63/50 quasi-equal major third
1.2631578 24/19 smaller undevicesimal major third
1.2641975 512/405 narrow diminished fourth
1.2656250 81/64 Pythagorean major third
1.2666666 19/15 undevicesimal ditone
1.2692307 33/26 tridecimal major third
1.2698412 80/63 wide major third
1.2727272 14/11 undecimal diminished fourth or major third
1.2777777 23/18 vicesimotertial major third
1.2800000 32/25 classic diminished fourth
1.2814453 6561/5120 acute major third
1.2828922 43046721/33554432 Pythagorean double augmented second
1.2857142 9/7 septimal major third, BP third
1.2962962 35/27 9/4-tone, septimal semi-diminished fourth
1.3000000 13/10 tridecimal semi-diminished fourth
1.3020833 125/96 classic augmented third
1.3061224 64/49 2 septatones or septatonic major third
1.3090909 72/55 undecimal semi-diminished fourth
1.3125000 21/16 narrow fourth
1.3153871 2097152/1594323 Pythagorean double diminished fifth
1.3168724 320/243 grave fourth
1.3183593 675/512 wide augmented third
1.3200000 33/25 2 pentatones
1.3333333 4/3 perfect fourth
1.3348388 10935/8192 fourth + schisma, 5-limit approximation to ET fourth
1.3500000 27/20 acute fourth
1.3515243 177147/131072 Pythagorean augmented third
1.3611111 49/36 Arabic lute acute fourth
1.3636363 15/11 undecimal augmented fourth
1.3653333 512/375 double diminished fifth
1.3714285 48/35 septimal semi-augmented fourth
1.3732910 5625/4096 double augmented third
1.3740000 687/500 11/4-tone
1.3750000 11/8 undecimal semi-augmented fourth
1.3846153 18/13 tridecimal augmented fourth
1.3857576 536870912/387420489 Pythagorean double diminished sixth
1.3888888 25/18 classic augmented fourth
1.3913043 32/23 23rd subharmonic
1.4000000 7/5 septimal or Huygens' tritone, BP fourth
1.4046639 1024/729 Pythagorean diminished fifth
1.4062500 45/32 diatonic tritone
1.4117647 24/17 1st septendecimal tritone
1.4141414 140/99 quasi-equal tritone
1.4142857 99/70 2nd quasi-equal tritone
1.4166666 17/12 2nd septendecimal tritone
1.4222222 64/45 2nd tritone
1.4238281 729/512 Pythagorean tritone
1.4285714 10/7 Euler's tritone
1.4375000 23/16 23rd harmonic
1.4400000 36/25 classic diminished fifth
1.4432537 387420489/268435456 Pythagorean double augmented third
1.4444444 13/9 tridecimal diminished fifth
1.4545454 16/11 undecimal semi-diminished fifth
1.4555555 131/90 13/4-tone
1.4563555 8192/5625 double diminished sixth
1.4583333 35/24 septimal semi-diminished fifth
1.4648437 375/256 double augmented fourth
1.4666666 22/15 undecimal diminished fifth
1.4693877 72/49 Arabic lute grave fifth
1.4798105 262144/177147 Pythagorean diminished sixth
1.4814814 40/27 grave fifth
1.4927113 512/343 3 septatones or septatonic fifth
1.4980000 749/500 ancient Chinese quasi-equal fifth
1.4983081 16384/10935 fifth - schisma, 5-limit approximation to ET fifth
1.5000000 3/2 perfect fifth
1.5151515 50/33 3 pentatones
1.5170370 1024/675 narrow diminished sixth
1.5187500 243/160 acute fifth
1.5204648 1594323/1048576 Pythagorean double augmented fourth
1.5238095 32/21 wide fifth
1.5277777 55/36 undecimal semi-augmented fifth
1.5306122 75/49 BP fifth
1.5360000 192/125 classic diminished sixth
1.5384615 20/13 tridecimal semi-augmented fifth
1.5423728 91/59 15/4-tone
1.5428571 54/35 septimal semi-augmented fifth
1.5555555 14/9 septimal minor sixth
1.5589773 67108864/43046721 Pythagorean double diminished seventh
1.5607376 10240/6561 grave minor sixth
1.5625000 25/16 classic augmented fifth
1.5714285 11/7 undecimal augmented fifth
1.5750000 63/40 narrow minor sixth
1.5757575 52/33 tridecimal minor sixth
1.5789473 30/19 smaller undevicesimal minor sixth
1.5802469 128/81 Pythagorean minor sixth
1.5820312 405/256 wide augmented fifth
1.5833333 19/12 undevicesimal minor sixth
1.5873015 100/63 quasi-equal minor sixth
1.5882352 27/17 septendecimal minor sixth
1.6000000 8/5 minor sixth
1.6018066 6561/4096 Pythagorean augmented fifth
1.6181728 16384/10125 double diminished seventh
1.6190476 34/21 supraminor sixth
1.6200000 81/50 acute minor sixth
1.6250000 13/8 tridecimal neutral sixth
1.6296296 44/27 neutral sixth
