Aaron Freed – Āleātōrium semper vincit

download/stream this album

download/stream this album

⟨aaronfreed.github.io/aleatoriumsempervincit.html⟩

Context

Āleātōrium semper vincit is Latin for The House Always Wins (or, literally, The Casino Always Wins). As is often the case, I’ve chosen a rather topical album title.

If you’ve glanced at my website lately, you probably know I’m a huge baseball fan. And if you’ve paid even cursory attention to sports lately, you certainly know the cancer of sports betting is now ubiquitous. No matter how well you know sports, reality is too stochastic to follow predictable patterns. A pitcher can give up two home runs (or fewer) in a half-season, then two in a single inning.⁽¹⁾ Perhaps he was tipping his pitches, or perhaps, as they say, baseball happens. The sports betting industry wants you to think you can beat the odds. But ultimately, the house always wins.

My previous album, Τὸν ὦνον πάντων καί τὴν ἀξίᾱν οὐδένων ⟨aaronfreed.github.io/tonononpanton.html⟩ (Attic Greek: The Price of Everything and the Value of Nothing), is over two hours long, and I’d put so much work into the Rubicon X soundtrack ⟨aaronfreed.github.io/rubiconxost.html⟩ that afterwards, I took a few months off from writing music. Thus, it felt like an appropriate time to start a new album. Two early tracks on this album were written for Rubicon X at almost the last minute (and thus after most of Τὸν ὦνον πάντων), but the rest of it will be new material.

Table of Contents

  1. Context
  2. Table of Contents
  3. Translations
  4. Track Commentary
    1. “Regressiō ad medium” (8:00) [DR12]
    2. «Ῥοδοδάκτυλος Ἠώς» (3:36) [DR12]
    3. «Ή αἰγίς τῆς Ἱππολύτης» (6:56) [DR11]
    4. «Λᾱ́ζᾱρος ἐκ μηχανῆς» (4:44) [DR10]
    5. 「折り紙のユニコーン」 (10:30) [DR13]
    Current total: 33:46 [DR12]
  5. Rights Notice
  6. Related Links
  7. Endnotes

Translations

Title Translations, or, “What Language Even Is That?” (Greek, Japanese, Latin Spanish)
#/Language Title Romanization (if applicable) Meaning
Latin Āleātōrium semper vincit The House Always Wins
1 Latin Regressiō ad medium Regression to the Mean
1.α Attic Greek Συμπτώματα Symptṓmata Symptoms
1.β Attic Greek Θεραπεία Therapeía Therapy
1.γ Attic Greek Ἀνάστασις Anástasis Anastasis
1.δ Attic Greek Αἰωνία παλιγγενεσία Aiōnía palingenesía Eternal Recurrence
2 Epic Greek Ῥοδοδάκτυλος Ἠώς Rhododáktylos Eṓs Rosy-Fingered Dawn
3 Attic Greek Ή αἰγίς τῆς Ἱππολύτης He aigís tês Hippolýtēs Hippolyta’s Aegis
4 Attic Greek Λᾱ́ζᾱρος ἐκ μηχανῆς Lā́zāros ek mēkhanês Lazarus from the Machine
5 Japanese 折り紙のユニコーン Origami no yunikōn Origami Unicorn
5.一 Japanese アンビエント Anbiento Ambient
5.二 Japanese ドゥームメタル Doūmu metaru Doom Metal
5.三 Japanese スラッシュメタル Surasshu metaru Thrash Metal

Track Commentary

  1. “Regressiō ad medium” (8:00) [DR12]

    Latin: “Regression to the Mean”

    A musical representation of recovery from mental disorders that takes an ABA′ form. Originally, this piece had three movements, each lasting exactly two minutes, with the third fading out by the six-minute mark to suggest that the process of recovery never ends. However, the event I wrote this for needed it to loop seamlessly, so I wrote a coda to facilitate this, resulting in four movements, all titled in Attic Greek:

    1. «Συμπτώματα» (“Symptoms”, «Symp­tṓ­ma­ta») (0:00-2:00) [DR11]

      • The A section, reflecting everyday lived experience prior to the start of therapy.
    2. «Θεραπεία» (“Therapy”, «The­ra­pe­ía») (2:00-4:00) [DR11]

      • The B section, representing the process of therapy.
    3. «Ἀνάστασις» (“Anastasis”, «Aná­sta­sis») (4:00-6:00) [DR11]

      • The A′ section, representing a somewhat (but not fully) recovered experience.
    4. «Αἰωνία παλιγγενεσία» (“Eter­nal Re­cur­rence”, «Aiōnía pa­lin­ge­ne­sía») (6:00-8:00) [DR11]

      • The coda, signifying recovery’s continuous, sometimes cyclical nature.

    The shifts between movements α, β, and γ are extremely obvious, but the shift between movement γ and δ is more a gradual transition than a sharp dichotomy. Make of that what you will. (I find it especially fitting that ἀνάστασις and παλιγγενεσία are at least plesionyms if not synonyms.)

    (The “alternate mixes” folder features a variant specifically made for looping. I also boosted the gain in the quiet parts, since the album mix’s loudness range of over 15 LU risks either the quiet parts being inaudible or the loud ones drowning out people’s conversations. The loop variant tones down the dynamic shifts substantially without eliminating them entirely, resulting in a loudness range of around 6 LU.)

    This track also expresses several additional interlinking concepts:

    1. I primarily wrote it in Dorian mode, an entirely symmetrical mode of the diatonic major scale (that is, its ascent and descent use the same intervals in the same order). It’s often used both in Gregorian chants and in nature documentaries, as it conveys a sense of awe and hope, but its root chord is also minor. The general mood conveyed here is that things aren’t great now, but there’s a chance they’ll improve.

      This is a deliberate aesthetic choice. A person must be receptive to therapy for it to be helpful. In short, if you refuse to accept that you need help, you won’t get better. By corollary, if you refuse to accept your own imperfections, you can’t hope to improve them. If you go to therapy believing it can work, it has a much greater chance of success. If you have to be dragged kicking and screaming, you might as well have stayed home. It’s a weirdly self-fulfilling prophecy.

      In short, I used Dorian mode because it reflects the necessary mindset for successful therapy.

    2. The bass line continuously ascends and descends, just as recovery is rarely a steady progression. It plateaus for several measures in one part of «Θεραπεία», reflecting how progress often feels like it’s come to a complete halt, before nonetheless resuming its ascent and descent.
    3. I used counterpoint to reflect dualities in neurodiverse and neurotypical communication and thought. The alternate melody in «Συμπτώματα» effectively flips its progession after the first use of counterpoint, only for the response to flip as well.

      Neurodiverse individuals’ nonverbal communication is often misread, causing underdiagnosis of mental disorders, problems in the workplace and with relationships, and more. Experience teaches us to mirror neurotypical body language (a type of masking, or concealing neurodivergent traits) if we wish to be understood. This stresses us out, often creating its own feedback loop (the more stressed we are, the worse we are at masking), in turn causing burnout and further complications (including several I already listed at the start of this paragraph).

    4. «Συμπτώματα» uses a very simple chord progression, mostly based on variants of A minor, G major, D major, and E major. (E major is this progression’s only chord that’s non-diatonic to Dorian mode.) «Θεραπεία» deliberately gets more avant-garde and increasingly diverges from the original chord progression, reflecting several ideas:

      1. the ebb and flow of trying to break out of a pattern
      2. how therapy can progress frustratingly slowly, or not even seem to be progressing at all
      3. how sometimes the first attempt at a therapy doesn’t even work
    5. The steadily increasing complexity and activity in the melody and harmony and the overall intensity of the arrangement represent how thoughts can become more layered and active as therapy proceeds – a common experience for neurodivergent individuals in particular.
    6. Just before the halfway mark in «Θεραπεία», most aspects of the arrangement drop out, then return. The idea here is that sometimes one must focus on one fundamental idea or behavior rather than try to fix everything at once. This is what ultimately leads to the breakthrough that leads into «Ἀνάστασις».
    7. «Θεραπεία» also explores the idea of the extinction burst – that is, a temporary increase in a behavior’s frequency after rewards or reinforcements for that behavior end.
    8. «Ἀνάστασις» largely uses A Mixolydian (A major and A Dorian’s effective midpoint in mood – and on the circle of fifths ⟨aaronfreed.github.io/musicalmodes.html⟩). Mixolydian, the diatonic scale’s darkest major mode, expresses a more wistful, nostalgic, or bittersweet mood than Ionian (standard major) due partly to its minor fifth (dominant) chord. We still periodically revert to Dorian to indicate that, while things have improved, progress requires constant effort. I also used two near-opposite methods to adapt the Dorian melodies and chords to Mixolydian:

      A minor, D major, E major, G major
      1. Merely by altering them to fit the A Mixolydian scale:

        • A major, D major, E minor, G major
      2. By inverting them using negative harmony (explained below):
        • A major, E minor, D minor, B minor

        (I’ve kept chords in the same places across transformations; italicized chords that are non-diatonic to their respective modes; and printed major chords in bold small caps. I’ve also ignored this song’s many seventh and ninth chords to keep this explanation simple.)

      Negative harmony is to music as negatives ⟨en.wikipedia.org/wiki/Negative_(photography)⟩ are to photography: to oversimplify, it flips melodies and chords, turning major chords minor, minor chords major, ascents into descents, and descents into ascents, ideally with the same root⁽²⁾. Mixolydian is Dorian’s negative harmony counterpart; thus, as seen above, the negative harmony chords are all diatonic to A Mixolydian except D minor – the negative harmony counterpart of E major, the original progression’s only chord that’s non-diatonic to A Dorian.