1.6326530 80/49 smaller approximation to neutral sixth
1.6333333 49/30 larger approximation to neutral sixth
1.6339869 250/153 17/4-tone
1.6363636 18/11 undecimal neutral sixth
1.6406250 105/64 septimal neutral sixth
1.6410256 64/39 39th subharmonic
1.6460905 400/243 grave major sixth
1.6470588 28/17 submajor sixth
1.6479492 3375/2048 double augmented fifth
1.6647868 32768/19683 Pythagorean diminished seventh
1.6666666 5/3 major sixth, BP sixth
1.6770186 270/161 Kirnberger's sixth
1.6800000 42/25 quasi-tempered major sixth
1.6842105 32/19 19th subharmonic
1.6855967 2048/1215 narrow diminished seventh
1.6875000 27/16 Pythagorean major sixth
1.6923076 22/13 tridecimal major sixth
1.7000000 17/10 septendecimal diminished seventh
1.7059558 4096/2401 4 septatones or septatonic major sixth
1.7066666 128/75 diminished seventh
1.7085937 2187/1280 acute major sixth
1.7105230 14348907/8388608 Pythagorean double augmented fifth
1.7142857 12/7 septimal major sixth
1.7280000 216/125 semi-augmented sixth
1.7297297 64/37 37th subharmonic
1.7311827 161/93 19/4-tone
1.7333333 26/15 tridecimal semi-augmented sixth
1.7361111 125/72 classic augmented sixth
1.7500000 7/4 harmonic seventh
1.7538495 8388608/4782969 Pythagorean double diminished octave
1.7558299 1280/729 grave minor seventh
1.7578125 225/128 augmented sixth
1.7600000 44/25 undecimal grave minor seventh
1.7777777 16/9 Pythagorean minor seventh
1.7818181 98/55 quasi-equal minor seventh
1.7857142 25/14 middle minor seventh
1.8000000 9/5 just minor seventh, BP seventh
1.8020324 59049/32768 Pythagorean augmented sixth
1.8125000 29/16 29th harmonic
1.8181818 20/11 large minor seventh
1.8204444 2048/1125 double diminished octave
1.8225000 729/400 acute minor seventh
1.8285714 64/35 septimal neutral seventh
1.8310546 1875/1024 double augmented sixth
1.8333333 11/6 21/4-tone, undecimal neutral seventh
1.8409090 81/44 2nd undecimal neutral seventh
1.8461538 24/13 tridecimal neutral seventh
1.8518518 50/27 grave major seventh
1.8571428 13/7 16/3-tone
1.8666666 28/15 grave major seventh
1.8728852 4096/2187 Pythagorean diminished octave
1.8750000 15/8 classic major seventh
1.8823529 32/17 17th subharmonic
1.8876404 168/89 quasi-equal major seventh
1.8888888 17/9 septendecimal major seventh
1.8947368 36/19 smaller undevicesimal major seventh
1.8962962 256/135 octave - major chroma
1.8984375 243/128 Pythagorean major seventh
1.9000000 19/10 undevicesimal major seventh
1.9047619 40/21 acute major seventh
1.9090909 21/11 undecimal major seventh
1.9166666 23/12 vicesimotertial major seventh
1.9200000 48/25 classic diminished octave
1.9221679 19683/10240 acute major seventh
1.9243383 129140163/67108864 Pythagorean double augmented sixth
1.9285714 27/14 septimal major seventh
1.9290123 625/324 octave - major diesis
1.9375000 31/16 31st harmonic
1.9393939 64/33 33rd subharmonic
1.9418074 32768/16875 octave - double augmentation diesis
1.9428571 68/35 23/4-tone
1.9440000 243/125 octave - maximal diesis
1.9444444 35/18 septimal semi-diminished octave
1.9496638 32768/16807 5 septatones or septatonic diminished octave
1.9531250 125/64 classic augmented seventh, octave - minor diesis
1.9600000 49/25 BP eighth
1.9660800 6144/3125 octave - small diesis
1.9683000 19683/10000 octave - minimal diesis
1.9687500 63/32 octave - septimal comma
1.9730807 1048576/531441 Pythagorean diminished ninth
1.9753086 160/81 octave - syntonic comma
1.9775390 2025/1024 2 tritones
1.9868214 390625/196608 octave - W�rschmidt's comma
1.9977442 65536/32805 octave - schisma
2.0000000 2/1 octave
2.0272865 531441/262144 Pythagorean augmented seventh
2.0833333 25/12 classic augmented octave
2.1250000 17/8 septendecimal minor ninth
2.1333333 32/15 minor ninth
2.1428571 15/7 septimal minor ninth, BP ninth
2.1648806 1162261467/536870912 Pythagorean double augmented seventh
2.2000000 11/5 neutral ninth
2.2222222 20/9 small ninth
2.2500000 9/4 major ninth
2.2857142 16/7 septimal major ninth
2.3333333 7/3 minimal tenth, BP tenth
2.5200000 63/25 quasi-equal major tenth, BP eleventh
2.7777777 25/9 classic augmented eleventh, BP twelfth