      I liked both Mixolydian progressions well enough, and the two express different enough moods that it felt worthwhile to use both. So I did.

    9. Even in the Dorian sections of «Ἀνάστασις», the counterpoint melody now generally runs in the same direction, indicating an improvement in overall communication.
    10. The several breaks in the melody are also deliberate. We all need breaks now and then.

    Lastly, some etymological and linguistic trivia:

    • Despite this piece’s Latin title, I named its movements in Greek because therapy, symptom, and anastasis (i.e., a rebirth or recovery from a debilitating condition) were already Greek.
    • I briefly considered retitling it «Ὑποστροφή εἰς μέσον» («Hypostrophḗ eís méson», Attic Greek for “Regression to the Mean”) so its title and its movements’ titles would use the same language, but it’d already been “Regressiō ad medium” for months by then, so I didn’t.
    • The coda’s title can also mean “Eternal Rebirth”. More specifically, palingenesis (literally, again-birth) can mean a spiritual rebirth, especially through reincarnation. However, I had a different sense of palingenesis in mind that Arthur Schopenhauer would’ve appreciated: history’s infinitely cyclical nature. (History’s motto, Schopenhauer wrote, should be “Eādem, sed aliter”, a Latin phrase meaning roughly likewise, but otherwise or the same, but different.)

    • I originally called the coda «Αἰωνία περίοδος» (romanized: «Aiōnía períodos»). Περίοδος, yet another word English has pilfered for its own nefarious purposes, is derived from περί- (perí-, around), and ὁδός (hodós, path or journey), so it literally means return trip or road around. Needless to say, I didn’t change the meaning much, but παλιγγενεσία’s specific connotation of historical cycles was too good to pass up.
    • I considered three other names for the coda. (The first two also mean “eternal recurrence”.)

      1. „ewige Wiederkunft“, Friedrich Nietzsche’s original German phrase
      2. “Recurrentia aeterna”, the Latin etyma of both words of the English translation
      3. «Καιρός ἐστί πλατύς κύκλος» («Kairós estí platýs kýklos»), Attic Greek for “Time is a flat circle” (a True Detective quote summarizing Nietzsche’s concept)

      I ultimately opted for linguistic consistency among movement titles, and as much as I love the True Detective quote, it was too long to feel consistent with the other movement titles. That said, I think another True Detective quote sums up this piece’s main theme far more succinctly than I have, and for some reason, I felt compelled to translate it to Greek:

      «Πρίν, ἦν μόνος ζόφος. Ἐᾱ́ν μοι ἐρωτᾷς, τό φῶς ἐστί νῑκῶν.» -Ρυστ Κωλ, Ὀρθός ντετέκτιβ

      «Prín, ên mónos zóphos. Eā́n moi erōtaîs, tó fôs estí nīkôn.» –Rust Kōl, Orthós ntetéktib

      “Once, there was only dark. If you ask me, the light‘s winning.” -Rust Cohle, True Detective

      Maybe, in a way, that contradicts the album title. Maybe, every now and then, we beat the odds.

      Just don’t bet your life savings on it.

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  2. «Ῥοδοδάκτυλος Ἠώς» (3:36) [DR12]

    Epic Greek: “Rosy-Fingered Dawn”
    Romanized: «Rhododáktylos Eṓs»

    Inspired by: Mahavishnu Orchestra, “Dawn” (The Inner Mounting Flame, 1971-08-14)

    (John McLaughlin)

    From my Rubicon soundtrack commentary:

    Inspired by Mahavishnu Orchestra’s “Dawn” and named for one of Homer’s ubiquitous epithets. It’s only partly coincidence that Christopher Nolan has brought Homer to the popular consciousness again: he’s never far from it. (I thought Nolan had filmed dramatic remakes of Monty Python and the Holy Grail scenes until my chronological error dawned⁽³⁾ on me midway into his film: Python had mined Homer for comedic inspiration. The smoking gun: the Trojan Rabbit. Holy Grail is The Odyssey as a comedy, recast as Arthurian legend.)

    This one came together extremely quickly; I think it took me about two hours of work. I considered expanding it further, but ultimately I think it works fine the way it is.

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  3. «Ή αἰγίς τῆς Ἱππολύτης» (6:56) [DR11]

    Attic Greek: “Hippolyta’s Aegis”
    Romanized: «He aigís tês Hippolýtēs»

    Inspired by: The Doors, “Riders on the Storm” (L.A. Woman, 1971-04-19)

    (John Densmore, Robby Krieger, Ray Manzarek, & Jim Morrison)

    A jazz fusion track heavily inspired by the Doors‘ “Riders on the Storm”. Frankly, I could do without its unsettling lyrics (not at all helped by Jim Morrison’s creepy whispered vocals underlying the verses, or him dying by the time it came out), but Ray Manzarek’s Fender Rhodes performance is exquisite – in particular, he brilliantly imitates raindrops falling in a manner both simple and immediately clear.