-----
James

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#27863

Show of hands, please.  How many of you view rec.puzzles in a
fixed-width font?  IIRC back in the 20th century, this newsgroup
had an over-riding default to fixed-width (Courier) because many
presentations relied on that.

 - - - - - - - - -

Music scales lead to interesting exercises in simple arithmetic.
Nothing new in this post, but it might be a starting point for those who've
never thought about the arithmetic of music intervals.

In the following diagram white keys on the piano are shown as AAA, BBB, etc.
Black keys are shown as ,,, or mmm.  The keys C major and A minor use
only white keys, but for ease of comparison we look at the key of C minor
where three black keys (E♭, A♭ and B♭) are used.

     CCC  ,,,  DDD  mmm  EEE  FFF  ,,,  GGG  mmm  AAA  mmm  BBB  CCC
      24  ...   27  28.8  30   32  ...   36  38.4  40  43.2  45   48
                        "3rd" "4th"    "5th"     "6th"     "7th"

Starting with C = 24 arbitrarily, the other white keys are simple integers:
Initially 24, 27, 30.  The numbers need to be sort of logarithmic,
so the whole step difference increases from 3 to 4: 32, 36, 40.
(The step from 30 to 32 is just a half step: There's no black key in between.)
Finally, the difference is 5 (40, 45) with the final 45, 48 another
half-step.

If you ever forget the simple rational numbers which define the Just Scale,
the above explanation for (24, 27, 30, 32, 36, 40, 45, 48) should
obviate the need for Google.  Unfortunately if you tune your piano
to produce the Just Scale for C Major, it won't be Just in any other key.

The black keys shown above for C Minor have ratios to C=24 that may look
uninviting (28.8/24 "minor 3rd", 38.4/24 "minor 6th", 43.2/24 "minor 7th")
but multiply throughout by 25/24 and get nice numbers: 6/5, 8/5, 9/5.
(In "principle."  Your instrument is very unlikely to be tuned this way.
6/5, 14/9, 16/9 is another solution for the black keys in C minor.)

The scales evolved, and some cultures used quite different scales.
Etc., etc.

James

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#27864

James Dow Allen <user4353@newsgrouper.org.invalid> posted:

> 
> Show of hands, please.  How many of you view rec.puzzles in a
> fixed-width font?  IIRC back in the 20th century, this newsgroup
> had an over-riding default to fixed-width (Courier) because many
> presentations relied on that.




https://rec.puzzles.narkive.com/

https://newsgrouper.org/rec.puzzles
https://csiph.com/group/rec.puzzles
       ------- BOth these sites let me switch easily between views.


> 
>  - - - - - - - - -
> 
> Music scales lead to interesting exercises in simple arithmetic.
> Nothing new in this post, but it might be a starting point for those who've
> never thought about the arithmetic of music intervals.
> 
> In the following diagram white keys on the piano are shown as AAA, BBB, etc.
> Black keys are shown as ,,, or mmm.  The keys C major and A minor use
> only white keys, but for ease of comparison we look at the key of C minor
> where three black keys (E♭, A♭ and B♭) are used.
> 
>      CCC  ,,,  DDD  mmm  EEE  FFF  ,,,  GGG  mmm  AAA  mmm  BBB  CCC
>       24  ...   27  28.8  30   32  ...   36  38.4  40  43.2  45   48
>                         "3rd" "4th"    "5th"     "6th"     "7th"
> 
> Starting with C = 24 arbitrarily, the other white keys are simple integers:
> Initially 24, 27, 30.  The numbers need to be sort of logarithmic,
> so the whole step difference increases from 3 to 4: 32, 36, 40.
> (The step from 30 to 32 is just a half step: There's no black key in between.)
> Finally, the difference is 5 (40, 45) with the final 45, 48 another
> half-step.
> 
> If you ever forget the simple rational numbers which define the Just Scale,
> the above explanation for (24, 27, 30, 32, 36, 40, 45, 48) should
> obviate the need for Google.  Unfortunately if you tune your piano
> to produce the Just Scale for C Major, it won't be Just in any other key.
> 
> The black keys shown above for C Minor have ratios to C=24 that may look
> uninviting (28.8/24 "minor 3rd", 38.4/24 "minor 6th", 43.2/24 "minor 7th")
> but multiply throughout by 25/24 and get nice numbers: 6/5, 8/5, 9/5.
> (In "principle."  Your instrument is very unlikely to be tuned this way.
> 6/5, 14/9, 16/9 is another solution for the black keys in C minor.)
> 
> The scales evolved, and some cultures used quite different scales.
> Etc., etc.
> 
> James

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#27865

James Dow Allen:
> Show of hands, please.  How many of you view rec.puzzles in a
> fixed-width font?

Of course, what else?  (Don't answer that.)
-- 
Mark Brader |"It's bad enough that this... font doesn't distinguish
Toronto     | between I and l, but I'd never had a problem with V before!"
msb@vex.net |                                       -- Steve Summit

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#27866

On Thu, 28 May 2026 12:30:54 GMT, James Dow Allen wrote:

> Show of hands, please.  How many of you view rec.puzzles in a
> fixed-width font?  IIRC back in the 20th century, this newsgroup had an
> over-riding default to fixed-width (Courier) because many presentations
> relied on that.

Yes, monospaced font used here for the last year, or so. I find USENET 
presents several challenges, and opportunities, with respect to 
presentation - emojis and the like.