    Much like its namesake, this keeps switching keys, running through a sequence of E Dorian, A melodic major, D Dorian, and G melodic major in the opening 𝟕
    𝟖
    section, then E Dorian, A Dorian, E Dorian, G major, and E Dorian in the later 𝟓
    𝟖
    and 𝟔
    𝟖
    sections (which hew significantly closer to the Doors’ original chord progression). The Doors’ track is already basically jazz fusion, but I played up the jazz influence further (for instance, their track doesn’t use a saxophone).

    This is still incomplete, hence the fadeout. (I avoid those unless I have a strong artistic reason to use them or intend a track to repeat. This record’s first three tracks still fade out – go figure.)

    To answer the inevitable question about why the first three tracks on this album are jazz, it’s mostly because Logic Pro recently upgraded its saxophone voice. Both Logic’s previous sax voices and, uncharacteristically, ComposerCloud+’s sax voices are extremely basic and lacking in articulations. Logic’s new sax voices are exquisite, and that’s not just me saying that – one of my friends, himself a saxophonist who attended Berklee College of Music, agrees. (And to prove the point, another friend who graduated from Oberlin thought yet another friend of ours had played the sax part on “Regressiō ad medium”.)

    The title is a bilingual pun. «Ἱππῆς ἐπί τῇ αἰγίδι» («Hippês epí teî aigídi»), this song’s original name, is Attic Greek for “Riders on the Storm”. I renamed it because αἰγίς (aigís) refers to both a storm and a shield, and ἵππος (híppos, horse) is the root of both ἱππεύς (hippeús, rider) and Ἱππολύτη (Hippolýtē, literally horse releaser), whom I find to be a fascinating mythological figure.

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  4. «Λᾱ́ζᾱρος ἐκ μηχανῆς» (4:44) [DR10]

    Attic Greek: “Lazarus from the Machine”
    Romanized: «Lā́zāros ek mēkhanês»

    In case anyone was expecting this entire record to be jazz, I threw this curveball: a drone track composed entirely of nightmarish CS-80 voices. From my Rubicon soundtrack commentary:

    We planned for “Lazarus ex machina” to have no music until near the end of development. Its layout is so convoluted that I couldn’t use players’ location as a cue to fade out the music as they approach the climactic Ascension theme, so I threw together this deliberately minimalistic drone track in about forty minutes (not an exaggeration).

    Ultimately, I thought of an approach that enabled us to fade out the music without referring to the player’s location, so we instead reused “Hastur” on this level. I wasn’t really bothered by that, since I wrote this so quickly. For the soundtrack, we still left it in the position of “Lazarus ex machina”, which would otherwise have no track to represent it.

    The familiar Latin phrase Deus ex machina (God from the machine) is calqued from the far less familiar Greek phrase ἀπὸ μεχανῆς θεός (apò mēkhanês theós); as ex is a cognate of ἐκ (ek), I changed it accordingly. I also reordered the words to fit the more familiar Latin order; both orders are valid in both languages. (Languages with noun cases, like Latin and Greek, usually have more flexible word orders than languages without them, like English.)

    The titular Lazarus was a man Jesus resurrected in the New Testament. Many people reading this may not have played Rubicon yet, so I won’t explain the story significance beyond that.

    The length is quite deliberate. In China, Japan, Korea, Vietnam, whose writing systems all use or used Chinese characters, the number four is culturally associated with death because 四 (four) and 死 (death) are or were homonyms or near-homonyms: sì and sǐ in Mandarin; shi for both in Sino-Japanese; sa for both in Sino-Ko­re­an; tứ and tử in Sino-Vietnamese. This may have led some languages to use other terms for the number four; for instance, modern Japanese often reads 四 as よ (yo) or よん (yon) instead of し (shi).

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  5. 「折り紙のユニコーン」(10:30) [DR13]

    Japanese: “Origami Unicorn”
    Rōmaji: 「Origami no yunikōn」
    「残念だ彼女生きないでも誰が」〜ガフ、『ブレードランナー』
    「Zan’nenda kanojo ikinai demo dare ga.」 ~Gafu, 「Burēdo Ran’nā」
    “Too bad she won’t live. But then again, who does?” -Gaff, Blade Runner
    1. 「折り紙のユニコーン(アンビエント)」(0:00-3:21)[DR11]

      Japanese: “Origami Unicorn (Ambient)”
      Rōmaji: 「Origami no yunikōn (Anbiento)」

      A Blade Runner-inspired piece that makes heavy use of CS-80 voices. Each of its movements can be divided into two halves. The first half consists of a drone on variants of C♯ major and variants thereof (e.g., C♯7add ♯11, C♯add ♭6, C♯maj7).