-- 
David Entwistle

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#27867

On Thu, 28 May 2026 12:30:54 GMT, James Dow Allen
<user4353@newsgrouper.org.invalid> wrote:

>
>Show of hands, please.  How many of you view rec.puzzles in a
>fixed-width font?  IIRC back in the 20th century, this newsgroup
>had an over-riding default to fixed-width (Courier) because many
>presentations relied on that.

As a fossil, I cannot think of any other way to do such stuff unless
there is no option. I prefer it in all my personal documents (my
thesis was done that way, on an electric typewriter, equations
and all). 

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#27868

Charlie Roberts <croberts@gmail.com> wrote or quoted:
>As a fossil, I cannot think of any other way to do such stuff unless
>there is no option. I prefer it in all my personal documents (my
>thesis was done that way, on an electric typewriter, equations
>and all). 

  I'm working on a Python program in my leisure to typeset paragraphs
  that way.

  Example input:

An expression like $x^y$ is called a power.
You can imagine specific values for $x$ and $y$,
as in $3^2$ or $2^3$.

  Example output:

                    y                                            
An expression like x  is called a power. You can imagine specific
                           2     3 
values for x and y, as in 3  or 2 .

  However, for now, my program only supports digits 0-9, letters
  a-z, and exponentiation "^" in formulas. This is a large
  project with paragraph wrapping and document formatting that
  still needs a lot of work before it is ready for practical use.

  Also not ready for most uses is my ASCII-art plotter, but here's
  an example output of an ellipse:


                                     ....''--'..
                                  ..''         `.
                               ..''             \.
                             ..'                 '
                           ..'                   ..
                         ..'                     .'
                        .'                       .
                       .'                        .
                     ..                          .
                    .'                          .'
                   .'                          ..
                  .'                           .
                 .'                           .'
                .'                           .'
               .'                           ./
               .                           ./
              .'                          .'
              .                         ..'
             .                         ..
             .                        .'
             .                      ..'
             '                    ..'
             .                  ..'
             '.               ..'
              \.           ..''
               `..     ...''
                  '-'''

  The client code to draw this ellipse is:

def plot_ellipse(img, slope, W, H, pad_w, pad_h):
    cx = pad_w + W // 2
    cy = pad_h + H // 2
    a, b = W // 4, H // 3
    theta = math.radians(30)

    coords = []
    for k in range(720):
        t = 2 * math.pi * k / 720
        ex, ey = a * math.cos(t), b * math.sin(t)
        rx = ex * math.cos(theta) - ey * math.sin(theta)
        ry = ex * math.sin(theta) + ey * math.cos(theta)
        px, py = cx + rx, cy - ry
        pxi, pyi = int(round(px)), int(round(py))
        coords.append((pxi, px, pyi, py))
    render_shape(img, slope, coords)

  "render_shape" is the call that renders the ellipse in ASCII
  while taking into account the slope at every point and choosing an
  appropriate character depending on the "sub-pixel" (smaller than a
  char) position and the slope ("/" is only used when its slope is
  similar to the slope of the curve at this point, for example).

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#27874

ram@zedat.fu-berlin.de (Stefan Ram) posted:

>   I'm working on a Python program in my leisure to typeset paragraphs
>   that way.
> 
...
>   Example output:
>                      y                                            
> An expression like x  is called a power. You can imagine specific

This reminds me of implementing Thai script on a personal computer.
Thai has above-vowels and below-vowels; so the 24 physical lines
of a PC monitor were reduced to 8 lines of text to leave room for
such vowels (and tone-marks).  Some systems combined the above- and
below-vowels onto a single line and got 12 lines of Thai text; but this
required software that checked for collision (i.e. below line N and above
line N+1 both had characters) and inserted spaces as needed.

I was first to implement Thai text on Macintosh.  The above/below
problem disappeared but there was another major problem.  Mac had 8
"text modes" including TextOr and textCopy.  TextCopy effectively did an
erase which defeated Thai script:  I solved this problem with a complex
change to the text drawing firmware.