      The back half adds synth arpeggiation. Also, while keeping the C♯ pedal tone employed throughout the song, it adds a series of chords descending the C♯ Phrygian scale. Most of these are major, apart from the G♯ and F♯ chords, which produces a very uncanny effect. As this progression repeats a few times, I also add a series of ascending bass notes. These notes vary somewhat between repetitions, although notably, each of the last two outlines a G♯ diminished seventh chord in its conclusion.

    2. 「折り紙のユニコーン(ドゥームメタル)」(3:21-6:33)[DR13]

      Japanese: “Origami Unicorn (Doom Metal)”
      Rōmaji: 「Origami no yunikōn (Doūmu metaru)」

      Now we add drums, bass, and guitar to ratchet up the tension. In this movement, the guitar simply plays slow power chords, hence the movement’s subtitle. Note that the actual doom metal section starts at 4:09, but it does so in the midst of the keyboard solo that opens this movement.

      Vangelis isn’t the only Greek composer that influenced this piece: I also quote the four-note motif from Risk of Rain, which Chris Christodoulou composed for until Stavros Markonis took over from him for the Alloyed Collective DLC. At this rate, I’m pretty sure the Risk of Rain motif is the melody I’ve quoted second-most often. (Just on this album, «Ή αἰγίς τῆς Ἱππολύτης» also quotes it.) Ironically, the one that beats it out is also best known for a four-note phrase: “Diēs Īrae”.

    3. 「折り紙のユニコーン(スラッシュメタル)」(6:33-10:30)[DR12]

      Japanese: “Origami Unicorn (Thrash Metal)”
      Rōmaji: 「Origami no yunikōn (Surasshu metaru)」

      This is where the tension explodes. The guitar and bass riffs here feel reminiscent enough of thrash metal to justify the subtitle (though it reverts to doom near the end). This track proved to me I need to use more power chords for rhythm parts – they have an intensity that single notes can’t match.

    Overall, despite the metal sections, this is a very moody, atmospheric, spacy-sounding track. Several of these chords (particularly the add ♭6 chord) are ubiquitous in space-related soundtracks for a reason. Despite that, it also conveys a sense of dread that, given how many of its chords are major, is almost uncanny. I think there are at least two reasons for this:

    1. There are enough dissonant intervals mixed in to override the traditional major feeling.
    2. The fact that the melodies outline the Phrygian scale (the diatonic scale’s darkest minor mode) and a diminished seventh chord provide a further sense of dread.

    As I doubt it’s obvious, I wish to acknowledge the influence of early game composers like Hirokazu Tanaka (Metroid, Mother, EarthBound) and Tim & Geoff Follin (Plok!, Solstice, Pictionary) on this track, and particularly my chord choices. (Tim, it was just Pictionary.)

    The title refers to the origami figures Gaff (Edward James Olmos) folds throughout the film, and specifically to the one he gives Rick Deckard (Harrison Ford) in its final scene. Evidently, Olmos ad-libbed the film’s famous final line; thus, its two most iconic lines (Roy Batty [Rutger Hauer]’s “tears in rain” monologue is of course the other) were at least partly written by their actors.

    (I admit to having been tempted to name this track 「折り紙の大谷翔平」, i.e., 「Origami no Ōtani Shōhei」, but the title is already obscure enough without me adding another layer of obscurity atop that.⁽⁴⁾ Plus, the song expresses too serious a mood to use such a silly pun as a title.)

    One final note: Those of you paying close attention may note that I call seven-accidental key signatures “pretentious garbage” in this very album’s endnotes. So why do I say this is in C♯ major, a seven-accidental key signature? Do I contradict myself? Very well then I contradict myself, (I am large, I contain multitudes.)

    The true reason is simple: it descends the C♯ Phrygian scale (C♯, D, E, F♯, G♯, A, B), which has but three accidentals. D♭ Phrygian (D♭, E𝄫, F♭, G♭, A♭, B𝄫, C♭) has nine – as far as I’m concerned, it doesn’t exist. I’m willing to tolerate C♯ major in this sort of context (and only this sort of context).

    Of course, I could’ve avoided this by using a key that required fewer accidentals, but I’d chosen the furthest key from G minor, the key of “Finis finalis” on Mūsica ex tempore malōrum ⟨aaronfreed.github.io/musicaextemporemalorum.html⟩ (to which this is notably stylistically similar), and I’d already locked myself into the key before deciding to use the Phrygian scale.

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Rights Notice

You may freely distribute this collection in unmodified form, provided that:

  1. This document is included, without any sort of abridgement or modification.
  2. You do not sell it for any reason.
  3. Under no circumstances may my music or writing be used to train large language models or generative “artifical intelligence”. I will construe attempts to do so as violations of my:

    • control over adaptations or derivatives of my work (as defined under U.S. copyright law);
    • moral rights to be attributed for my work and to preserve its integrity (outside the U.S.);

    and I will deal with violations using every legal avenue available to me.