You ask "Wouldn't French and German have the same problem with their
umlauts, etc.?"  Yes ... IF they were implemented in the way Apple
so proudly bragged about.  They were NOT -- they just had separate font
characters for each combo, exactly as PC did it.  (Thai couldn't use
that approach: it has far more than 256 combos.)

Mike Terry <news.dead.person.stones@darjeeling.plus.com> posted:
> I don't see how a newsgroup can have an "over-riding default" to any particular font.  The choice of 
> font is made when displaying an article by each newsreader client.

I THINK I vaguely recall a reader that defaulted to Courier on rec.p
and maybe 1 or 2 other groups BUT my memory is failing fast.  It would have
been either 'news' on Unix or (less likely?) the "Groups" site once
supported by Do_Be_Evil_Inc.  I don't recall needing a click; maybe it
was an option in a company-wide '.newsrc' file, if there was such a thing.


Meanwhile:  Nobody got a giggle from my brief comment on the music scale?  :-{

-----
James

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#27875

Speaking of Unicode and and sub- and super-scripts, why are there
not Unicode characters for these?

I want to post a puzzle involving the molecule C6H12O6 but without
Stefan's subscript generator what am I to do?  Unicode has LOTS of
almost-useless characters; why no useful subscripts?

> 
> ram@zedat.fu-berlin.de (Stefan Ram) posted:
> 
> >   I'm working on a Python program in my leisure to typeset paragraphs
> >   that way.
> > 
> ...
> >   Example output:
> >                      y                                            
> > An expression like x  is called a power. You can imagine specific
> 

Speaking of Thai script and Unicode, if I were angry at Stefan
I might attack his post with above-vowels!  Let's see if it works:
   ส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋ส  คููููููููููููููููููููููููููููููู มึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึทมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึทมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึทมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึท

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#27876

James Dow Allen <user4353@newsgrouper.org.invalid> posted:

@David -- You just posted the Usenet rules.  Am I in violation if I try
to attack the header?  มึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึ      ดึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึ   ดึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึ 

> Speaking of Unicode and and sub- and super-scripts, why are there
> not Unicode characters for these?

Here a few of the useless Unicode digits:
       (C⑹H⑿O⑹)   (C⑥H⑫O⑥)   (C⓺H⑿O⓺) 
Why not simple subscripts?
> 
> > 
> > ram@zedat.fu-berlin.de (Stefan Ram) posted:
> > 
> > >   I'm working on a Python program in my leisure to typeset paragraphs
> > >   that way.
> > > 
> > ...
> > >   Example output:
> > >                      y                                            
> > > An expression like x  is called a power. You can imagine specific
> > 
> 
> Speaking of Thai script and Unicode, if I were angry at Stefan
> I might attack his post with above-vowels!  Let's see if it works:
>    ส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋สส๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋๋ส  คููููููููููููููููููููููููููููููู มึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึทมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึทมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึทมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึมึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึึท

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#27877

James Dow Allen <user4353@newsgrouper.org.invalid> wrote or quoted:
>Speaking of Thai script and Unicode, if I were angry at Stefan
>I might attack his post with above-vowels!  Let's see if it works:

  Well, you already "attacked" my post by quoting it with Newsgrouper
  which seems to "indent" quoted lines by different amounts:

|>                      y
|> An expression like x  is called a power. You can imagine specific

  . The "y" was indented by one more space than the "A", so the "y"
  is shifted right compared with my post. So, such newsreaders are
  not "monospaced-font-friendly".

  I was not able to see your "attack" effect in my usual newsreader,
  but pasting the Unicode text into the text field of a modern browser.
  This is the best way to read Unicode text, because modern browsers
  support more Unicode symbols than text editors or newsreaders.

<!DOCTYPE html>
<html lang="en">
<head>
    <meta charset="UTF-8">
    <title>Unicode Textfield</title>
</head>
<body>
    <textarea rows="10" cols="50"></textarea>
</body>
</html>

  But since the subject is "musical notes", from a 2025 post of mine,


|            | 
|            |                                             |
|         | O-  |                                          |
|         |     |          |                         |  OOO
|       -O---- O----   ----| -----   -------------| -| -------|
|                   |   | O   |                |  | O         |
|  _    ------------|  -|-----|---   -------| -| O--------- OO
| . .            OOO   O     O           |  | O    
| . .   ------------   -----------|  -| -| O------------------
| .-.                             |   | O      
|| g |  ------------   --------OOO   O------------------------
| ._.                
|  |    ------------   -----------   -------------------------
|   | 
| ''

  .

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#27869

On 28/05/2026 13:30, James Dow Allen wrote:
> 
> Show of hands, please.  How many of you view rec.puzzles in a
> fixed-width font?  IIRC back in the 20th century, this newsgroup
> had an over-riding default to fixed-width (Courier) because many
> presentations relied on that.
> 

I've always used fixed-width fonts for Usenet and Email

I don't see how a newsgroup can have an "over-riding default" to any particular font.  The choice of 
font is made when displaying an article by each newsreader client.  Also, when posting an article, 
the article headers are set by a combination of the newsreader client and the receiving NNTP server, 
neither of which is under control of the rec.puzzles newsgroup.  The best a group can do is present 
recommended practices for users of the group.

If I post e.g. a table relying on fixed spacing, I sometimes add a note [view with fixed-width font] 
or similar.

Mike.

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#27870

On Fri, 29 May 2026 22:38:28 +0100, Mike Terry wrote:

> The best a group can do is present recommended practices for users of
> the group.

Yes, indeed. Does rec.puzzles have a charter?

I'm sure most, if not all, readers of rec.puzzles are aware of the 
following documents. These documents don't cover fonts, but do set guiding 
principles for the use of USENET. I believe some of that guidance was 
forgotten with the introduction of USENET web interfaces like Yahoo Groups 
and then Google Groups. The RFCs are shining examples of technical 
documents written with clarity and simplicity.

RFC 1036: Standard for interchange of USENET messages:

https://www.rfc-editor.org/info/rfc1036/

Usenet Organization and Etiquette:

https://smfr.org/mtnw/docs/Usenet.html

Anyhow, back to puzzling...

-- 
David Entwistle

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#27911

James Dow Allen <user4353@newsgrouper.org.invalid> writes:
> Show of hands, please.  How many of you view rec.puzzles in a
> fixed-width font?  IIRC back in the 20th century, this newsgroup
> had an over-riding default to fixed-width (Courier) because many
> presentations relied on that.

Never anything but. Through familiarity, some of my favourite fonts
are fixed-width ones, so anything else would now uglify usenet for
me. Lose-lose.

Phil
-- 
We are no longer hunters and nomads. No longer awed and frightened, as we have
gained some understanding of the world in which we live. As such, we can cast
aside childish remnants from the dawn of our civilization.
-- NotSanguine on SoylentNews, after Eugen Weber in /The Western Tradition/

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#27913

Phil Carmody <pc+usenet@asdf.org> posted:

> James Dow Allen <user4353@newsgrouper.org.invalid> writes:
> > Show of hands, please.  How many of you view rec.puzzles in a
> > fixed-width font?  IIRC back in the 20th century, this newsgroup
> > had an over-riding default to fixed-width (Courier) because many
> > presentations relied on that.
> 
> Never anything but. Through familiarity, some of my favourite fonts
> are fixed-width ones, so anything else would now uglify usenet for
> me. Lose-lose.
> 
> Phil

I am delighted to read the strong support for fixed-width fonts
shown by most rec.puzzlers.

When I was working at Milošević Labs (not its real name) I eavesdropped
on a memo from Mr. Pattern_Recognition Himself complaining about
"James and his Courier font"!

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