Please ask ⟨aaronfreed.github.io/aboutme.html⟩ if you wish to use my music for your own creative endeavors. I’ll grant most noncommercial uses, but I want to know who’s using it and for what.

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Contents · Tracklist · Discography · Portfolio · Contact me · Website index

Endnotes

Endnotes
# Note
1. Am I referring to Chris Sale or Blake Snell here? Yes. (The fact that there are two applicable examples in the 2026 National League playoffs alone goes to prove my point.)
2.

This isn’t possible in all scales, most notably Locrian mode, the diatonic scale’s sole mode:

  1. That negative harmony doesn’t transform into another mode of the diatonic scale.
  2. Whose root moves in the transformation (a semitone upward, to be precise).
  3. That doesn’t have a note a perfect fifth above its root note.

#3 is one cause of #1 and #2. Another is negative harmony reflecting notes on a line halfway between the root and the dominant (i.e., 3½ semitones, or half a perfect fifth, above the root). Different notes are swapped for different roots.

I’ve yet to find a way to explain negative harmony that isn’t at least slightly confusing, so bear with me. In fact, I’m going present several tables to explain the concept.

Negative harmony output (no flats in input)
Scale
Root
Input Note
A A♯ B C C♯ D D♯ E F F♯ G G♯
A E E♭ D D♭ C B B♭ A A♭ G G♭ F
A♯ G♭ F E E♭ D D♭ C B B♭ A A♭ G
B A♭ G G♭ F E E♭ D D♭ C B B♭ A
C B♭ A A♭ G G♭ F E E♭ D D♭ C B
C♯ C B B♭ A A♭ G G♭ F E E♭ D D♭
D D D♭ C B B♭ A A♭ G G♭ F E E♭
D♯ E E♭ D D♭ C B B♭ A A♭ G G♭ F
E G♭ F E E♭ D D♭ C B B♭ A A♭ G
F A♭ G G♭ F E E♭ D D♭ C B B♭ A
F♯ B♭ A A♭ G G♭ F E E♭ D D♭ C B
G C B B♭ A A♭ G G♭ F E E♭ D D♭
G♯ D D♭ C B B♭ A A♭ G G♭ F E E♭
Negative harmony output (no sharps in input)
Scale
Root
Input Note
A B♭ B C D♭ D E♭ E F G♭ G A♭
A E D♯ D C♯ C B A♯ A G♯ G F♯ F
B♭ F♯ F E D♯ D C♯ C B A♯ A G♯ G
B G♯ G F♯ F E D♯ D C♯ C B A♯ A
C A♯ A G♯ G F♯ F E D♯ D C♯ C B
D♭ C B A♯ A G♯ G F♯ F E D♯ D C♯
D D C♯ C B A♯ A G♯ G F♯ F E D♯
E♭ E D♯ D C♯ C B A♯ A G♯ G F♯ F
E F♯ F E D♯ D C♯ C B A♯ A G♯ G
F G♯ G F♯ F E D♯ D C♯ C B A♯ A
G♭ A♯ A G♯ G F♯ F E D♯ D C♯ C B
G C B A♯ A G♯ G F♯ F E D♯ D C♯
A♭ D C♯ C B A♯ A G♯ G F♯ F E D♯

A few notes on the above tables:

  • The first column of each row lists the root of the scale being transformed; the columns list the transformed note. At their intersection lies that note’s negative harmony counterpart for that root.
  • The red notes are the transformations for the note a perfect fifth above the scale root (usually the fifth note of the scale, except in Locrian mode).
  • The blue notes are the scale roots’ transformations. Blue and red notes swap places.
  • The red and blue borders represent notes outside the scale’s standard perfect fifth.
  • The red and blue color-coding refers to the Doppler effect.
  • The green border represents the axis of reflection for a given root.
  • If you’re not in Locrian mode and your key signature has sharps or flats, you may need to switch to the other accidental type to avoid awful key signatures. (You may also need to transform a root to its enharmonic equivalent, e.g., G♭ to F♯.)

    What do I mean by awful key signatures? Glad you asked.

    G♭ Lydian (G♭ A♭ B♭ C D♭ E♭ F) turns into F♯ Phrygian (F♯ G A B C♯ D E) because G♭ Phrygian is G♭ A𝄫 B𝄫 C♭ D♭ E𝄫 F♭. For those in the cheap seats, we just replaced two sharps with four flats and three double-flats. Ain’t nobody got time for that.

    (At least 𝄫 has a clear meaning. 𝄪, the double-sharp, is an abomination unto Nuggan.)

    Lydian (+5×♭) and Phrygian (+5×♯) are likeliest to experience this problem, followed by Ionian (+3×♭) and Ionian (+3×♯). It rarely affects Mixolydian (+♭) or Dorian (+♯). Locrian, meanwhile, changes its root, so key signature nonsense is the least of our worries there.

Some readers might find it more legible to base transformations on scale degrees, like so:

Negative harmony output (no sharps in input)
Scale
Root
Input°
1 ♭2 2 ♭3 3 4 ♭5 5 ♭6 6 ♭7 7
A E D♯ D C♯ C B A♯ A G♯ G F♯ F
B♭ F E D♯ D C♯ C B A♯ A G♯ G F♯
B F♯ F E D♯ D C♯ C B A♯ A G♯ G
C G F♯ F E D♯ D C♯ C B A♯ A G♯
D♭ G♯ G F♯ F E D♯ D C♯ C B A♯ A
D A G♯ G F♯ F E D♯ D C♯ C B A♯
E♭ A♯ A G♯ G F♯ F E D♯ D C♯ C B
E B A♯ A G♯ G F♯ F E D♯ D C♯ C
F C B A♯ A G♯ G F♯ F E D♯ D C♯
G♭ C♯ C B A♯ A G♯ G F♯ F E D♯ D
G D C♯ C B A♯ A G♯ G F♯ F E D♯
A♭ D♯ D C♯ C B A♯ A G♯ G F♯ F E
Out° 5 ♯4 4 3 ♯2 2 ♯1 1 7 ♯6 6 ♯5
Negative harmony output (no flats in input)
Scale
Root
Input°
1 ♯1 2 ♯2 3 4 ♯4 5 ♯5 6 ♯6 7
A E E♭ D D♭ C B B♭ A A♭ G G♭ F
A♯ F E E♭ D D♭ C B B♭ A A♭ G G♭
B G♭ F E E♭ D D♭ C B B♭ A A♭ G
C G G♭ F E E♭ D D♭ C B B♭ A A♭
C♯ A♭ G G♭ F E E♭ D D♭ C B B♭ A
D A A♭ G G♭ F E E♭ D D♭ C B B♭
D♯ B♭ A A♭ G G♭ F E E♭ D D♭ C B
E B B♭ A A♭ G G♭ F E E♭ D D♭ C
F C B B♭ A A♭ G G♭ F E E♭ D D♭
F♯ D♭ C B B♭ A A♭ G G♭ F E E♭ D
G D D♭ C B B♭ A A♭ G G♭ F E E♭
G♯ E♭ D D♭ C B B♭ A A♭ G G♭ F E
Out° 5 ♭5 4 3 ♭3 2 ♭2 1 7 ♭7 6 ♭6
  • Scale degrees are based on Ionian (major): the note between its second and third notes can be either “♭3” or “♯2”. Don’t read too much into the nomenclature: key signature matters as much as mode when it comes to picking a transformation.
  • The magenta and green lines reflect to the same notes on different octaves.

Note that a scale containing A can have at most two flats (B♭, E♭) or four sharps (F♯, C♯, G♯, D♯). The reason is relatively straightforward: A♭ would be the next flat, and A♯ the next sharp. No mode of the diatonic major scale containing either of those notes can also contain A♮. (It can, however, contain G♯ and A, or A and B♭, although not all three.)

As it happens, though, A♯ still pertains to a scale transformation rooted on A:

  1. A Locrian contains both A and E♭. Flats should transform into either sharps or naturals.
  2. Relatedly, A Locrian becomes A♯ Locrian. (Originals on left, transformations on right.)
    Negative harmony: Scale transformations on A
    Original5432176Transformed1234567
    A Locrian E♭ DCB♭ AGFA♯ LocrianA♯ BC♯ D♯ EF♯ G♯
    A Phrygian EDCB♭ AGFA Lydian ABC♯ D♯ EF♯ G♯
    A Aeolian EDCBAGFA Ionian ABC♯ DEF♯ G♯
    A Dorian EDCBAGF♯ A Mixolydian ABC♯ DEF♯ G
    A MixolydianEDC♯ BAGF♯ A Dorian ABCDEF♯ G
    A Ionian EDC♯ BAG♯ F♯ A Aeolian ABCDEFG
    A Lydian ED♯ C♯ BAG♯ F♯ A Phrygian AB♭ CDEFG

    (We could also call it B♭ Locrian, but we’d have to rename E to F♭ and B to C♭. Gross. I [begrudgingly] tolerate E♯ and C♭ in six-accidental key signatures, as they prevent a flurry of unnecessary accidentals in notation, but seven-accidental key signatures are pretentious garbage that I’ll only tolerate when they prevent the use of double-flats or double-sharps, as on this very album’s「折り紙のユニコーン」.)

A few other notes on the above table:

  • The notes shift places between pre- and post-transformation scales. I list the original scale in descending order, starting on the fifth scale degree; this results in the transformation appearing in the more familiar ascending order, starting on the root.
  • Besides Locrian, each mode and its mirror image in mood (sorry, not sorry) are paired.
    • Dorian is the brightest minor mode; Mixolydian is the darkest major mode.
    • Aeolian’s root, subdominant, and dominant chords are all minor; Ionian’s are major.
    • Phrygian is the darkest minor mode; Lydian is the brighest major mode.
    • Locrian is the only mode with a diminished root chord, so the normal rules don’t apply to it, and applying negative harmony to it just gives us Locrian in a different key.
  • Dorian’s interval composition is symmetrical from the root, but it becomes Mixolydian rather than itself because we’re reflecting it on a line 3½ semitones above the root.
3.

Sorry.

Well, OK, I’m slightly sorry.

4.

In case you don’t follow baseball, Shōhei Ohtani ⟨fangraphs.com/players/shohei-ohtani/19755⟩ is routinely called a unicorn, largely because he’s the first major-leaguer to pitch and hit at his level since racial integration (1947-1959). Imagine Roger Clemens ⟨fangraphs.com/players/roger-clemens/815⟩ hitting like Barry Bonds ⟨fangraphs.com/players/barry-bonds/1109⟩, but without the (alleged) steroid use, and you’d basically have Ohtani. Or, for Europeans out there, imagine Lionel Messi ⟨fbref.com/en/players/d70ce98e/Lionel-Messi⟩ and Manuel Neuer ⟨fbref.com/en/players/8778c910/Manuel-Neuer⟩ being the same player. The few two-way players who ever even approached Ohtani’s level all played before integration:

  • Albert Spalding ⟨fangraphs.com/players/al-spalding/1012300⟩ (1871-1876)
  • John Montgomery Ward ⟨fangraphs.com/players/john-ward/1013615⟩ (1878-1884)
  • Bob Caruthers ⟨fangraphs.com/players/bob-caruthers/1002026⟩ (1884-1892)
  • Charlie Ferguson ⟨fangraphs.com/players/charlie-ferguson/1003987⟩ (1884-1887)
  • George Sisler ⟨fangraphs.com/players/george-sisler/1012021⟩ (1915-1916)
  • Bullet Rogan ⟨seamheads.com/NegroLgs/player.php?playerID=rogan01bul⟩ (1916-1928)
  • Babe Ruth ⟨fangraphs.com/players/babe-ruth/1011327⟩ (1918-1919)
  • Martín Dihigo ⟨seamheads.com/NegroLgs/player.php?playerID=dihig01mar⟩ (1925-1946)
  • Lázaro Salazar ⟨seamheads.com/NegroLgs/player.php?playerID=salaz01laz⟩ (1930-1946)
  • Ray Brown ⟨seamheads.com/NegroLgs/player.php?playerID=brown01ray⟩ (1931-1945)
  • Leon Day ⟨seamheads.com/NegroLgs/player.php?playerID=day--01leo⟩ (1937-1943)
  • Jimmie Foxx ⟨fangraphs.com/players/jimmie-foxx/1004285⟩ (1945)

(Hall of Famers are printed in bold small capitals. I’ve listed players’ years as active two-way players, not their full careers. Links for Negro League players go to Seamheads, the most complete Negro League reference; other players’ links go to FanGraphs.)

And let’s not kid ourselves: modern Major League Baseball’s level of athleticism utterly eclipses what players faced prior to its racial integration. Babe Ruth may have the second-highest slugging percentage of all time, but he was hitting against plumbers and electricians. Would he still be a great hitter today? Probably. Would his career slugging percentage be .690? (Nice.) Absolutely not.

(MLB recognized seven Negro Leagues as major in 2020 and integrated their stats in 2024; thus, Josh Gibson ⟨seamheads.com/NegroLgs/player.php?playerID=gibso01jos⟩ holds career slugging percentage and batting average records that, respectively, Ruth and Ty Cobb ⟨fangraphs.com/players/ty-cobb/1002378⟩ once held. Both would likely support this; Cobb spoke openly in favor of racial integration, and Ruth’s widow Claire said he’d have enthusiastically cheered Henry Aaron ⟨fangraphs.com/players/hank-aaron/1000001⟩’s pursuit of his home run record.)

My book Honor lūdī (Latin: Honor of the Game or The Game of Honor) ranks dozens of two-way players ⟨aaronfreed.github.io/baseball.html#twowayplayers⟩ and explains why stat comparisons break down across eras ⟨aaronfreed.github.io/baseball.html#pitchingwasworse⟩ (and, relatedly, why no one hits .400 anymore). I’d change certain aspects of the former if I were rewriting it today (most notably, I’d emphasize league- and park-adjusted stats like ERA+ and wRC+ more in my rankings), but it’s still a solid overview.

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